POWER TRANSFER ON MULTIPLE FIBER COUPLERS
DEDI IRAWAN
UNIVERSITI TEKNOLOGI MALAYSIA
POWER TRANSFER ON MULTIPLE FIBER COUPLERS
DEDI IRAWAN
A dissertation submitted in partial fulfilment of the
requirements for the award of the degree of
Master of Science (Physics)
Faculty of Science
UNIVERSITI TEKNOLOGI MALAYSIA
April 2010
iii
To my dearest father and mother, Bustami and Suarni
Also to my beloved brothers, Eka Nazra, Hardi Nurmansyah,
and my sister, Umi Salmah
iv
ACKNOWLEDGEMENT
First and foremost I thank to Allah Subhanahua Ta’ala, the most Gracius and
always gives me blessedness and the strength to complete my thesis.
The work done on this thesis over the past year would have not been possible
without the help of several people in and outside the Advanced Photonic Science
Institute (APSI), Faculty of Science UTM. To begin with, I would like to thank my
supervisor, Dr Saktioto who was great help and giving motivation, support and his
knowledge on my research.
Despite all of this help, I would have never made it through one and half years of
UTM without all of the friends I have made here. It has been a delight to know every one
of them. I would like to especially thank my best friends who were also my classmate,
Jauharah, Bibi Aishah, Izyan, Izzati, and Norehan for always working hard together and
give me information in Campus. My friends, Ikhwan, Suzairi, Iraj, Hamdi, Kasif and
Imran, thanks for your discussion. I would like to thank my brother, Eka Nazra Bustami
who always helps me, both support and motivation to do my masters degree. It is not
possible to write all of my friend, I only be able to thanks for all.
Finally, I also would like to thank the Government of Malaysia, University
Teknologi Malaysia and the Government of Riau Indonesia for the financial support in
my study.
v
ABSTRACT
Fiber coupler is a great utilization that can be used to split one optical power to the
more than two power outputs. Power transfer in fiber coupler depends on not only
the geometry arrangement of the waveguides, but also the coupling coefficient.
Power transfers between the waveguides are calculated using matrix transfer method.
By introducing power input to the one of the waveguides, assuming cross section,
power propagation, and fiber axis are held constants, at coupling constant coupling
coefficient. It shows that coupling length of NX2, linear order NX3, and triangular
NX3
z=
fiber
π
2 2κ
couplers
are
odd-multiple
of
z = π / 2κ ,
z = π / 2κ , and
respectively. If the coupling coefficient is varied along the coupling
region, phase needed to transfer power to the others waveguide changes. The
calculation of power transfer has been shown in 3D with variation of coupling
coefficient from 0.1 mm −1 to 1.5 mm−1 . It describes that the higher the coupling
coefficient, the shorter the coupling length or phase that required to transfer power
completely to the others waveguides. The good agreement in minimum coupling
length or minimum phase of NX2 and NX3 shows that the coupling length of linear
order NX3 is larger with a factor
2 than the coupling length of NX2. However, the
coupling length of triangular order NX3 is shorter with a factor 1 / 2 than the
coupling length of NX2 fiber couplers.
vi
ABSTRAK
Gentian pengganding merupakan penggunaan yang boleh digunakan untuk
membahagikan satu kuasa optik kepada lebih daripada dua kuasa yang dihasilkan.
Penghantaran kuasa dalam gentian pengganding bukan hanya bergantung kepada
struktur
geometri
pemandu
gelombang,
tetapi
juga
pekali
pengganding.
Penghantaran kuasa antara pemandu gelombang dihitung menggunakan cara
hantaran matrik. Kuasa masukan hanya di kenakan ke salah satu dari pemandu
gelombang. Keratan rentas, pemala perambatan, paksi gentian adalah dianggap
malar. Hasil perhitungan telah memperlihatkan bahawa panjang pengganding bagi
NX2, susunan linear NX3, dan segitiga NX3 gentian pengganding adalah
penggandaan ganjil dari z = π / 2κ , z = π / 2κ , dan z =
π
2 2κ
masing-masing.
Pekali pengganding merupakan parameter yg mempengaruhi hantaran gelombang
diantaran pemandu gelombang. Hasil dari pengiraan penghantaran kuasa di
tunjukkan
dalam
3
dimensi
0.1 mm −1 to 1.5 mm−1 .
Iya
dengan
mengubah
menunjukkan
bahawa
pekali
pengganding dari
semakin
tinggi
pekali
pengganding, semakin pendek panjang pengganding yang di perlukan untuk
memindahkan semua kuasa kepada pemandu gelombang yang lain. Panjang
pengganding dan fasa yang minimum bagi gentian pengganding susunan linera NX3
lebih besar dengan factor
2 dari pada panjang pengganding bagi NX2. Sedangkan
panjang pengganding bagi susunan segitiga NX3 lebih pendek dengan faktor 1 / 2
berbanding gentian pengganding NX2.
vii
TABLE OF CONTENTS
TITLE
CHAPTER
1
2
PAGE
DECLARATION
ii
DEDICATION
iii
ACKNOWLADGEMENT
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF FIGURES
ix
LIST OF SIMBOLS
xi
LIST OF APENDICES
xiii
INTRODUCTION
1.1
Introduction
1
1.2
Background of the study
2
1.3
Problem statement
3
1.4
Objectives of the study
3
1.5
Scope of the study
3
1.6
Signification of the study
4
1.7
Organization of theses report
4
LITERATURE REVIEW
2.1
Introduction
5
2.2
Modes in Fiber
7
2.3
Coupled-mode Theory for the Optical Waveguides
10
2.4
Fiber Coupler Parametric And Coupling Ratio
12
viii
2.5
3
2.4.1 Propagation Constans
12
2.4.2 Coupling Coefficient
13
2.4.3 Coupling Length
14
2.4.4 Coupling Ratio
14
The Matrix Transfer Method
15
RESEARCH METHODOLOGY
3.1
Introduction
17
3.2
Matrix transfer methods of optical directional coupler
18
with two and three waveguides
3.3
Calculation of power transfer in fiber couplers with
22
constant coupling coefficient
3.4
Calculation of power transfer in fiber couplers with
24
variable coupling coefficient
3.5 Derivation of coupling Length of NX2 and NX3 Fiber
26
Couplers.
4
RESULTS AND DISCUSSION
4.1
Introduction
28
4.2
Power transfer on Fiber Couplers with Constant
28
Coupling Coefficient
4.3
Power transfer on Fiber Couplers with Variation of
31
Coupling Coefficient
5
CONCLUTION
5.1
Conclusion
37
5.2
Future work
38
REFERENCES
39
APPENDICES
41 - 51
ix
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
2.1
Basic structure of fiber optics
7
2.2
Ray propagation in fiber optics
8
2.3
Plot of b versus V for one type of fiber
9
2.4
Cross section of two fiber
10
2.5
Transfer matrix of 2X2 directional coupler
16
3.1
Two cylindrical 2X2 directional fiber coupler
18
3.2
NX3 fiber coupler, (a) Three single-mode fiber in 3X3
directional coupler, (b) Cross section of linear order
placement, (c) Cross section of triangle order placement.
19
3.3
A flow chart to simulate power transfer in fiber optical
coupler with constant coupling coefficient
23
3.4
A flow chart to calculate power transfer in optical
directional coupler with variation of coupling
coefficient
25
3.5
A flow chart to plot coupling length and phase versus
coupling coefficient
27
4.1
Power transfer of waveguide 1 and waveguide 2 in
coupling region
29
4.2
Power transfer of waveguide 1, waveguide 2, and
waveguide 2 in coupling region of linear order NX3
fiber coupler
30
4.3
Power transfer of waveguide 1, waveguide 2, and
waveguide 2 in coupling region in triangular order NX3
fiber coupler
31
x
4.4
Power transfer in waveguide 1, and waveguide 2 at
coupling region of NX2 fiber coupler as function of
phase and coupling coefficient
32
4.5
Power transfer in waveguide 1, waveguide 2 and
waveguide 3 at coupling region of linear order NX3
fiber coupler as function of phase and coupling
coefficient
33
4.6
Power transfer in waveguide 1, waveguide 2 and
waveguide 3 at coupling region of triangular order NX3
fiber coupler as function of phase and coupling
coefficient
34
4.7
Plot of coupling length versus coupling coefficient
35
4.8
Plot of minimum phase versus coupling coefficient
36
xi
LIST OF SYMBOLS
D
-
Dimension
N
-
The number of fiber
dB
-
Decibel
n1
-
Refractive index of core
n2
-
Refractive index of cladding
am
-
Acceptance angle
θC
-
Critical angle
NA
-
Numerical Aparture
V
-
V-parameter
λ
-
Wave length
k
-
Wave number
b
-
Normalized propagation constant
β
-
Propagation constant
Vc
-
Cut-off frequency
E
-
Electric field
z
-
Direction of propagation
κ
-
Coupling coefficient
A
-
Power Amplitude
ψ
-
Scalar wave equation solution
a
-
Core radius
U
-
Normalized lateral phase constant
K
-
Hankel Function
w
-
Normalized attenuation constant
d
-
Separation of fiber axis
xii
P
-
Power Propagation
CR
-
Coupling Ratio
WG
-
Waveguide
L
-
Interaction length
lC
-
Coupling length
xiii
LIST OF APPENDICES
APPENDIX
A
TITLE
Code to Calculate Power Transfer in NX2 Fiber
PAGE
41
Couplers with Constant Coupling Coefficient
B
Code to Calculate Power Transfer in both Linear and
43
Triangular Order NX3 Fiber Couplers with Constant
Coupling Coefficient
C
Code to Calculate Power Transfer in NX2 Fiber
46
Couplers with Variation of Coupling Coefficient
D
Code to Calculate Power Transfer in both Linear and
48
Triangular Order NX3 Fiber Couplers with Variation
of Coupling Coefficient
E
Code of Calculation of Minimum Coupling Length as
Function of Coupling Coefficient
51
1
CHAPTER 1
INTRODUCTION
1.1
Introduction
An optical fiber is a glass/ plastic fiber that carries information in terms of
electromagnetic waves over longer distances. Fiber optics have been developed to
obtain power losses as small as possible. Single mode fiber is one kind of fiber
optics that allows a single mode of light propagating in the core of the fiber. Its
tremendous information-carrying capacity and low intrinsic loss have made singlemode fiber the ideal transmission medium for a multitude of applications.
Optical directional coupler uses single-mode fibers as the waveguides. It has
been fabricated and developed for industrial application such as optical switches and
tunable filter in telecommunication and device sensor. Fiber couplers may refer as
devices that are used to combine and split optical signal in the optical system.
Fiber coupler is an optical directional coupler that consists of two or more
fibers arrangement. The most common optical directional coupler is NX2 fiber
couplers, where the two waveguides are in parallel arrangement. The NX3 fibers
coupler has been also designed using three single-mode fibers as waveguides. they
are made by fusion technique. During the fusion process, it is not easy to maintain
coupling length of the fiber couplers. Sometimes the coupling length exceeds the
desire one and resulting undesired coupling ratio between the waveguides. In order to
make it better, and more effective, the study about fiber coupler becomes necessary.
2
A matrix transfer method is a mathematical theory to calculate power
coupling distribution in fiber couplers. This power distribution as a function of
coupling ratio can be controlled by adjusting coupling length and coupling
coefficient which can be varied along the interaction region. The calculation of
power transfer gives important information in coupling region such as power transfer
between the waveguides. Then, the coupling length is determined in order to obtain
output characteristics due to appropriate application such as splitters, routers, etc.
1.2
Background of Study
Fiber couplers are passive devices in which a wide range of parametric
changes occurs during fiber fusion. One of the main phenomena which occurs in
optical coupler is the coupling of mode in space. Coupling length and coupling
coefficient contribute to power transfer along coupled fiber. In order to explain these
phenomena, characteristics of NX2 and NX3 fiber couplers will be studied and
makes the fiber couplers more efficiency in coupling length.
Power transfer characteristics among N parallel single mode optical fibers
have been investigated using coupled mode theory (Wang et al, 2007). The analysis
shows that power transfer of N fibers is periodical during coupling among parallel
single mode optical fibers. NXN optical directional couplers with a linear array
arrangement have been studied by assuming that the coupling coefficient is constant
(Pan et al, 2000).
In most investigation, the power transfer between the waveguide are
investigated when the fibers are in parallel or linear array arrangements and with
constant coupling coefficient. However, it is interesting to investigate the power
transfer characteristics between the waveguides with different of coupling
coefficients.
In this research, coupled mode theory is modified to be a set of matrix
transfer equation that it will be used to determine power transfer in NX2 and NX3
3
single-mode fiber couplers. The coupling length is then analyzed by varying
coupling coefficient along the fiber.
1.3
Problem Statement
In order to obtain the desired output characteristics of the fiber coupler, the
power transfer will be calculated. Experimentally, it is difficult to maintain the
coupling length of fiber couplers due to changes of geometry of fiber during the fiber
fusion. The calculation of the power distribution and the coupling length in order to
address this condition, the matrix technique is introduced, then becomes important.
1.4
Objectives of Study
The objective of this research is to investigate power transfer in NX2 and
NX3 fiber couplers for constant and variable coupling coefficients, and the effect of
order placement for obtaining the power output characteristics by using a matrix
transfer for power propagation computation. It is modeled in 2D and 3D. The
relationship between the coupling length and coupling coefficient is also
investigated.
1.5
Scope of Study
This research focuses on the study of power transfer in interaction region of
multiple fiber couplers. The calculation of power transfer is done with the
assumption that the fiber couplers have constant cross section. The propagation
constants and the separation between the waveguides are maintained constant. The
calculations are performed under two conditions, both constant, and for variable
coupling coefficient. The coupling length is determined to observe the effect of the
coupling coefficient to the power transfer characteristic of the fiber couplers. The
4
kind of multiple fiber couplers that will be investigated are NX2 and NX3 of linear
and triangular order arrangement.
1.6
Significance of Study
The calculations of power transfer in linear order of MXN fiber couplers have
been done with assumption that the coupling coefficient is constant. The calculation
with variable coupling coefficient along the fiber will provide a better understanding
of power transfer on fiber couplers
1.7
Organization of Thesis report
This thesis described five chapters which is as follows. Chapter 1 is the
research framework. This chapter contains some discussion on the introduction to the
study, a description to the problem, the objectives of the study, the scope of study,
the significance of the study and finally the chapter organization.
Chapter 2 will brief about the theory that support this research that has been
done related and used to the study. Chapter 3 will describe a complete calculation on
the research methodology. The coupled-mode theory is expand and expressed in term
of matrix transfer that it will be used to calculated power transfer in fiber couplers.
Chapter 4 is the result and analysis of the calculation based on Chapter 3.
These results are shown in simulation and its analysis which are explained in terms
of the relevant theory. Finally, chapter 5 gives the conclusion of this research and the
future works.
5
CHAPTER 2
LITERATURE REVIEW
2.1
Introduction
The development of optical fiber began in nineteenth Century, when the
French Chappe brothers invented the first optical telegraph. Then, in 1880,
Alexander Graham Bell invented the Photophone, which transmitted a voice signal
on a beam of light. At the same time, William Wheeler invented a system of light
pipes lined with a highly reflective coating that illuminated homes by using light
from an electric arc lamp placed in the basement and directing the light around the
home with the pipes.
In the twentieth century, fiber optic technology experienced a phenomenal
rate of progress. In 1930, Heinrich Lamm was the first person to transmit an image
through a bundle of optical fibers. It was continued by Abraham Van Heel that
covered a bare fiber glass or plastic with a transparent cladding of lower refractive
index. In early 1964, a paper was published by Charles Kao and George Hockham
from Standard Communications Laboratories in England which demonstrated,
theoretically, that light loss in existing glass fibers could be decreased dramatically
by removing impurities. In 1970, the goal of making single mode fibers with
attenuation less then 20dB/km was reached by scientists at Corning Glass Works.
The study of the possibilities of using a dielectric rod as waveguide was not
until 1966 that Kao patented the principle of ‘information transmission through a
6
transparent dielectric medium. But, the use of glass fibers actually became a viable
proposition. Today, fiber’s optical performance is approaching the theoretical limits
of silica-based glass materials. This purity, combined with improved system
electronics, enables fiber to transmit digitized light signals hundreds of kilometers
without amplification. When compared with early attenuation levels of 20 dB per
km, today’s achievable levels of less than 0.35 dB per km at 1310 nanometers (nm)
and 0.25 dB per km at 1550 nm.
Nowadays, 850 nm, 1310 nm, and 1550 nm systems are all manufactured and
deployed along with very low-end, short distance, systems using visible wavelengths
near 660 nm. The important factor of optical fiber development is the increase in
fiber transmission capacity. There are extraordinary possibilities for future fiber optic
applications because of the fiber optic technology’s immense potential bandwidth, 50
THz or greater. Already, the push to bring broadband services, including data, audio,
and especially video, into the home is well underway. To manage these optical
communication networks are needed several basic operations of fiber optics such as
switching, routers, splitters, and wavelength demultiplexing (Okayama, 1991).
Optical directional coupler is a great devices and it has been used in many
application such as optical switches and tunable filter in telecommunication and
device sensor. Typically, fiber couplers made by familiar technique of fused
tapering, exhibit low loss. But the coupling ratio depends on the wavelength passing
through the fiber coupler (Campbell et al, 1993).
In many applications, it is needed to branch one optical fiber into more than
two outputs. NXM fiber coupler is a capable one to divide optical power from N
source into M output ports by controlling the coupling ratio. Where N is the number
of input ports, and M is the number of fused fiber.
In a single optical wavelength system application, typically only one of the
waveguides is used as the input. The input power will be distributed to the others
waveguides. It is possible to determine power propagation on multiple fiber couplers
which consist of N waveguides with single input power. Multiple fiber couplers can
be used to split an optical power into desire output characteristic.
7
2.2
Optical Fiber Parameters
Fiber optics is a tool to bring the data or signal in big capacity over longer
distances. The design of fiber optics has developed both commercial and advance
quality. The simplest fiber optic consists of a core with a refractive index n1 which is
higher than the refractive index n2 of the surrounding region, the cladding as shown
in Figure 2.1.
Core, n1
Cladding, n2
n1 > n2
Figure 2.1. Basic structure of fiber optics
The signal propagating inside the fiber is in term of ray optics. The ray of
light propagation experiences total internal reflection at the core and cladding
interface. For total internal reflection condition, the incident angle φ at the corecladding surface must exceed the critical angle θC . For that surface as described in
Figure 2.2, and hence θ to be less than θ m =
π
2
− θC , thus the angle of incident a
must be less than certain angle am (Khare, 2004).
normal
θm
am
a
θ
θC φ > θ C
Core (n1 )
Cladding (n2 )
Figure 2.2. Ray propagation in fiber optics
8
From Figure 2.2, the critical angle θC is defined as follows.
 n2 

 n1 
θ c = sin −1 
(2.1)
Based on Snell’s law at the core medium (air) interface, there is relationship as,
na sin am = n1 sin am = n1 cos θC
2.2
substituting Equation (2.1) in to Equation (2.2) yields the Numerical Aparture (NA)
of the fiber, which determines the light gathering capacity of the fiber (Khare, 2004).
NA = na sin am = ( n12 − n22 )
Defining refractive index different as ∆ =
1
2
2.3
n1 − n2
, NA can be expressed as follows.
n1
NA=n 1 2∆
2.4
A mode of the fiber describes the possible path of light propagating along the
fiber (Khare, 2004). By solving wave propagation in cylindrical waveguides, and λ
denotes wavelength, the number of modes of an optical fiber is determined by Vparameter.
V=
2π a
λ
n12 − n22
2.5
Normalized propagation constant b is defined as a certain propagation
coefficient value β which can vary from a minimum of n2 k (propagation in the
cladding) to a maximum value of n1 k (propagation constant in the core). Figure 2.3
shows b value of modes vary with V, where each mode can exist above a certain
value of V.
(β k )
b=
− n22
n12 − n22
2
2.6
9
It can be seen in Figure 2.3 that only certain modes are allowed to propagate
along the fiber, and for value of V below 2.405, only one mode can be exist.
Otherwise, for value of V above 2.405 the optical fiber is called as Multimode Fiber.
The condition for β = β2 is known as cut-off of a mode, where at β = β2 normalized
propagation constants b = 0, w = 0, and u = V = Vc , with Vc is the cut-off frequency
(Khare, 2004).
1
08
01
06
11
b(v)
21
02
04
31
12
02
0
2
4
6
8
10
V
Figure 2.3. Plot of b versus V for one type of fiber (Casmier and Carolyn, 2005)
2.3
Coupled-mode Theory for the Optical Waveguides
Coupled mode theory is a basic concept to describe optical directional
couplers. The essence of coupled-mode theory is clearly one treats the composite or
compound waveguide structure as a collection of simpler waveguides, which modes
associated with each individual waveguides being perturbed by the presence of the
others or any additional non uniformity (Litle and Hung, 1995). These perturbation
lead to coupling and exchange of power among guided modes.
Assuming two waveguides as two dielectrics are parallel, far apart, and
isolated. Each waveguides support a number of guided modes possibility a
continuum radiation mode (Chen, 2007), as shown in Figure 2.4.
10
waveguide 1
waveguide 2
d
Figure 2.4. Cross section of two fiber
The guided portion of two waveguides is expressed mathematically as the
source-free solution to the Maxwell’s equation, which is sum of the modes. If
e1 ( x, y) and e2 ( x, y) are the fields, and β1 and β2 are propagation constants, the
fields guided by the composite waveguide structure can be expressed as follows.
E( x, y ) = a1 ( z )e1 ( x, y )e − j β1z + a2 ( z )e1 ( x, y )e − j β 2 z
2.7
E( x, y ) = ∑ ai ( z )Ei ( x, y )e − j βi z
2.8
or in general
i
where E( x, y, z ) = ei ( x, y )e − j β1 z .
When the waveguide 1 e1 ( x, y )e − j β1z physically extend to waveguide 2 and
e 2 ( x, y )e − j β2 z to waveguide 1, amplitude functions can be determined as the
following equations.
a1 ( z ) = ( as 0 e − jσ z + aa 0 e − jσ z )e + jδ z
2.9a
a2 ( z ) = ( as 0 rs e − jσ z + aa 0 ra e − jσ z )e + jδ z
2.9b
11
Where σ = κ1κ 2 + δ 2 , and rs , a denote two root constants of c and d . If the two
equations above are differentiated, and A1 ( z ) and A2 ( z ) are introduced in to these
equation such that A1 ( z ) = a1 ( z )e − j β1 z and A2 ( z ) = a2 ( z )e − j β2 z , as a result given as
coupled mode equations (Wang et al, 2007).
dA1 ( z )
= − j β1 A1 ( z ) − jκ12 A2 ( z )
dz
2.9a
dA2 ( z )
= − j β 2 A2 ( z ) − jκ1 A1 ( z )
dz
2.9b
For optical fiber couplers consist of M fiber, mode equation can be expressed as
follows.
dAm ( z )
= − j β m Am ( z ) − jκ m ( m −1) Am −1 ( z ) − jκ m ( m +1) Am +1 ( z )
dz
2.10a
dAn ( z )
= − j β n An ( z ) − jκ n ( n −1) An −1 ( z )
dz
2.11b
Where the z-axis in Cartesian coordinate is taken to be parallel to the fiber axes, n
and m refer to the mth fiber and nth fiber respectively. Then β denotes propagation
constants, and κ is the coupling coefficient.
2.4
Fiber coupler parametric and coupling ratio
Power transfer in a directional coupler is affected by the propagation
constants, coupling coefficient, coupling length, and coupling ratio. Each parameter
is determined from the wave equation with approximation and boundary conditions.
12
2.4.1 Propagation Constants
Assume two identical single mode fibers in Figure 2.4 are optically well
separated. The fields of one fiber in isolation are not a good approximation to the
fields of composite waveguide, so that the contribution from one fiber to the field at
the center of the second fiber is small (Snyder, 2003).
The scalar wave equation solutions of the fiber isolation are respectively
ψ 1,ψ 2 , and the composite waveguide solution is ψ ( x, y ) , the symmetry of composite
waveguide leads to the two fundamental solutions for ( x, y ) (Ctyroky and Thylen,
1994).
ψ + = ψ 1 +ψ 2 ,
ψ − = ψ 1 −ψ 2
2.12
The propagation constants associated with ψ + and ψ − are β+ and β− respectively,
and β is the propagation constant common to ψ 1 and ψ 2 . Since the two fibers are
separated
infinitely,
it
can
be
approximated
β = β± , ψ = ψ ± , n = n1 , and ψ = ψ 1 , and obtained

( n − n )ψ ψ
∫

β± = β + k 
1
1
2
A∞

dA

∫ ψ ψ dA
1
1
β+ = β− = β or
2.13
A∞
Where n is the composite profile, and A∞ is the infinite cross section. Noting that ψ 2
is exponentially small over the core of the unperturbed fiber.
The integral of ψ 1,ψ 2 can be negligible compared with the integral of ψ 1 in
2
the dominator. In the nominator n − n1 vanishes over the core of the unperturbed
fiber, so the dominant contribution to the integral comes from the core of perturbing
fiber 2 (Snyder, 1983). Under this assumption, and based on Equation (2.13),
propagation constants can be approximated as follows.
β1 = β ± κ ,
2.14
13


where κ =k  ∫ n − n1 ψ 1ψ 2 dA
A ∞

(
)
∫ψ
2
1
dA .
A∞
2.4.2 Coupling Coefficient
The Coupling coefficient is a parameter that effect power propagation in
optical fiber coupler. Noting n − n1 in Equation (2.14) is non zero only over the core
of the second fiber, and that the integral in the dominator is proportional to the
normalization of N (Wang, 2008). Since r1φ1 and r2φ2 denote cylindrical fiber, the
coupling coefficient κ is expressed as follows.
2π a
K 0 (W )
1  ∆ U2
r K (Wr / a ) J 0 (Ur2 / a )dr2 dφ2
 
2
π a  2  V K1 (W ) J 0 (U ) ∫0 ∫0 2 0 1
12
κ=
2.15
Solving Equation (2.15) using Bessel function method yields the coupling coefficient
for directional couplers (Okamoto, 2006).
( 2∆ )
κ=
12
a
U 2 K 0 (Wd / a)
V 3 K12 (W )
2.16
Where a is the core radius, d is separation of fiber axis, K0 , and K1 are the zeroth
12
and first order of Hankel function, V = 2π a /  λ ( n12 − n22 )  is the normalized


frequency, U and W are normalized lateral phase constant and normalized
attenuation constant respectively. Equation (2.16) describes that the coupling
coefficient is a function of normalized frequency, which depends on refractive index
of core and cladding and separation of the fiber axis (Saktioto, 2008).
14
2.4.3 Coupling Length
In almost all of the applications, the knowledge of coupling lengths of the
directional couplers is essential in the design of these devices. Coupling length, lc is
defined as the minimum length that required to transfer power completely from one
waveguide to the others waveguides (Chen, 2007).
2.4.4 Coupling Ratio
Coupling ratio is defined as the fraction of the power available at the output
ports (Khare, 2002). If P1 , and P2 are the power in waveguide 1 and waveguide 2
respectively, the coupling ratio can be written mathematically as follows,
 P2 
Coupling Ratio (CR) (%)= 
 X 100
 P1 + P2 
2.17
or it can be expressed in dB.
 P2 
Coupling Ratio (CR) (dB)= -10 log10 

 P1 + P2 
2.18
The coupling ratio can be controlled by spacing adjustment. The power
transfer ratio can also be controlled from nearly 0% (actually, less than 0.01% has
been obtained) up to nearly 100% (less than a 0.5-dB loss) (Jeunhomme, 1983).
2.5
The Matrix Transfer Method.
The Matrix Transfer Method is typically used to calculate the path and final
location of light rays in geometric optics, but can also be used to analyze propagation
of electromagnetic through a layered medium. It is a powerful tool for the analysis of
15
periodic structure (Makino, 1995). In the majority of this research it will be studying
how to use matrix transfer in geometric optics.
The plane wave method was the first technique used to solve the full vector
photonic band structure problem and has probably become the most widely used due
to its ease of implementation and wide applicability. However, it cannot be used in
every situation. Pendry and Mac Kinnon developed an alternative technique known
as the transfer matrix method by applying a procedure that had been very successful
in the field of disordered systems.
The method (which is explained in depth elsewhere in this thesis) essentially
converts Maxwell’s equations into a set of difference equations in real space and then
rearranges those equations into the form of a matrix transfer. This matrix transfer
relates the electric and magnetic fields in one fiber to the fields in an adjacent fiber.
By comparing the transfer matrix equation to Bloch’s law it can be seen that the band
structure is given simply by the eigenvalues of the transfer matrix. This approach has
several key advantages over the plane wave method. The main advantage is that the
coupled modes equation determined from Maxwell’s equation can be rearranged in a
matrix transfer model.
The transfer matrices of fiber coupler are different for every condition, such
as active and passive waveguide, crossing, phase shifter, splitter, combiner, and
directional coupler (Marz, 1994). The following explanations are only concentrated
in directional couplers.
If A1 (0) and A2 (0) are the amplitudes of power input, and A1 ( z ) and A2 ( z )
are the amplitudes of power output of waveguide 1 and waveguide 2 respectively, the
transfer matrix for elementary components of 2X2 fiber coupler is shown in Figure
2.5.
A1 (0 )
A1 ( z )
κ
A 2 (0 )
L
A2 ( z )
Figure 2.5. Transfer matrix of 2X2 directional coupler (Marz, 1994).
16
Defining M as the matrix transfer between input and output port of the fiber
couplers, it can be written as follows.
 A ( z)
M = 1 *
 − A2 ( z )
A2 ( z ) 

A1 ( z )* 
(2.19)
The modes of dielectric waveguides with arbitrary profiles are computed in
this research by a two-dimensional Fourier expansion. The scalar-wave equation is
converted into a matrix eigenvalue equation by an expansion of the unknown field in
a complete set of orthogonal functions. This expansion is applied to convert a linear
partial differential equation into a matrix eigenvalue equation.
17
CHAPTER 3
RESEARCH METODHOLOGY
3.1
Introduction
To study fiber optical coupler, is needed several methodologies. It will be
explained in this chapter to describe power transfer between two or more
waveguides, and to identify effect of change of coupling coefficient in fiber optical
coupler.
This research deals with modeling and simulating power propagation between
the fibers in fiber optical coupler. Moreover, the effect of variation coupling
coefficient on optical fiber coupler will be designed in 3D. It will be done and
concentrated on the NX2 and NX3 fiber optical coupler.
To observe power transfer in 1X2 and 1X3 fiber optical coupler, matrix
transformation is required which was built from eigenvalue and eigenvector. Then,
there is assumed that input power is fed to one waveguide only.
The data processing and numerical analysis were done using computer
software MATLAB 6.0. The user of this program can perform operations in two
ways. First is the interactive mode in which all commands are entered directly in the
command window. Secondly, we can run a MATLAB program stored in script file.
For this study, we are using the interactive mode and with extension “.m”.
18
3.2
Matrix Transfer Method of Optical Directional Coupler with Two and
Three Waveguides
The matrix transfer has been used to represent the solution of the coupled
mode equation by 2X2 transfer matrix, which relates the forward and backward
propagating field amplitude (Makino, 1995). Consider a fiber coupler consist of two
waveguides namely waveguide 1, WG1 and waveguide 2, WG2, as two cylindrical
directional coupler as shown in the following Figure 1.
Single-mode fiber
WG1
L
WG2
Figure 3.1: Two cylindrical 2X2 directional fiber coupler
For a directional coupler with uniform cross section and constant spacing, the
coupled differential Equations (2.10a) and (2.10b) can be written in term of matrix
form.
= − κ
κ (3.1)
and then,
υ + υ = υ + υ (3.2)
Equation (3.2) is the solution of Equation (3.1) and shows the normal mode
theory of two waveguides directional coupler. Given [ ] and [ ] are
the eigenvalues, and are amplitude, and are the propagation constant of
the normal modes. This equation demonstrates the propagation of the two normal
modes. For directional couplers which have identical waveguide, the solution in
matrix form given by the associated eigenvectors of [1 1] and [1 − 1] is
19
 A1 ( z )   as 0 e − j β s z + aa 0 e − j βa z 

=

− jβs z
− aa 0 e − j β a z 
 A2 ( z )   as 0 e
(3.3)
The interference of the symmetric and antisymmetric normal modes leads to
the periodic exchange between the two waveguides. The transfer matrix of 2X2
optical couplers described by the initial power amplitudes fed to the waveguide 1,
A1 (0) and waveguide 2, A2 (0) and the power amplitudes at certain position of
waveguide 1, A1 ( z ) and waveguide 2, A2 ( z) can be written as the follows
cos & − sin &
(
="
+κ − sin &
'
σ
−
+κ,
sin &
cos & −
σ
'
(
0
sin & 0
.
(3.4)
Equation (3.4) is a symmetry matrix form and it shows that power propagations in
fiber coupler are cosine function.
A directional coupler with three waveguides can be arranged in two order
placement as shown in Figure 3.2.
Single-mode fiber
Input beam
WG1
L
WG2
WG3
(a)
WG1
WG1
ʘ
Input beam
WG2
(b)
WG3
WG2
ʘ
Input beam
WG3
(c)
Figure 3.2. NX3 fiber coupler, (a) Three Single-mode fiber in 3X3 directional
coupler, (b) Cross section of linear order placement, (c) Cross section of
triangle order placement at fiber coupling.
20
The coupled mode equation of 3X3 fiber coupler is generalized becomes:
 A1 ( z ) 
 β1
d 

A2 ( z )  = − j  k 21

dz
 A3 ( z ) 
 k31
k13   A1 ( z ) 
k 23   A2 ( z ) 
β 3   A3 ( z ) 
k12
k2
k32
(3.5)
Where β1, β2 , and β3 are the propagation constant of isolated waveguide 1,
waveguide 2, and waveguide 3, and κ12 , κ13 , κ 21 , κ 2 , κ 23 , κ 31 , κ 32 are the coupling
constants between waveguide 1, waveguide 2, and waveguide 3. The amplitudes of
the field guided by three waveguides can be written as the following:
Ai ( z ) = Pi e j β a z + Qi e j β b z + Ri e j β c z ; i = 1,2 and 3
(3.6)
Consider a directional coupler has three identical and equally spaced
waveguides, β1 = β2 = β3 = β , and κ 21 , = κ 23 = κ . Since the interactions occur
between nearest waveguides, κ13 and κ31 can be ignored, and Equation (3.5) is
written as:
 A1 ( z ) 
β
d 

A2 ( z )  = − j  k
dz 
 A3 ( z ) 
 0
0   A1 ( z ) 
k   A2 ( z ) 
β   A3 ( z ) 
k
k
k
(3.7)
based on eigenvectors of matrix 3X3, [V ] can be constructed as:
 1
1
[V ] =  2
2
 1
By defining [U ] = [V ]
equation:
−1
−1
2
−1
 1
2 

1
−1
0  , or [V ] =  −1
2

− 2 
 2
2
2
0
1 

−1 

− 2 
(3.8)
[ A] , Equation (3.7) can be transformed to matrix differential
 β + 2k
U 1 ( z ) 
d 
 =−j 0
U
(
z
)

2

dz 
 0
U 3 ( z ) 

0
β − 2k
k
0  U 1 ( z ) 

0  U 2 ( z ) 
β  U 3 ( z ) 

(3.9)
21
solution of Equation (3.7) is the following:
− j ( β + 2k ) z 
U1 ( z )  U1 (0)e
U ( z )  = U (0)e − j ( β − 2k ) z 
 2   2
U 3 ( z )   U1 (0)e − j β z 


(3.10)
finally, Substituting Equation (3.7) to [ A] = [V ][U ] yields.
 A1 ( z ) 
− jβ z
 A ( z) = e
 2 
2
 A3 ( z ) 


−j
 U1 (0)e


 1 
 
+j
 2  + U 2 (0)e
 1 
 
2kz
 −1 
 2
 
 
2
+
(0)
U
3
 
 0   (3.11)
 −1 
 2
 
 
2kz
Matrix transfer 3X3 is constructed from Equation (3.11).
(
)
1
 2 cos 2κ z + 1

 A1 ( z ) 
j


− jβ z 
sin 2κ z
 −
 A2 ( z )  = e
2

 A3 ( z ) 
1
 2 cos 2κ z − 1

(
)
−
j
2
sin 2κ z
cos 2κ z
−
j
sin 2κ z
2
(
)
(
)
1
cos 2κ z − 1
2
j
−
sin 2κ z
2
1
cos 2κ z + 1
2
  A1 ( 0 ) 


  A2 ( 0 )  (3.12)
  A3 ( 0 ) 




Equation (3.12) is a set of matrix transfer that describes power propagation in
3x3 fiber coupler. It is can be used to calculate power transfer between waveguides
both linear and triangular order placement of NX3 fiber couplers since the input
power to the input port are given. For example, in this research the input power is fed
to one of the three waveguides whatever the outer waveguide or the center
waveguide.
22
3.3
Calculation of Power Transfer on Multiple Fiber Couplers with Constant
Coupling Coefficient
The are several parametric affect the Power propagation inside the fiber,
those are refractive index, propagating constant, and coupling coefficient. Each
constant have important relationship between them. In this case each waveguides are
assumed having the same space constants.
By assuming input power is fed to the waveguide only, power transfer can be
determined using matrix transformation. For 1X2 fiber optical coupler, by inserting
power (say in to waveguide 1), then A1 (0) = 1 and A2 (0) = 0 , so power amplitude
can be yielded from Equation (3.4)
= cos & −
= −
'
(
+κσ
sin &
sin &
(3.13a)
(3.13b)
To calculate the power transfer between waveguide 1 and waveguide 2 using
set formulation as in the flowchart of Figure 3.3 (Appendix A):
23
nco , ncl , z
β =κ xn
δ = ∆β 2
σ=
( κ1 xκ 2 ) + δ 2
Phase= κ z π
Phase =0 − 0.8 × 10−6
no
yes
Figure 3.3. A flow chart to simulate power transfer in fiber optical coupler with
constant coupling coefficient
In the same way, NX3 fiber optical coupler is assumed that power is fed to
the one waveguide only. First, if power is introduced to the waveguide 1 or
waveguide 3, and 0 = 1, 0 = 0 0 = 0. Substituting these values to
24
the Equation (3.12), the power output amplitude A1 ( z ), A2 ( z ) and A3 ( z ) can be
written as.
= 1−1 + 234√2κ7 = −
√
1489√2κ7 0 = 11 + 234√2κ7 Second, if power is
( A2 (0) = 1,
fed to
(3.14a)
(3.14b)
(3.14c)
the center or waveguide 2, then
and A1 (0) = A3 (0) = 0 ) , similarly, substituting it to the Equation (3.12)
yields:
A1 ( z ) = A3 ( z ) = − j
A2 ( z ) =
(
)
1
sin 2κ z e − j β z
2
(3.15a)
)
(3.15b)
(
1
cos 2κ z e − j β z
2
The calculation of power transfer in both linear and triangular order can be
used a flow chart given in Figure 3.3 (Appendix B).
3.4
Calculation of Power Transfer on Multiple Fiber Couplers with
Variation of Coupling Coefficient
As described in the Chapter 2, the coupling coefficient is one of parametric
that influences power propagation along the fibers. Power transfer of both NX2 and
NX3 fiber couplers is calculated still using the same method and assumption as
before, but the coupling coefficient is varied along the waveguides. The flow chart is
shown in Figure 3.4, (see Appendix C and D)
25
κ
nco , ncl , z
β =κ xn
δ = ∆β 2
σ=
( κ1 xκ 2 ) + δ 2
Phase= κ z π
κ = 0.1x103 − 1.5x103
no
yes
Figure 3.4. A flow chart to calculate power transfer in optical directional coupler
with variation of coupling coefficient
26
3.5
Derivative of Coupling Length of NX2 and NX3 Fiber Couplers
As mentioned in Chapter 2, the coupling length is the length of interaction
region to transfer power from one waveguide to others waveguides. Since the power
is fed to the one waveguide only, the coupling length can be calculated using power
amplitude equations. By setting Equation (3.13b) equals to 1, this means that the
power is transferred completely to waveguide 2, the coupling length determined as
follows.
A1 ( z ) = −
κ
sin σ z = 1
σ
(3.16)
Equation (3.16) is solved by assuming δ = 0 , and sin σ z must be 1 , or
z=
nπ
, n = 1,3,5,… Taking n=1, then z is defined as the minimum coupling length,
2κ
lC that required to transfer power completely from the waveguide 1 to the
waveguide 2 in NX2 fiber coupler.
lC =
π
2κ
(3.17)
Similarly, the minimum coupling length of linear order NX3 fiber coupler
that required to transfer power completely from waveguide 1 to waveguide 3 is
determined by setting Equation (3.14c) equal to 1 as the following.
A3 ( z ) =
As a result, z =
(
)
1
cos 2κ z + 1 e − j β z = 1
2
(3.18)
nπ
, for n = 1,3,5,... the minimum coupling length is given as
2κ
follows.
lC =
π
2κ
(3.19)
27
Similarly, the minimum coupling length for triangular order fiber coupler is
determined from Equation 3.15a, and makes it equal to 1 2 due to power transferred
50% : 50% to the two others waveguides.
A1 ( z ) = A3 ( z ) = − j
Solving Equation (3.20) yields z =
(
)
1
1
sin 2κ z e − j β z =
2
2
(3.20)
nπ
, for n = 1,3,5,... The minimum coupling
2 2κ
length is given as follows.
lC =
π
2 2κ
(3.21)
From coupling length equations, it can be seen that coupling length is a
function of the coupling coefficient. It can be calculated using the flow chart given in
Figure 3.5, (Appendix E).
κ = 0.1 − 1.5 (mm −1 )
Phase =0 − 0.8 × 10 −6
Figure 3.5. A flow chart to plot coupling length and phase versus coupling
coefficient
28
CHAPTER 4
RESULT AND DISCUSSION
4.1
Introduction
This chapter describes results of calculation and simulation which have been
conducted in the previous chapter. Power transfer between waveguides for both NX2
and NX3 fiber couplers has been calculated using matrix transfer method with
assumption that the propagation constants, cross section, and coupling length are
held constant.
4.2
Power Transfer on Fiber Couplers with Constant Coupling Coefficient
Power transfer between the waveguides in both NX2 and NX3 fiber couplers
has been determined based on coupled mode theory and expressed as matrix transfer
model. Equation (3.4) is the matrix transfer for NX2 fiber coupler. It is symmetry
matrix form which describes that the two waveguides are in parallel arrangement or
sit side by side. The power propagation amplitudes of each waveguides are given by
Equation (3.13). Power input is 1 Watt and fed to the one waveguide only. The
following is the explanation of obtained results and discussions. Since the power is
fed to one of the two fibers, and assuming the propagation constants, cross section,
29
coupling length, and coupling coefficient are held constants, power transfer in
coupling region of NX2 fiber coupler can be shown in Figure 4.1.
Figure 4.1. Power transfer of waveguide 1 and waveguide 2 in coupling region
As coupling length or phase increases, power in the waveguide 1 decreases
gradually and it is transferred to the waveguide 2. Power of waveguide 1 will be
transferred completely to the waveguide 2 when the power of waveguide 1 is
minimum. This occurs at certain condition, which is odd multiple of z = nπ / 2κ . For
n=1, the value of z = π / 2κ , which is minimum coupling length determined in
section (3.5.).
For NX3 fiber couplers, the three waveguides can be arranged in two order
arrangement. First, the three waveguides are parallel and sit side-by-side as a linear
order shown in Figure 3.2a. Since power is fed to one of outer waveguides
(waveguide 1 or waveguide 3), the power input in waveguide 1 will be distributed to
the waveguide 2 and waveguide 3 as shown in Figure 4.2.
30
Figure 4.2. Power transfer of waveguide 1, waveguide 2, and waveguide 3 in
coupling region of triangular NX3 fiber coupler
When the power input is launched to the waveguide 1, it will decrease
gradually and transferred to the waveguide 2 and continue to the waveguide 3. The
power input will be transferred completely to the waveguide 3 when the power of
waveguide 1 and waveguide 2 are minimum as it can be seen in Figure 4.2.
4.2 In
example
mple at phase = 1, when power in both waveguide 1 and waveguide 2 equal to
zero, the waveguide 3 is maximum or equal to 1. This condition occurs periodically
at z = nπ / 2κ , where for n=1, the value of z = π / 2κ is the minimum coupling
length to transfer power from waveguide 1 to the waveguide 3 as given by Equation
(3.17).
Secondly, the three waveguides of NX3 fiber couplers are in triangular order
arrangement as shown in Figure 3.2c. When the power is fed to the waveguide
wavegu
2
only, the power input will be distributed symmetrically to the two others waveguides
with the same value as given by Equation (3.15a).
31
Figure 4.3. Power transfer of waveguide 1, waveguide 2, and waveguide 2 in
coupling region in triangular order NX3 fiber coupler
Figure 4.3 shows that power in waveguide 1 and waveguide 3 are
overlapping. It means that power launched to the waveguide 2 is always distributed
to the two others waveguides with the same value which are 50% and 50%.
50% Power
input will be transferred completely at z =
π
2 2κ
. It can be seen in Figure 4.3, in
example at phase = 0.5, power in waveguide 1 is minimum,
minimum, otherwise the two others
waveguides are maximum. This kind of fiber couplers can be used as a power
divider.
4.3
Power Transfer on Fiber Couplers with Variation of Coupling
Coefficient
The
he power transfer of both NX2 and NX3 fiber couplers has been
investigated with the assumption that power coupling parameters are constant. In this
following calculation, the coupling coefficient is varied along the waveguides. But,
32
the coupling coefficient between the waveguides is assumed the same and
interactions occur between the nearest waveguides only.
Power transfers between the waveguides in both NX2 and NX3 fiber couplers
are calculated using matrix transfers which are given by Equation (3.4) and Equation
(3.12) respectively with variable coupling coefficients.
Coupling coefficients along the interaction region are adjusted by changes
refractive index of core and cladding. The changes will affect the coupling length of
fiber couplers.
(µ m−1 )
Figure 4.4. Power transfer in waveguide 1, and waveguide 2 at coupling region of
NX2 fiber coupler as function of phase and coupling coefficient
Figure 4.4 shows the power transfer in coupling region of NX2 fiber coupler
with variable coupling coefficients. It shows a good agreement between coupling
coefficient and phase as given by Equation (4.3). Increasing the coupling coefficients
will make phase of power transfer becomes shorter.
33
The effect of changes of coupling coefficient as applied in NX2 fiber coupler
also behaves in NX3 fiber couplers. By introducing power to the waveguide 1 in
linear order NX3 fiber coupler, and varying coupling coefficients along the fiber
coupler, power propagation in coupling region can be shown in 3D.
(µ m−1 )
Figure 4.5. Power transfer in waveguide 1, waveguide 2 and waveguide 3 at
coupling region of linear order NX3 fiber coupler as function of phase
and coupling coefficient
The relationship between power in waveguide 1, waveguide 2, and
waveguide 3 which are given by matrix transfer of 3x3 fiber couplers are much
dependent on the coupling coefficient. If the coupling coefficient increases, the phase
of power propagation decreases as it can be observed in Figure 4.5.
If the power is fed into waveguide 2 in triangular order NX3 fiber coupler,
and varying coupling coefficient along the coupling region, power propagation in
coupling region can be shown in 3D.
34
(µ m−1 )
Figure 4.6. Power transfer in waveguide 1, waveguide 2 and waveguide 3 at
coupling region of triangular order NX3 fiber coupler as function of
phase and coupling coefficient
Figure 4.6 shows that the input power in waveguide 2 is distributed to the
waveguide 1 and waveguide 3. Increasing coupling coefficient both NX2 and NX3
fiber coupler from 0.1 mm−1 to 1.5 mm−1 gives significant change in the minimum
coupling length, and of course, it will change the minimum phase needed to transfer
power as shown in Figure 4.7 and Figure 4.8 respectively.
Since the coupling length of NX2 and NX3 (both linear and triangular order)
defined as minimum length that is required to transfer power completely from one
waveguide to others waveguides, there are good agreement in coupling length and
phase between them. Based on the coupling length equations which have been
determined in Chapter 3, they can be compared as follows.
35
lC ( NX2 ) : lC ( NX3 linear order ) : lC ( NX3 triangular orde )
lC =
π
π
π
:
:
2κ
2κ 2 2κ
(4.1)
(4.2)
The relationship between coupling length and coupling coefficient of NX2
and NX3 both linear and triangular order arrangement of fiber couplers can be shown
in Figure 4.7.
Figure 4.7.
4.7 Plot of coupling length versus coupling coefficient
Based on equation (4.2), it can be seen
seen clearly in Figure 4.7 that the coupling
length of the linear order NX3 is larger with a factor
2 than the coupling length of
NX2, but the coupling length of triangular order NX3 is shorter with a factor 1/ 2
than the coupling length of NX2 fiber coupler. Phase of power propagation in fiber
couplers define as follows (Chen, 2006).
2006)
Phase = κ z /π
(4.3)
Substituting coupling length
le
given by Equation (4.2) into
to Equation (4.3), the
relationship between phase of NX2 and NX3 fiber couplers is shown in Figure 4.8.
36
Figure 4.88. Plot of minimum phase versus coupling coefficient
Figure 4.8 shows that minimum phase of linear order NX3 is larger with a
factor
2 than phase of NX2, but phase of triangular order NX3 is shorter with a
factor 1/ 2 than phase of NX2 fiber as shown
shown in Figure 4.7. In example, at
κ = 1500 µ m−1 , minimum phase to transfer power completely for NX2, linear order
NX3, and triangular order NX3 are 0.5, 0.7, and 0.35 as shown in Figure 4.1, Figure
4,2, and Figure 4.3 respectively.
37
CHAPTER 5
CONCLUSION
5.1
Conclusion
The results of the calculation of power transfer in fiber couplers have shown
that power transfer distribution depends on not only the geometry order of the fibers,
but also on the coupling coefficient along the interaction region. When the power is
fed to one of two waveguides in NX2 fiber coupler, the power input will be
distributed in to another waveguide, and 100% complete transfer occurs at oddmultiple of z = π / 2κ .
For NX3 fiber coupler, there are two situations. If power is fed to the outer
waveguide as a linear order, at odd-multiple of z = π / 2κ , the power input will be
completely transferred 100% to the crossover waveguides. If the power is fed to the
center waveguides as triangular order, the power input will be distributed
symmetrically to the two others waveguides with coupling ratio of 50% at oddmultiple of z =
π
2 2κ
.
The coupling length of the linear order NX3 is larger
2
times than the
coupling length of the NX2 fiber coupler. The coupling length for triangular order
NX3 is shorter by a factor 1 2 than the NX2 fiber coupler.
38
The power transfer of both NX2 and NX3 fiber couplers have been described
in 3D. The changes of the coupling coefficient affect the coupling length of the fiber
couplers. The higher the coupling coefficients, the shorter the coupling length is
needed to transfer power to the others waveguides.
5.2
Future Work
The calculation which has been done in this research is for the two and three
waveguides only, and with several assumptions. The important thing is assuming the
coupling coefficients between the waveguides are equal, and the interaction of light
occurs between the nearest waveguides only. It is suggested that in the future to
examine the interaction of light in many fibers such as NXM and various
arrangements.
39
REFERENCES
Alan W. Snyder and Jhon D. Love, Optical Waveguide Theory. London, (1983).
B. E. little and W. P. Huang, Coupled-Mode Theory for Optical Waveguides,
Progress in Electromagnetics Research, PIER 10, 217-270 (1995)
Chen, C, Foundations for Guided-Wave Optics, Perdue University, West Lafayette,
Indiana, (2007)
Casimer M. D and Carolyn J. S. D., Fiber Optics Essential. Distinguished Engineer,
IBM Corporation, Poughkeepsie, Academic Press, London (2006)
H. Kogelnik and R. V. Schmidt, Switched directional couplers with alternating ∆,
IEEE Journal of Quantum Electronics, VOL. QE-12, no. 7, 396-401 (July
1976).
H. Okayama, T. Ushikubo, and T. Ishida, Directional coupler with reduced voltageproduct, IEEE J. Lightwave Technolgy, Vol. 9, pp. 1561-1566 (1991).
J. Ctyroky and L. Thylen, Analysis of a directional coupler by coupled modes of a
single waveguide, Opt. Lett., Vol. 19, pp. 1621-1623 (1994).
J. Pan and M Shih, 1XN and NXN Fiber Optics Couplers, IEEE J. patent number
6,049,6452000.
J. Ctyroky and L. Thylen, Analysis of a directional coupler by coupled modes of a
single waveguide, Opt. Lett., Vol. 19, pp. 1621-1623 (1994).
Jeunhomme L.B., M, Dekker, Single Mode Fiber Optics. Principle and
Applications, Marcel Dekker Inc., New York, 1983
Khare R. P, Fiber Optics and Optoelectronics, Oxford University Press. India
(2004)
Makino, T., Transfer matrix Method with Application to Distributed Feedback
40
Optical Devices, Progress in Electromagnetics Research, PIER 10, 271-319
(1995)
Nor Faridah Hanim, Saktioto, Jalil Ali, Rosly Abdul Rahman. ”Modeling of
Induced Voltage Silicon Dioxide Fiber Coupling”, Journal of Nonlinear Optic
Physics and Materials, World Scientific Publishing Company. Vol.18, No3,
pp. 481-488, 2009.
Okamoto, K., Fundamental of Optical Waveguides.Okamoto Laboratory Ltd
Ibaraki, Japan (2006).
Saktioto, Jalil Ali, Mohamed Fadhali, Theoretical and empirical comparison of
coupling coefficient and refractive index estimation for coupled waveguide
fiber, Journal of Aplied Science and Engineering Technology. (2008).
Saktioto, Ali, J., Fadhali, M., Rahman, R.A., Zainal, J. Modeling of
coupling coefficient as a function of coupling ratio”, Proceedings of
SPIE – The International Society for Optical Engineering 7155, art. No.
71551P (2008)
Saktioto, Ali, J., Fadhali, M., Zainal, J. Estimation of coupling power parameter of
fused coupled fibers, Proceedings of SPIE – The International Society for
Optical Engineering 6793, art. No. 679309 (2008)
Saktioto, Power Parameters of Fusion Fiber Coupling. Thesis of Doctor of
Philosophy Universiti Teknologi Malaysia (2008).
Wang, Y., Xue, D., Lu, X., Power transfer characteristics among N parallel
single-mode optical fibers, Science Direct. Zhejiang University (2007)
Wang, X., Xiong, C., Luo, J., Coupling coefficients evaluation of a directional
coupler using gain guided and index antiguided fibers. Journal of Optic
Comunication, Elsevier (2008)
41
APPENDIX A
Code to Calculate Power Transfer in NX2 Fiber Couplers with Constant Coupling
Coefficient
kappa1=1.5e3;
kappa2=1.5e3;
n1=1.4677;
n2=1.4624;
B1=kappa1*n1;
B2=kappa2*n2;
DB=B1-B2;
delta=DB/2;
d=delta/sqrt(kappa1*kappa2);
sigma=sqrt((kappa1*kappa2)+delta^2);
%coupling length range
step=5000;
zal1=1e-8;
zal2=7.5e-6;
zal=[zal1:7.489e-6/step:zal2] ;
M=1000;
%normal mode amplitude CHECK IT AGAIN THE AMPLITUDE!
as=1;
aa=1;
rs=1;
ra=1;
for (r=1:step+1)
F=[1 0;0 0];
z=zal(r);
%amplitude function
a1z=(((as*exp(-i*sigma*z))+((aa*exp(i*sigma*z))*exp(i*delta*z))));
a2z=(((as*rs*exp(-i*sigma*z))+((aa*ra*exp(i*sigma*z))*exp(i*delta*z))));
%power function
A1=a1z^2;
A2=a2z^2;
for (s=1:M)
%matrix transform 2X2
f11=(cos(sigma*z)-i*(delta/sigma)*(sin(sigma*z)));
f12=-i*(kappa2/sigma)*(sin(sigma*z));
f21=-i*(kappa1/sigma)*(sin(sigma*z));
f22=(cos(sigma*z)+i*(delta/sigma)*(sin(sigma*z)));
ff=[f11 f12;f21 f22];
F=ff*F;
42
end
%function of phase, CHECK THIS EQUATION AGAIN!
kappa3=1.5e3;
phase=kappa3*z/pi;
Ampli(r)=F(1,1)+F(1,2);
%amplitude input function
Poweri(r)=(abs(Ampli(r)))^2;
%power input function
Amplo(r)=F(2,1)+F(2,2);
%amplitude output function
Powero(r)=(abs(Amplo(r)))^2;
%power output function
Amplitot(r)=F(1,1)+F(1,2)+F(2,1)+F(2,2);
%amplitude total
function
Poweritot(r)=(abs(Amplitot(r)))^2; %power total function
Poweritot2(r)=(Poweri(r)+Powero(r));
%power total function
end
figure
plot(kappa3*zal/pi,Poweri,'-')
grid on, box on
title('Power propagation input of coupled single mode fiber')
xlabel('Phase')
ylabel('Power (W)')
hold on
plot(kappa3*zal/pi,Powero,':')
grid on, box on
title('Power propagation output of coupled single mode fiber')
xlabel('Phase')
ylabel('Power (W)')
legend('Power input','Power output')
43
APPENDIX B
Code to Calculate Power Transfer in both Linear and Triangular Order NX3 Fiber
Couplers with Constant Coupling Coefficient
kappa1=1.5e3;
kappa2=1.5e3;
kappa=1.5e3;
n1=1.4677;
n2=1.4624;
B1=kappa1*n1;
B2=kappa2*n2;
B=kappa1*n1;
DB=B1-B2;
delta=DB/2;
d=delta/sqrt(kappa1*kappa2);
sigma=sqrt((kappa1*kappa2)+delta^2);
%coupling length range
step=5000;
zal1=1e-8;
zal2=7.5e-6;
zal=[zal1:7.489e-6/step:zal2] ;
M=1000;
%normal mode amplitude CHECK IT AGAIN THE AMPLITUDE!
as=1;
aa=1;
rs=1;
ra=1;
i=sqrt(-1);
for (r=1:step+1);
F=[1 0 0;0 0 0;0 0 0];
z=zal(r);
%amplitude function of linear order NX3 fiber coupler
a1z=(1/2)*((cos((sqrt(2))*kappa*z))-1)*(exp((-i)*B*z));
a2z=(-i/(sqrt(2)))*(sin((sqrt(2))*kappa*z))*(exp((-i)*B*z));
a3z=(1/2)*((cos((sqrt(2))*kappa*z))+1)*(exp((-i)*B*z));
%amplitude function of triangular NX3 fiber coupler
%a1z=(-i)*(1/(sqrt(2)))*(sin((sqrt(2))*kappa*z))*(exp((-i)*B*z));
%a2z=(cos((sqrt(2))*kappa*z))*(exp((-i)*B*z));
%a3z=(-i)*(1/(sqrt(2)))*(sin((sqrt(2))*kappa*z))*(exp((-i)*B*z));
44
%power function
A1=a1z^2;
A2=a2z^2;
A3=a3z^2;
for (s=1:M)
%matrix transform 3x3
f11=(1/2)*((cos((sqrt(2))*kappa*z))+1)*(exp((-i)*B*z));
f12=(-i/(sqrt(2)))*(sin((sqrt(2))*kappa*z))*(exp((-i)*B*z));
f13=(1/2)*((cos((sqrt(2))*kappa*z))-1)*(exp((-i)*B*z));
f21=(-i/(sqrt(2)))*(sin((sqrt(2))*kappa*z))*(exp((-i)*B*z));
f22=(cos((sqrt(2))*kappa*z))*(exp((-i)*B*z));
f23=(-i/(sqrt(2)))*(sin((sqrt(2))*kappa*z))*(exp((-i)*B*z));
f31=(1/2)*((cos((sqrt(2))*kappa*z))-1)*(exp((-i)*B*z));
f32=(-i/(sqrt(2)))*(sin((sqrt(2))*kappa*z))*(exp((-i)*B*z));
f33=(1/2)*((cos((sqrt(2))*kappa*z))+1)*(exp((-i)*B*z));
ff=[f11 f12 f13;f21 f22 f23;f31 f32 f33];
F=ff*F;
end
%function of phase, CHECK THIS EQUATION AGAIN!
kappa3=1.5e3;
phase=kappa3*z/pi;
Ampli1(r)=F(1,1)+F(1,2)+F(1,3);
%amplitude input function
Poweri1(r)=(abs(Ampli1(r)))^2; %power input function
Amplo2(r)=F(2,1)+F(2,2)+F(2,3);
%amplitude output function
Powero2(r)=(abs(Amplo2(r)))^2; %power output function
Amplo3(r)=F(3,1)+F(3,2)+F(3,3);
%amplitude output function
Powero3(r)=(abs(Amplo3(r)))^2; %power output function
Amplitot(r)=F(1,1)+F(1,2)+F(1,3)+F(2,1)+F(2,2)+F(2,3)+F(3,1)+F(3,2)+
F(3,3); %amplitude total function
Poweritot(r)=(abs(Amplitot(r)))^2; %power total function
Poweritot2(r)=(Poweri1(r)+Powero2(r)+Powero3(r));
function
end
figure
%power total
45
plot(kappa3*zal/pi,Poweri1,'k')
grid on, box on
title('Power propagation input of three waveguides single mode
fiber')
xlabel('Phase')
ylabel('Power (W)')
hold on
plot(kappa3*zal/pi,Powero2,'r')
hold on
plot(kappa3*zal/pi,Powero3,'b')
grid on, box on
title('Power propagation output of three waveguides single mode
fiber')
xlabel('Phase')
ylabel('Power (W)')
legend('WG1','WG2','WG3')
46
APPENDIX C
Code to Calculate Power Transfer in NX2 Fiber Couplers with Variation of Coupling
Coefficient
clear all
%coupling length range
step=600;
time=100;
zal1=1e-6;
zal1=1e-8;
zal2=7.5e-6;
zal=[zal1:7.489e-6/step:zal2] ;
M=10;
%normal mode amplitude CHECK IT AGAIN THE AMPLITUDE!
as=1;
aa=1;
rs=1;
ra=1;
AAAA = [];
BBBB = [];
CCCC = [];
kappat=0.1e3:0.01390e3:1.5e3;
for (r=1:step+1)
F=[1 0;0 0];
z=zal(r);
for t=1:time+1
kappa=kappat(t);
n1=1.4677;
n2=1.4624;
B1=kappa*n1;
B2=kappa*n2;
DB=B1-B2;
delta=DB/2;
sigma=sqrt((kappa*kappa)+delta^2);
%amplitude function
a1z=(((as*exp(-i*sigma*z))+((aa*exp(i*sigma*z))*exp(i*delta*z))));
a2z=(((as*rs*exp(-i*sigma*z))+((aa*ra*exp(i*sigma*z))*exp(i*delta*z))));
%power function
A1=a1z^2;
A2=a2z^2;
for (s=1:M)
%matrix transform 2X2
47
f11=(cos(sigma*z)-i*(delta/sigma)*(sin(sigma*z)));
f12=-i*(kappa/sigma)*(sin(sigma*z));
f21=-i*(kappa/sigma)*(sin(sigma*z));
f22=(cos(sigma*z)+i*(delta/sigma)*(sin(sigma*z)));
% f11=(cos(sigma*z)-i*(d/(sqrt(1+d^2)))*(sin(sigma*z)));
% f12=-i*((exp(((sinh(c*d))^1)/2))/(sqrt(1+d^2)))*(sin(sigma*z));
% f21=-i*((exp(((sinh(c*d))^1)/2))/(sqrt(1+d^2)))*(sin(sigma*z));
% f22=(cos(sigma*z)+i*(d/(sqrt(1+d^2)))*(sin(sigma*z)));
ff=[f11 f12;f21 f22];
F=ff*F;
end
%function of phase, CHECK THIS EQUATION AGAIN!
phase=kappa*z/pi;
phaset(t) = phase;
Ampli(r)=F(1,1)+F(1,2);
%amplitude input function
Poweri(r)=(abs(Ampli(r)))^2;
%power input function
Amplo(r)=F(2,1)+F(2,2);
%amplitude output function
Powero(r)=(abs(Amplo(r)))^2;
%power output function
Amplitot(r)=F(1,1)+F(1,2)+F(2,1)+F(2,2);
%amplitude total
function
Poweritot(r)=(abs(Amplitot(r)))^2; %power total function
Poweritot2(r)=(Poweri(r)+Powero(r));
%power total function
AAAA=[AAAA;kappa phase Poweri(r)];
BBBB=[BBBB;kappa phase Powero(r)];
end
end
plot3(AAAA(:,1),AAAA(:,2),AAAA(:,3),'.r')
box on
grid on
title('Power propagation output of coupled single mode fiber')
xlabel('kappa')
ylabel('Phase')
zlabel('Power (W)')
legend('Power input','Power output')
hold on
plot3(BBBB(:,1),BBBB(:,2),BBBB(:,3),'.b')
box on
grid on
title('Power propagation output of coupled single mode fiber')
xlabel('kappa')
ylabel('Phase')
zlabel('Power (W)')
legend('Power input','Power output')
hold on
48
APPENDIX D
Code to Calculate Power Transfer in both Linear and Triangular Order NX3 Fiber
Couplers with Variation of Coupling Coefficient
clear all;
clc;
%
%
%
%
%
%
%
%
%
%
n1=1.4677;
n2=1.4624;
B1=kappa1*n1;
B2=kappa2*n2;
DB=B1-B2;
delta=DB/2;
% c=DB/(kappa1-kappa2)
d=delta/sqrt(kappa1*kappa2);
sigma=sqrt((kappa1*kappa2)+delta^2);
%coupling length range
step=300;
time=100;
zal1=1e-8;
zal2=7.5e-6;
zal=[zal1:7.489e-6/step:zal2];
M=7;
%normal mode amplitude CHECK IT AGAIN THE AMPLITUDE!
as=1;
aa=1;
rs=1;
ra=1;
i=sqrt(-1);
AAAA = [];
BBBB = [];
CCCC = [];
RRRR = [];
% kappat=0.1e3:0.00028e3:1.5e3;
% kappat=0.1e3:0.00775e3:1.5e3;
% kappat=0.1e3:0.0275e3:1.5e3
kappat=0.1e3:0.01390e3:1.5e3;
phaset = zeros;
for (r=1:step+1)
F=[1 0 0;0 0 0;0 0 0];
z=zal(r);
%kappa=1.5e3;
for t=1:time+1
kappa=kappat(t);
49
n=1.4677;
B=kappa*n;
RRRR=[RRRR;kappa B];
%amplitude function
a1z=(1/2)*((cos((sqrt(2))*kappa*z))-1)*(exp((-i)*B*z));
a2z=(-i/(sqrt(2)))*(sin((sqrt(2))*kappa*z))*(exp((-i)*B*z));
a3z=(1/2)*((cos((sqrt(2))*kappa*z))+1)*(exp((-i)*B*z));
%amplitude function of triangular NX3 fiber coupler
%a1z=(-i)*(1/(sqrt(2)))*(sin((sqrt(2))*kappa*z))*(exp((i)*B*z));
%a2z=(cos((sqrt(2))*kappa*z))*(exp((-i)*B*z));
%a3z=(-i)*(1/(sqrt(2)))*(sin((sqrt(2))*kappa*z))*(exp((i)*B*z));
%power function
A1=a1z^2;
A2=a2z^2;
A3=a3z^2;
for (s=1:M)
%matrix transform 2X2
f11=(1/2)*((cos((sqrt(2))*kappa*z))+1)*(exp((-i)*B*z));
f12=(-i/(sqrt(2)))*(sin((sqrt(2))*kappa*z))*(exp((i)*B*z));
f13=(1/2)*((cos((sqrt(2))*kappa*z))-1)*(exp((-i)*B*z));
f21=(-i/(sqrt(2)))*(sin((sqrt(2))*kappa*z))*(exp((i)*B*z));
f22=(cos((sqrt(2))*kappa*z))*(exp((-i)*B*z));
f23=(-i/(sqrt(2)))*(sin((sqrt(2))*kappa*z))*(exp((i)*B*z));
f31=(1/2)*((cos((sqrt(2))*kappa*z))-1)*(exp((-i)*B*z));
f32=(-i/(sqrt(2)))*(sin((sqrt(2))*kappa*z))*(exp((i)*B*z));
f33=(1/2)*((cos((sqrt(2))*kappa*z))+1)*(exp((-i)*B*z));
ff=[f11 f12 f13;f21 f22 f23;f31 f32 f33];
F=ff*F;
end
%function of phase, CHECK THIS EQUATION AGAIN!
phase=kappa*z/pi;
phaset(t) = phase;
Ampli(r)=F(1,1)+F(1,2)+F(1,3);
%amplitude input
function
Poweri(r)=(abs(Ampli(r)))^2;
Amplo1(r)=F(2,1)+F(2,2)+F(2,3);
%power input function
%amplitude output
function
Powero1(r)=(abs(Amplo1(r)))^2;
Amplo2(r)=F(3,1)+F(3,2)+F(3,3);
%power output function
%amplitude output
function
Powero2(r)=(abs(Amplo2(r)))^2;
%power output function
50
Amplitot(r)=F(1,1)+F(1,2)+F(1,3)+F(2,1)+F(2,2)+F(2,3);
%amplitude total function
Poweritot(r)=(abs(Amplitot(r)))^2; %power total function
Poweritot2(r)=(Poweri(r)+Powero1(r)+Powero2(r));
total function
%power
AAAA=[AAAA;kappa phase Poweri(r)];
BBBB=[BBBB;kappa phase Powero1(r)];
CCCC=[CCCC;kappa phase Powero2(r)];
end
end
% figure(1)
plot3(AAAA(:,1),AAAA(:,2),AAAA(:,3),'r')
grid on
title('Power propagation output of coupled single mode fiber')
xlabel('kappa')
ylabel('Phase')
zlabel('Power (W)')
legend('WG1','WG2','WG3')
hold on
% figure(2)
plot3(BBBB(:,1),BBBB(:,2),BBBB(:,3),'.b')
box on
grid on
title('Power propagation output of coupled single mode fiber')
xlabel('kappa')
ylabel('Phase')
zlabel('Power (W)')
legend('WG1','WG2','WG3')
hold on
% figure(3)
plot3(CCCC(:,1),CCCC(:,2),CCCC(:,3),'.g')
box on
title('Power propagation output of coupled single mode fiber')
xlabel('kappa')
ylabel('Phase')
zlabel('Power (W)')
legend('WG1','WG2','WG3')
grid on,
51
APPENDIX E
Code to Calculate Minimum Coupling Length as Function of Coupling Coefficient
step=5000;
kappa=0.1e3:1.4e3/step:1.5e3; %coupling coefficient
PPPP = [];
QQQQ = [];
RRRR = [];
for (r=1:step+1);
k=kappa (r);
zNX2=pi/(2*k);
zNX3100=pi/((sqrt(2))*k);
zNX3010=pi/(2*(sqrt(2))*k);
PPPP=[PPPP;k zNX2];
QQQQ=[QQQQ;k zNX3100];
%Coupling length of NX2
%Coupling length of linear order
NX3
RRRR=[RRRR;k zNX3010];
order NX3
%Coupling length of triangular
PhaseNX2=kappa*(zNX2)/pi;
%Phase of NX2
PhaseNX3100=kappa*(zNX3100)/pi; %Phase of linear order NX3
PhaseNX3010=kappa*(zNX3010)/pi; %Phase of triangular NX3
end
figure (1)
plot(PPPP(:,2),PPPP(:,1 ),'k')
hold on
plot(QQQQ(:,2),QQQQ(:,1 ),'b')
hold on
plot(RRRR(:,2),RRRR(:,1 ),'r')
grid on
xlabel('coupling length')
ylabel('coupling coefficient')
figure (2)
plot(PhaseNX2,kappa,'k')
hold on
plot(PhaseNX3100,kappa,'b')
hold on
plot(PhaseNX3010,kappa,'r')
grid on
xlabel('phase')
ylabel('coupling coefficient')