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

ii

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


7.1.4 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


7.2.4 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


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


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.


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 ,


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


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


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


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>> ;


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]


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


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


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,


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.


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.


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).


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.


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.



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.




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


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.


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


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


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


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-


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


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


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


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.


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


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


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.


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


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


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.


40

3.8 References


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.


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


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


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


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.


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.


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


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


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


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


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.


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 ,


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


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.


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


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.


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


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.


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


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.


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


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.


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.


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.


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


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).


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


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


67

Ref-

dn4

L10

000

c10

--

10.

wmf


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.


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


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


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.


69

Ref-

dn4

L10

000

c10

--

10.

wmf


Re

flect

an

ce (

p.

u.)

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1


different lengths: )(5000 mL µ= (red solid line), )(7000 mL µ= (green dashed


447.1=effn , 0004.0=nδ , )(550.1 mD µλ = 100−=F .


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


different values of the index changed nδ .


70

Ref-

dn4

L10

000

c10

--

10.

wmf


Re

flect

an

ce (

p.

u.)

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1


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.


71

Ref-

dn4

L10

000

c10

--

10.

wmf


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


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


( F )x axis

100=F150=F200=F

100=F150=F200=F


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


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


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.


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


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.


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.


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


( F )Null x axis

100=F150=F200=F

100=F150=F200=F


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.


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


447.1=effn , 0004.0=nδ , )(550.1 mD µλ = .


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.


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


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


( F )Null x axis

100=F150=F200=F

100±=F150±=F200±=F


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.


81


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


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


447.1=effn , 0004.0=nδ , )(550.1 mD µλ =


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


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.


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


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 .


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


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.


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


87


Ref

lect

ance

(p.

u.)

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1


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)


Tim

e de

lay

(ps)

130

120

110

100

90

80

70

60

50

40

30

20

10


Tim

e de

lay

(ps)

120110100

908070605040302010

0-10-20

(b)


Dis

pers

ion

(ps/

nm)

2,000

1,500

1,000

500

0

-500

-1,000

-1,500

-2,000


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


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


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


89


Re

flect

ance

(dB

)

-10

-20

-30

-40

-50

-60

-70

-80

-90

(a)


Tim

e de

lay

(ps)

110100

908070605040302010

0-10

(b)


90


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


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)


92


Dis

pe

rsio

n (p

s/nm

)150

100

50

0

-50

-100

-150

-200

(c)


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


Re

flect

ance

(dB

)

-10

-20

-30

-40

-50

-60

-70

-80

(a)


93


Tim

e de

lay

(ps)

110

100

90

80

70

60

50

40

30

20

10

0

-10

(b)


Dis

pe

rsio

n (p

s/nm

)

140120100

80604020

0-20-40-60-80

-100-120-140

(c)


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


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


95


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.


96


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.


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.


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 =


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.


99

5.9 References




no.8, 1997, pp. 1277-1294.



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.



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


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.


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.


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]:


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.


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


105


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.


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


Tim

e de

lay

(ps)

120

100

80

60

40

20

0

-20

Figure 6 - 3 (b) The time delay


107


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].


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


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


109


Tim

e de

lay

(ps)

70

65

60

55

50

45

40

35

30

25

20

Figure 6 - 4 (c) The time delay


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 .


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 .


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


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 .


Re

flect

ance

(p.

u.)

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0


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.


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)


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 .


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 .


3 dB

ban

dwid

th (

nano

met

re)

4

3

2

1

Figure 6 - 8 (a) without preconditioning


114


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.


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)


Tim

e de

lay

(ps)

110100

908070

605040302010

0-10-20

(b)

Figure 6 - 9 (a) The reflectance spectrum and (b) the time

delay.


116


Dis

pers

ion

(ps/

nm)

1,400

1,200

1,000

800

600

400

2000

-200

-400

-600

-800

-1,000

-1,200

(c)


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 .


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.


118


Re

flect

an

ce (

p. u

.)0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1


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.


119


Tim

e de

lay

(ps)

140

120

100

80

60

40

20

0

-20

(a)


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.


120


Dis

pers

ion

(ps/

nm)

2,000

1,500

1,000

500

0

-500

-1,000

-1,500

-2,000

(a)


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


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.


122

6.7 References

1. A. Othonos, “Fibre Bragg gratings”, Review of Scientific Instruments, vol.68,

no.12, 1997, pp. 4309-4341.


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


7.1.4 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


7.2.4 Simulation results ............................................................................ 132

7.2.5 Applications ..................................................................................... 137

7.3 CONCLUSION.............................................................................................. 137

7.4 REFERENCES ............................................................................................. 138


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


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].


125


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.


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


Phase-shift

(φ )π Null π π

21

π23

Table 7 - 1 Simulation parameters of phase-shifted gratings


126


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)


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 .


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 .


128


Tra

nsm

ittan

ce(p

. u.

)0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

(a)


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 = .


129

7.1.4.2 Phase-shifted gratings with different phase shifts


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)


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 = .


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


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


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).


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,


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.


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.


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


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.


134

7.2.4.1 Sampled gratings with different grating lengths


Ref

lect

ance

(p.

u.)

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

(a)


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= .


135

7.2.4.2 Sampled gratings with different sampling periods


Ref

lect

ance

(p.

u.)

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

(a)


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 .


136

7.2.4.3 Sampled gratings with different duty cycles


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)


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= .


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.


138

7.4 References


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.




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


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


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


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.


143

8.3 References


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

Bibliography

1. 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.

2. 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.

3. 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.

4. 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.

5. G. Booch, “Object-oriented development”, IEEE Transactions on Software

Engineering, vol.12, 1986, p. 211.

6. G. Booch, “Object-oriented analysis and design with applications”,

(Benjamin-Cummings), 1994.

7. 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.

8. 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

gratings”, Proc SPIE. Fibre Optic and Laser Sensors XIV, vol.2839, 1996, pp. 8-

19.

9. 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.

BIBLIOGRAPHY

145

10. G. Buzzi-Ferraris, “Scientific C++: building numerical libraries the object-

oriented way”, (Addison-Wesley), 1993.

11. 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




13. A. Eliens, “Principles of object-oriented software development”, (Addison-

Wesley), 1994.

14. 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.

15. T. Erdogan, “Fibre grating spectra”, Journal of Lightwave Technology,

vol.15, no.8, 1997, pp. 1277-1294.

16. D. M. Etter, “Engineering problem solving with MATLAB”, (Prentice Hall),

1993.

17. C. R. Giles, “Lightwave applications of fibre Bragg gratings”, Journal of

Lightwave Technology, vol. 15, no.8, 1997, pp. 1391-1404.

18. 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.

19. C. R. Giles and J. M. P Delavaux, “Repeaterless bidirectional transmission

of 10 Gb/s WDM channels”, in ECOC’95, Brussels, 1995, paper PD2.

20. 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.

21. 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.

22. K.O. Hill, Y. Fujii, D.C. Johnson, and B.S. Kawasaki, “Photosensitivity in

optical fibre waveguides: application to reflection filter fabrication”, Applied

BIBLIOGRAPHY

146

Physics Letters, vol. 32, no.10, 1978, pp.647-649.

23. 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.

24. 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.

25. 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.

26. 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.

27. J.F. Kaiser, "Nonrecursive digital filter design using the 0I - sinh window

function", IEEE Symp. Circuits and Systems, April 1974, pp. 20-23.

28. R. Kashyap, “Photosensitive optical fibres: devices and applications”,

Optical Fibre Technology, 1994, pp.17-34

29. 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.

30. A. D. Kersey, “A review of recent developments in fibre optic sensor

technology”, Optical Fibre Technology, vol.2, 1996, pp. 291-317.

31. 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


32. 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.

33. 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-

BIBLIOGRAPHY

147

185.



35. H. Kogelnik,"Coupled wave theory for thick hologram gratings", Bell System

Technical Journal, vol.48, no.9, 1969, pp. 2909-2949.

36. 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.

37. B. H. Lee, Y. Chung, and U. Paek, "Fiber comb filters based on fiber

gratings", 5F/13/1999 , COOC99, pp19-20.

38. V. Loan and F. Charles, “Introduction to scientific computing: a matrix-vector

approach using MATLAB”, (Prentice Hall), 1999.

39. 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.

40. 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.

41. 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.



43. P. Norton “Peter Norton’s guide to Delphi 2 premier title”, (SAMS), 1996.



45. F. Ouellette, “Dispersion cancellation using linearly chirped Bragg grating

filters in optical waveguides,” Optics Letters, vol.12, no.10, 1987, pp. 847-849.

46. F. Ouellette, “Optical equalization with linearly tapered two-dissimilar-core

fibre”, Electronics Letters, vol.27, 1991, pp. 1668-1669.

47. L. Poladian, "Analysis and modeling of group delay ripple in Bragg

BIBLIOGRAPHY

148

gratings". In Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides of

OSA Technical Digest (1999), pp. 258-260.

48. 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.

49. 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.

50 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.

51. 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.


Hall, London), 1983, pp542.

53. S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Merged Sagnac-

Michelson interferometer for distributed disturbance detection”, Journal of


54. B. Stroustrup, “The C++ programming language”, (Addison-Wesley), 1991.

55. 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.

56. 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.

57. 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.

58. V. Tzolov, D. Feng, S. Tanev, and Z. Jakubczyk, "Modeling tools for

integrated and fiber optical devices", Integrated Optics Devices III, Photonics

BIBLIOGRAPHY

149

West 99, San Jose, CA, 1999, pp. 23-29.

59. 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.

60. 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.

61. 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.

62. D. A. Young, “Object-oriented programming with C++ and OSF/Motif”,

(Prentice Hall), 1992.

63. 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.

64. 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.

65. http://www.optiwave.com/.

66. http://www.apollophoton.com/.

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

Documents

Fibre Bragg Grating - University of Johannesburg