OPTICAL POWER SPLITTER BASED ON MULTIMODE
INTERFERENCE (MMI)
ABDULAZIZ MOHAMMED ALI AL-HETAR
A project report submitted in partial fulfillment of
the requirements for the award of the degree of
Master of Engineering
(Electrical-Electronics and Telecommunications)
Faculty of Electrical Engineering
Universiti Teknologi Malaysia
MAY 2007
iv
DEDICATION
Especially dedicated to my beloved parents, my wife, my daughters
Aya and Doa’a, and all my family members for their support.
v
ACKNOWLEDGEMENT
In the name of ALLAH, the Most Beneficent, the Merciful.
Foremost, all praise to ALLAH for the incredible gift endowed upon
me and for give me the health and strength to pursue with this study
and enable me to prepare this thesis.
I deeply appreciate the inspirations and guideline that I have
received from my supervisor Assoc. Prof. Dr. Norazan bin Mohd
Kassim for his personal kindness, skill, patience, valuable advice and
encouragement throughout this project.
I would like take this opportunity to thanks Dr. Haniff for his
helpful and fruitful discussion during this work.
I would also like to thank my parents and my wife for their
moral support on me.
In the, again all praise for the Almighty ALLAH SWT for his
blessings and uncountable awards toward me in spite of all of my
weaknesses and serious faults. Hundreds of millions Darood on our
Holy Prophet Mohamed (Peace upon Him).
vi
ABSTRACT
The challenge in optical access networking is to bring optical
fibers as close to the end-users as possible. One way to realize this
economically is to employ the passive devices. Therefore, it is
necessary to use plenty of passive optical power splitters in the
central office for distribution purposes. Some of the important
characteristics of such splitter are low loss, compactness, and a low
price. To achieve this description fully, Multimode Interference (MMI)
optical power splitters based on self-imaging meet all these
requirements,
with
considering
the
ability
to
optimize
the
performance of optoelectronic device over range of operational
parameters. These parameters are important to reduce device length
and provide increase component density. Mathematical model was
used in MATLAB software and verified it by BPM-CAD to design and
optimize 1X2, 1X4, 1X8, 1X16 and 1X32 power splitter based on
MMI.
vii
ABSTRAK
Cabaran rangkaian laluan optik adalah di dalam pendekatan
gentian optik dengan pengguna. Satu daripada cara ekonomi untuk
merealisasikannya adalah dengan menggunakan peranti pasif. Oleh
itu, adalah perlu untuk menggunakan banyak pemisah punca kuasa
optik di dalam pejabat utama bagi tujuan penyebaran Sebahagian
daripada
ciri-ciri
penting
pemisah
tersebut
adalah
sedikit
kehilangan, padat dan harga yang murah. Untuk memenuhi apa
yang dinyatakan di atas, pemisah punca kuasa optic gangguan
pelbagai mod (MMI) berasaskan gambaran diri perlu memenuhi
semua
keperluan,
dengan
mengambil
kira
pengoptimuman
keupayaan peranti optoelektronik alam julat parameter operasi.
Parameter ini adalah penting untuk mengurangkan panjang peranti
dan menyediakan peningkatan ketumpatan komponen. Suatu model
matematik yang menggunakan perisian MATLAB dan telah disahkan
oleh BPM-CAD untuk mencipta dan mengoptimumkan penggunaan
IX2, IX4, IX8, IX16 dan IX32 pemisah punca kuasa berdasarkan MMI.
LIST OF CONTENTS
CHAPTER
1
2
TITLE
PAGE
TITLE
ii
DECLARATION
iii
DEDICATION
iv
ACKNOWLEDGMENT
v
ABSTRACT
vi
ABSTRAK
vii
LIST OF CONTENTS
viii
LIST OF TABLES
xi
LIST OF FIGURES
xii
LIST OF SYMBOLS
xvi
LIST OF ABBREVIATIONS
xviii
INTRODUCTION
1.1
Background
1
1.2
Problem Statement
3
1.3
Objective Of The Project
4
1.4
Scope Of The Project
4
1.5
Methodology Of The Project
4
OPTICAL WAVEGUIDE
ix
2.1
2.2
Introduction
6
Optical Waveguide
2.2.1
Introduction
2.2.2
7
Electromagnetic Analysis Of The
Planar Waveguide
The Longitudinal Wavevector: β
2.2.3
3
7
8
11
2.2.4
Eigenvalues For The Slab Waveguide
12
2.2.5
The Symmetric Waveguide
13
2.2.6
Effective Index
15
2.3
Materials
16
2.4
S-Bend
17
MULTIMODE INTERFERENCE (MMI) PRINCIPLE
3.1
Introduction
19
3.2
The Self-Imaging Principle
20
3.3
Multimode Waveguide
20
3.3.1 Number Of Guided Modes In
A Waveguide
3.4
4
21
3.3.2 Propagation Constant
22
3.3.3 FIELD Distribution Inside Waveguide
24
3.3.3.1 General Interference
27
3.3.3.2
29
Symmetric Interference
Cascaded Multimode Interference
29
MATHEMATICAL MODEL
4.1
Introduction
31
4.2
Model Propagation Analysis
32
4.2.1 Guided Modes
32
x
4.3
5
6
4.2.2 Fields in the MMI Waveguide
36
Summary
41
SIMULATION BY BPM-CAD
5.1 Introduction
43
5.2 Waveguide Coupler
43
5.3 Guide Mode
45
5.4 Simulation
48
5.5 Analysis
57
5.6 Cascade MMI
60
CONCLUSIONS AND FUTURE WORK
6.1 Conclusions
62
6.2 Future Work
64
REFERENCES
APPENDICES A-C
65
68 - 77
xi
LIST OF TABLES
TABLE NO.
2.1
TITLE
PAGE
Classification of optical waveguides according
to the number of dimensions
2.2
Refractive indices for common materials
4.1
The maximum waveguide width before another
7
17
mode becomes supported for a waveguide with
nf = 3.45189744 and nc = 3.36755329.
4.2
Results of mathematical model for 1x2, 1x4, 1x8,
1x16 and 1x32 power splitter based on MMI
5.1
41
Gap between two waveguides vs. amount of
coupling power
5.2
33
45
Comparison among (Soldano and Pennings, 1995)’s
formula, our mathematical model and the optimum
length by BPM-CAD
58
xii
LIST OF FIGURES
FIGURE NO.
TITLE
1.1
Distribution the fiber to the home
2.1
The planar slab waveguide consists of three
PAGE
2
materials; the index of refractive (nf) is larger
than the surrounding substrate (ns) and
cover (nc) indices.
2.2
8
Transverse electric (TE) and Transverse
magnetic(TM) configuration A cross indicates
the field entering the page
2.3
β and k are the longitudinal and transverse
component, respectively, of the wavevector K
2.4
2.5
2.6
9
11
This ray and wave picture shows the
electromagnetic field as a function of β
12
The symmetric waveguide
14
A buried dielectric waveguide can be
decomposed into two spatially orthogonal
waveguide: a horizontal and a vertical slab
waveguide
15
xiii
2.7
Geometry of the s-bend waveguide
3.1
Two-dimensional representation of a step-index
18
multimode waveguide; (effective) index lateral
profile (left), and top view of ridge boundaries
and coordinate system (right)
3.2
22
Example of amplitude-normalized lateral field
profiles Ev ( x ) . Corresponding to the first 7 guided
modes in a step-Index multimode waveguide
3.3
26
Multimode waveguide showing the input field
E ( x,0 ) a mirrored single image at ( 3Lπ ) , a direct
single image at 2 ( 3Lπ ) . And tow-fold images at
1
3
( 3Lπ ) and ( 3Lπ )
2
2
28
3.4
Structure of the cascaded 1X 2 splitter
30
4.1
Electrical field profiles ( E y ) for symmetric TE modes for
a given structure of WM =14.3 μ m , λ =1.55 μ m ,
n f =3.4519, nc = 3.3675. E y has been normalized
to1 and x-axis in m
4.2
35
Electrical field profiles ( E y ) for symmetric TE modes
together for a given structure of
WM =14.3 μ m , λ =1.55 μ m , n f =3.4518, nc = 3.3675.
4.3
E y has been normalized to1 and x-axis in m
36
General shape of 1xN power splitter based MMI
37
xiv
4.4
Normalize field intensity at the beginning and ending
of1x2MMI coupler
4.5
Normalize field intensity at the beginning and ending
of1x4MMI coupler
4.6
41
The index profile of two slab waveguides separated
by d
5.2
40
Normalize field intensity at the beginning and ending
of1x32 MMI coupler
5.1
39
Normalize field intensity at the beginning and ending
of1x16 MMI coupler
4.8
38
Normalize field intensity at the beginning and ending
of1x8MMI coupler
4.7
37
44
Electrical field profiles ( E y ) for symmetric TE modes for
a given structure of WM =14.3 μ m , λ =1.55 μ m ,
n f =3.4519, nc = 3.3675. E y has been normalized to 1
and x-axis in μ m
5.3
47
E y E field profiles for symmetric TE modes together for
a given structure of WM =14.3 μ m , λ =1.55 μ m ,
n f =3.4518, nc = 3.3675. E y has been normalized to 1
and x-axis in m
5.4
47
Schematic layouts of MMI splitters:
(a) 1×2, (b) 1×4, (c) 1×8, (d) 1×16, (e) 1 x 32
51
xv
5.5
BPM-CAD analyses of MMI optical splitters:
(a) 1×2, (b) 1×4, (c) 1×8, (d) 1×16, (e) 1×32
5.6
53
Excess loss and imbalance versus length of MMI
section with λ =1.55 μ m for (a) 1X2 at width = 14.3 μ m ,
(b) 1X4 at width= 28.61 μ m , (c) 1X8 at width=49.05 μ m ,
(d) 1X16 at width=100.145 μ m and (e) 1X32 at width=
202.335 μ m
57
5.7
Structure of the cascaded 1X 2 splitter
61
5.8
BPM-CAD analyses of cascade MMI
61
xvi
LIST OF SYMBOLS
SYMBOL
DESCRIPTION
β
-
Propagation coefficient
εr
-
Relative permittivity
εo
-
Free space permittivity
μo
λo
c
-
Free space permeability
-
Optical wavelength in free space
γ
-
Speed of light in free space
-
Attenuation coefficient
K
-
Transverse wavevector
E
-
Electric field
H
-
Magnetic field
P
-
Power carrier in waveguide
nf
-
Refractive index for the core
nc
-
Refractive index for the clad
neff
-
Effective refractive index
ko
-
Vacuum wavevector
Rc
-
Curvature radius
O
-
S-bend offset
ω
-
Angular frequency
xvii
he
-
Effective width of the multimode region
h
-
Physical width of the multimode region
Lπ
-
The beat length
xviii
LIST OF ABBREVIATIONS
1D
-
One-Dimensional
2D
-
Two-Dimensional
3D
-
Three-Dimensional
AlGaAs
-
Aluminium Gallium Arsenide
BPM
-
Beam Propagation Method
EIM
-
Effective index method
FTTH
-
Fiber to the home
GaAs
-
Gallium arsenide
InGaAsP
-
Indium Gallium Arsenide Phosphide
InP
-
Indium Phosphide
MMI
-
Multi-mode interference
PDS
-
Passive double star
PICs
-
Photonic integrated circuits
PL
-
Propagation Loss
SiO2
-
Silica- Silicon Dioxide
TE
-
Transverse Electric
TM
-
Transverse Magnetic
WDM
-
Wavelength divisions multiplex
xix
LIST OF APPENDICES
APPENDIX
A
TITLE
MATLAB code solves for the longitudinal
propagation constants, β, of the guided modes
of a waveguide using the Newton-Raphson
method
B
MATLAB code Describe the electrical field
intensity for each mode
C
68
73
MATLAB code Describe the electrical field
intensity for the input and the each output
ports after a certain length from the input port
77
CHAPTER 1
INTRODUCTION
1.1
Background
The world we live in today requires communication on many
levels, the most basic of which is human to human. Indeed invention
of the telegraph and the telephone were the answers to this need that
changed our world and resulted in the creation of widespread
telecommunication network. A similar revolution has occurred in
electronics computing devices that have resulted in the need for
communication from computer to computer. An ever increasing
demand for the acquisition, processing and sharing of information
concerning the world and its future course, today, these needs are
echoed in the demand for higher bandwidth in telecommunication
network and similarly in computing as a demand for higher
processing speeds.
Today’s evolving telecommunication networks are increasingly
focusing on flexibility and reconfigurability, which requires enhanced
functionality of photonic integrated circuits (PICs) for optical
communications. In addition, modem wavelength demultiplexing
(WDM) systems will require signal routing and coupling devices to
have large optical bandwidth and to be polarization insensitive. Also
small device dimensions and improved fabrication tolerances are
2
required in order to reduce process costs and contribute to largescale PIC production.
The challenge in optical access networking is to bring optical
fibers as close to the end-users as possible that called Fiber to the
home (FTTH). One way to realize this economically is to employ the
passive double star (PDS) topology [I]. Therefore, it is necessary to use
plenty of passive optical power splitters in the central office for
distribution purposes as depicted in Figure 1.1. Some of the
important characteristics of such splitter are low loss, compactness,
compatibility with optical single-mode fibers, uniform distribution of
the output power on the output waveguides and a low price.
Figure 1.1:
Distribution the fiber to the home.
A power splitter 1×2 is usually a symmetric element which
equally divides power from a straight waveguide between two output
waveguides. The simplest version of a power splitter is the Y-branch,
which is easy to design and relatively insensitive to fabrication
tolerances. Nevertheless, the curvature radii of the two branches, as
3
well as the junction, must be carefully designed in order to avoid
power losses. Also, if the two branches are separated by tilted
straight waveguides, the tilt angle must be small, typically a few
degrees [2].
A different version of a power splitter is the multi-mode
interference element (MMI. This name comes from the multi-modal
character of the wide waveguide region where the power split takes
place. The advantage of this design is the short length of the MMI
compared to that of the Y-branch. Although the dimensions of the
MMI are not critical, allowing wide tolerances, this element must be
designed for a particular wavelength. The two power splitters which
have been described are symmetric, and thus 50% of the input power
was carried by each output waveguide. Nevertheless, asymmetric
splitters can also be designed for specific purposes. In addition, it is
possible to fabricate splitters with N output waveguides, and in that
case the element is called a 1 × N splitter.
MMI devices are important components for photonic and
optoelectronic integrated circuits due to their simple structure, low
loss, and large optical bandwidth. These structures provide power
splitting or combining.
1.2 Problem Statements
Power splitting is a basic function of the integrated optics. It
plays a central role in passive optical distribution networks.
Furthermore, the device should meet practical requirement such as
type of material, small size and wavelength dependency.
4
1.3 Objective of the Project:
The main objective of this project is to determine the
specification of the optical power splitter based on MMI coupler to
achieve acceptable power splitter design.
1.4
Scope of the project
In order to achieve the objective of this project, the following scope of
work has been identified which comprises of:
-
A
mathematical
model
using
numerical
analysis
was
formulated to calculate the optimum width and length of the
1x2, 1x4, 1x8, 1x16, and 1x32 power splitter based on MMI
and plot electric field intensity for input and each output
signal.
-
Simulate of the 1x2, 1x4, 1x8, 1x16, and 1x32 power splitter
based on MMI by BPM-CAD.
1.4
Methodology of the Project
To carry out this project, the following methodology is designed:
•
Build up the our mathematical model in Matlab software
to
design 1X2, 1X4, 1X8, 1X16, and 1X32 power splitter based on
multimode interference(MMI), which includes the following
tasks:
o Get the eingenvalues for propagation coefficient.
5
o Find the electrical field intensity for each mode inside the
waveguide.
o Determine the dimension of the device according to:
-
The numbers of output port.
-
Type of materials.
-
Wavelength.
o Find the output optical power for each port and mention
of the Excess loss and Imbalance.
•
Use the approximate formula (Soldano’s formula) to get
propagation constant and find all parameters as previous point.
o Get the eigenvalues for propagation coefficient.
o Find the electrical field intensity for each mode inside the
waveguide.
o Determine the dimension of the device according to:
-
The numbers of output port.
-
Type of materials.
-
Wavelength.
o Find the output optical power for each port and mention
of the Excess loss and Imbalance.
•
Simulate
all
those
devices
by
BPM-CAD
to
verify
our
mathematical model and compare with approximate formula.
CHAPTER 2
Optical waveguide
2.1
Introduction
This chapter, which is the first chapter in the literature review,
is highlighted in brief the basic concepts of optical waveguide. In this
chapter we will describe the theory of optical waveguides using the
electromagnetic theory of light. We start by describing the basic
geometries found in waveguide structure, planar waveguides. Then,
we introduce the concept of optical mode, and the types of modes
that can be supported by a planar structure. Using the Maxwell’s
equations we will obtain the wave equation for planar waveguides,
and then we will solve it for the simplest case of step-index planar
waveguides, considering TE polarized modes. After that we will get
the formula that calculates the number of modes in the waveguide
and the equations that describe the electrical field intensity at any
point inside the symmetrical waveguide. What is the technique that
converts from two dimensional structures to one dimensional
structure? It is the effective index method (EIM) which will be
explained in this chapter. Finally, the refractive indices for common
materials will be presented and type of materials which be used in
our waveguide. The purpose of this chapter is to understand the
concept of the forming the modes inside the waveguide.
7
2.2 Optical waveguide
2.2.1 Introduction
The basic element in integrated photonic technology is the
optical waveguide. A Waveguide can be defined as an optical
structure that allows the confinement of light within its boundaries
by total internal reflection. A first waveguide classification can be
made by looking at the number of dimensions in which the light is
confined table 2.1.
The table shows the three basic types of
waveguides depending on their number of dimensions for light
confinement: while a planar waveguide (or 1D waveguide) confines
the radiation in one dimension, channel waveguides (or 2D
waveguides) confine the light in two dimensions and Photonics
crystals (or 3D waveguides) confine the light in three dimensions [2].
Table2.1
Classification of optical waveguides according to the
number of dimensions
Dimensions of light
confinement
Classification of optical waveguides
1D
Planar waveguides
2D
Channel waveguides
Optical fibers
3D
Photonics crystals
In this project, I will concentrate of the planar waveguide (or slab
8
waveguide) in the figure 2.1; it is an idealization of real waveguide [2].
However, the one-dimensional analysis is directly applicable to many
real problems, and the techniques form the foundation for further
understanding. We will begin by solving the wave equation using
X
nc
h
Z
nf
Y
ns
Figure 2.1
The planar slab waveguide consists of three materials;
the index of refractive (nf) is larger than the surrounding
substrate (ns) and cover (nc) indices.
the boundary condition for the slab waveguide structure. This will
lead naturally to the concept of modes. A waveguide structure can
support only a discrete number of guided.
2.2.2 ELECROMAGNATIC ANALYSIS OF THE PLANAR
WAVEGUIDE
The electromagnetic theory of light applied to a planar
waveguide, because it is the simplest structure to be analyzed from
the point of view of its mathematical description, and from it the
general features related to more complex waveguide geometry can be
understood.
Starting
from
Maxwell’s
equations
and
from
the
constitutive relations, we will obtain the wave equations for TE and
TM propagation that govern light behavior in planar waveguides.
These wave equations will be solved for the general case of
asymmetric step-index planar waveguides. We must consider two
9
possible electric field polarizations, transverse electric or transverse
magnetic [3].
Transverse Electric
Transverse Magnetic
X
Z
Figure2.2
Transverse electric (TE) and Transverse
magnetic(TM) configuration A cross indicates the
field entering the page.
In the TE case, the field is polarized along y axis (into the page) of
Figure2.2. To find the allowed modes of the waveguide, we must find
solve the wave equation in each dielectric region, and then use the
boundary conditions to connect these solutions [3]. The field electric
take the form:
E y ( x, z ) = E y ( x )e −
jβ z
(2.1)
β is a propagation coefficient along the z direction, from Maxwell’s
equations we obtain an equation involving only the Ey component of
the electric field :
∂ 2E
∂x
2
y
+
(k
2
o
n 2i − β
2
)E
y
= 0
(2.2)
The choice of ni depends on the position x. For x >0, we would use nc ,
while 0>x>-h, we would use n f , etc. The general solution to equation
2.2 will depend on the relative magnitude of β with respect to a
vacuum wavevector of magnitude ko , where | ko |= ωo /c. Consider
10
the case where β > koni , the transverse wave equation 2.2 will have a
general solution with a real exponential form:
E y ( x ) = Eoe
± β 2 −k 2o n 2i x
for β > ko ni
(2.3)
Where Eo is the field amplitude at x = 0. In the case where β < k o n i ,
the solution has an oscillatory form:
Ey ( x) = Eoe
± k2on2i −β 2 x
forβ < koni
(2.4)
So depending on the value of β, the solution can be either oscillatory
or exponentially decaying. If β >
coefficient,
γ
koni
we define an attenuation
as:
γ =
β 2 − k 2on 2i
(2.5)
Then we define a transverse wavevector, K, as:
K =
k 2o n 2i − β 2
(2.6)
So E y ( x ) = Eoe ± jK x . Using equation (2.6), we see that β and k can
be geometrically related to the total wavevector K, k= ko ni , in the
guiding film.
K2 = k2 + β 2
X
k
Z
β
Figure 2.3
β and k are the longitudinal and transverse
component, respectively, of the wavevector K.
11
β and k are called the longitudinal and transverse wavevectors,
respectively inside the guiding film.
2.2.3 THE LONGITUDINAL WAVEVECTOR: β
The longitudinal wavevector β is used to identify individual
modes. β is defined as the Eigen value of the mode. Figure 2.4 plots
the transverse electric field distribution in a slab waveguide for
various values of β [4]. A guided wave must satisfy the condition that
kons < β < kon f
(2.7)
Where it is assumed that nc ≤ ns . This is a universal condition for any
dielectric waveguide, regardless of geometry.
nc
nf
ns
β
ko nc
Figure 2.4
ko ns
ko n f
This ray and wave picture shows the electromagnetic
field as a function of β
12
2.2.4 EINGEN VALUES FOR THE SLAB WAVEGUIDE
To find the values of β that lead to allowed solutions to the
wave equation. By solving the differential equations (2.2)–(2.3), the
electric field in the cover, film and substrate regions can be expressed
as:
E
y
= A e −γcx
x>0
⎡
⎤
γ
Ey = A ⎢cos ( k f x ) − c sin ( k f x ) ⎥
kf
⎢⎣
⎥⎦
-h < x< 0
⎡
⎤
γ
Ey = A⎢cos( k f h) + c sin( k f h)⎥ eγ s( x+h)
kf
⎣⎢
⎦⎥
(2.8)
x < -h
Where A is the amplitude at the x = 0 interface. Equation (2.8)
describes the amplitude of the electric field in all regions waveguide.
The negative values of x must be used in the guiding and substrate
layer-otherwise the formula will give nonsensical values. This
equation is very handy for plotting out the mode profiles of guided
modes. The propagation and decay constant, γ c , γ s , and k f all depend
on β which still undefined.
γ
=
β
2
c
− ko2nc2
γ
=
β
2
s
− ko2ns2
β =
2
ko n
2
f
− k
(2.9)
2
f
The is a boundary condition ∂E y / ∂x at x = -h , gives an equation for
β.
13
∂Ey
∂x
|x=−h = A⎡⎣k f sin ( k f h) − γ c cos ( k f h) ⎤⎦
(Film term
(2.10)
⎡
⎤
γ
= A ⎢cos ( k f h ) + c sin ( k f h ) ⎥ γ s
kf
⎢⎣
⎥⎦
(Substrate term)
Divide both side of equation by cos ( k f h ) to get the Eigen value
equation for β .
ta n (h k
f
γ
)=
k
c
+ γ
⎡
γ cγ s
⎢1 −
kf2
⎢⎣
f
(2.11)
s
⎤
⎥
⎥⎦
This is a transcendental equation that must be solved numerically or
graphically. All terms depend on the value of β . It is called the
characteristic equation for the TE modes of slab waveguide. Solution
this equation will yield the Eigen values, βTE that correspond to
allowed TE modes in the waveguide. To complete the solution, the
coefficient, A, should be related to a physical parameter. In practice,
A is related to the power carried in the waveguide [5].
Pz =
2
⎛ β ⎞ ∞
1 ∞
E
H
dx
E
=
⎜
⎟
y
x
y
∫−∞ dx
2 ∫−∞
⎝ 2ωμ o ⎠
(2.12)
So the amplitude of A is the normalize each mode according to
equation (2.12).
A = 1/
⎛
β
⎜
⎝ 2ω μ
o
⎞
⎟
⎠
∫
∞
−∞
2
E
y
dx
(2.13)
2.2.5 THE SYMMETRIC WAVEGUIDE
A symmetric waveguide has a guiding film with index n f and
thickness h is surrounded on both sides by an index ns .It convenient
14
to place the coordinate system in the middle of this waveguide since
the field will reflect the symmetry of the structure.
X
ns
nf
h
Z
ns
Figure 2.5
The symmetric waveguide
The general field description of TE mode within the symmetric
structure is [4]:
E
y =
A e
Ey = A
− γ
(x −
h / 2
co s ( k f x )
)
for x ≥ h/2
co s ( k f h / 2 )
sin ( k f x )
o rA
sin ( k f h / 2 )
E y = ± A e γ (x+ h / 2 )
for –h/2 < x< h/2 (2.14)
for x ≤ - h/2
The characteristic eigenvalue equation for TE modes in A symmetric
waveguide is [5]:
ta n ( k f h / 2 ) =
γ
for even (cos) modes
kf
(2.15)
=
γ
−
k
for odd (sin) modes
f
The characteristic equation for TM modes is
2
⎛n ⎞ γ
tan ( k f h / 2 ) = ⎜ f ⎟
⎝ ns ⎠ k f
for even (cos) modes
(2.16)
⎛ n
= ⎜ s
⎜ n
⎝ f
⎞
⎟⎟
⎠
2
k
γ
f
for odd (sin) modes
15
That is transcendental equation for propagation constant β , for
symmetric modes. All the eingenvalues derived can be solved either a
numerical or graphical methods.
2.2.6
EFFECTIVE INDEX METHOD
The effective index method is developed [6]. It converts a two-
dimensional structure into one dimensional structure. Figure 2.6
shows the buried rectangular waveguide and how to convert it from
two to one dimension.
Y
X
n3
a
n5
n1
n
3
Y
n4
n
n
n2
1
+
n
n4
5
neff
2
Figure 2.6 A buried dielectric waveguide can be decomposed into
two spatially orthogonal waveguide: a horizontal and a
vertical slab waveguide
The thin one dimensional slab waveguide can be analyzed in terms of
TE or TM modes to find the allowed value of
β for the wavelength
and mode of interest. Once β is found , the effective index of the slab
is determined through the expression
n
e ff
=
β
k
(2.17)
o
Where ko is the vacuum wavevector of the light being guided. After
16
this effective index is determined, we return to the original structure
and stretch it along thick axis (in this case vertical ), forming a slab
waveguide in x direction. The modes for this waveguide can now be
found, only instead of using the original value of the index for the
guiding film, the effective index found in the first step must be used.
The value of
β found from the last step is the true value for the
mode. The number of guided modes is limited by relation,
ko ns < ko neff < ko n f and increases as the core film thickness increase, as
the wavelength decreases and as the cor-cladding refractive index
difference [4] from the relation and equation (2.17).
2.3 MATERIAL
The choice of the right material is very important since the
properties of these materials have major influence on the dimensions,
performance, and losses. In integrated optics, for optical device the
following properties of the materials are very important.
-
Low optical loss.
-
Easy to handle (deposition, etching, etc.).
Table 2.2 shows some materials with own refractive indices and
concentrated quantities of the doping elements [5]. In our model,
ALGaAs is used in the core and cladding of the waveguide, sure with
difference refractive indices. The refractive index should be easily
tuned. So we choose the refractive index for the core nf = 3.45189744
and clad nc = 3.36755329. The waveguide that has these refractive
indices is called strong guiding waveguide [16].
17
Table2.2 Refractive indices for common materials
Material
Refractive index
Comments
GaAs
3.48043422
Pure GaAs
AlGaAs
3.44339260
Doping by 6.5% of Al
AlGaAs
3.45189744
Doping by 5 % of Al
AlGaAs
3.39552660
Doping by 15 % of Al
AlGaAs
3.36755329
Doping by 20 % of Al
(Sio2)
1.444618
Pure Silica
Silica with
6.3% Geo2
1.454052
Doping by 6.3 % of Geo2
Silicon
3.5
InGaAsP
3.3989
InP
3.1645
Aluminosilicate
(sol-gel)
1.4846
2.4 S-BEND WAVEGUIDE
Bends in dielectric waveguide are an important building
block for nearly all type of photonic integrated circuit (PICs)[17]. Bend
waveguides are the key elements of PICs device that perform guiding,
coupling, switching, multiplexing, demultiplexing, and splitting of
optical signal. The bend waveguides are used for an optical path
direction changes such as in practical implementation of directional
coupler based devices.
The bending section of directional coupler should be slowly
converging or diverging curve to minimize the radiation losses (Kim &
Kim, 1998). The slow convergence bending sections result in a longer
effective coupling length for the device. As the curvature radius Rc
18
decreases, an optical path direction is changed at a shorter
propagation distance and the smaller device length of the bend
waveguide. On the contrary, optical bending loss increase as the
curvature radius Rc decreases.
Since bending the waveguides will cause bending loss, the
bending radius should be optimized to minimize the bending loss. As
shown in Figure 2.7 the s-bend can physically be defined by two
parameters that are the bend radius, Rc and the offset, O
Rc
O/2
L
Figure 2.7:
Geometry of the s-bend waveguide
The offset is defined as the displacement of the channel in the upper
direction. These two parameters determine the length of the device.
The device length, L is related to the radius of curvature and offset
[18].
1
2
⎛ 2 ⎛
O ⎞ ⎞2
L = 2 ⎜ Rc − ⎜ Rc − ⎟ ⎟
⎜
2 ⎠ ⎠⎟
⎝
⎝
(2.18)
CHAPTER 3
MULTIMODE INTERFERENCE (MMI) PRINCIPLE
3.1
Introduction
In line with the growth of high throughput optical fiber
networks,
it
is
becoming
increasingly
important
to
provide
interconnects with large optical bandwidth at the network nodes. For
such applications, the devices are required to be suitable for photonic
integration which leads to reduction of packaging cost, volume and
insertion loss for cascading optical devices.
Power Splitting is a basic function of the integrated optics; such
devices play a central role in passive optical distribution networks, in
complex photonic integrated circuits, as well as in advanced active
optical components. Various solutions have been proposed and
realized to split or to combine optical signals. Especially, in the last
few years, multimode interference (MMI) couplers [7] have attracted
considerable interest due to advantages characteristics such as
compactness, relaxed fabrication tolerance, and large bandwidth, as
well as polarization insensitivity when strongly guided structure are
used [8]. Meanwhile, they have found application as splitters and
combiners [9], power splitter with arbitrary splitting ratio [10] and
mode converter [11].
20
In this chapter, an MMI principle is described in detail. Which
include A multimode waveguide, which is the core structure of MMI,
is introduced, followed by the self-imaging effect that lead to the
working principle of MMI.
The purpose of this chapter is to
understand the working principle of MMI based optical devices.
3.2
THE SELF-IMAGING PRINCIBLE
BRYNGDAHL first suggested the use of light pipes to form
multiple self-images of symmetric objects [7], Ulrich extended the
concept to the replication of images of random objects in multimode
waveguides and to the possibility of fiber interferometer based on
analogous phenomena [13]. The principle of self-imaging can be
stated [12] as follows: Self-imaging is a property of multimode
waveguides by which an input field profile is reproduced in single or
multiple images at periodic intervals along the propagation direction
of the guide.
3.3
MULTIMODE WAVEGUIDE
As discussed in chapter 2, the physical parameters of the
waveguide determine the number of modes that propagate inside the
waveguide. In order to launch light into and recover light from this
multimode waveguide, a number of access waveguides (usually single
moded) are placed at its beginning and at its end [12]. Such devices
are generally referred to as N x M MMI couplers, where N and M are
the number of input and output access waveguides, respectively.
The MMI device consists of a large multimode section with an
21
input and output access waveguide. The constructive interference
between the waveguide modes causes a ‘self-imaging’ phenomenon
which is the basic principle of operation of MMI-based devices. Since
the excited modes in the MMI section propagate with different phase
velocities, they interfere with each other to form one or more
interference pattern, the so-called ‘multiple images’ which are
dependent on the position along the waveguide section. Self-imaging
may exist in three-dimensional multimode structures, one from many
techniques can be used to represent three dimensions by two
dimensions.
The effective index method (EIM) is a simple and useful
technique to do that and may be used to calculate propagation
constant for simple structure which has been discussed in chapter 2.
3.3.1 NUMBER OF GUIDED MODES IN A WAVEGUIDE
The
waveguide
supports
a
different
number
of
modes
depending on its thickness. If we had adjusted the relative indices
between the layers, we would have found that the mode number
varied which has been discussed in chapter 2. The lowest order mode
has a K vector that is nearly parallel to the Z axis and shown in figure
2.3.
βlowest −order ≈ kn f
(3.1)
The highest order mode will have a wavevector at nearly the critical
angle.
β highest − order ≈ kn f cos θ critical ≈ kn s
(3.2)
The rest of the modes will have eingenvalues for β that fall between
these two extremes. To get an idea of the number of modes in the
waveguide, recall the general eingenvalues equation (2.11) for TE
22
modes.
3.3.2 PROPAGATION CONSTANT
We have discussed
in chapter 2 how to get the propagation
constant of the waveguide by using numbers of equation after that
apply the numerical analytic for these equations to get the values of
β v where v 0,1,2,…,(m-1):
m the number of modes inside the
waveguide and each mode has own β . Moreover, there is an
approximated way which formulated from equations (2.6), (2.9) and
(2.11) [12]. Figure.3.1 shows a step-index multimode waveguide of
width h , ridge (effective) refractive index n f and cladding (effective)
refractive index n s .
X
X
h
Y
Z
n s nf
Figure 3.1: Two-dimensional representation of a step-index
multimode waveguide; (effective) index lateral profile
(left), and top view of ridge boundaries and coordinate
system (right).
23
The waveguide supports m lateral modes , with mode numbers v = 0,
at a free-space wavelength λo . According to the
1, . . ., (m - 1)
equation (2.6) in Chapter 2, the eigenvalue equation for each mode v,
with propagation constant, βv can be written as:
K vy2 + β v2 = k o2 n 2f
(3.3)
With
ko =
2π
(3.4)
λo
The approximated value of wavevector is formulated by (Soldano et.
al. 1995) [12] as:
K vy =
( v + 1 )π
(3.5)
h ev
The effective width hev can be approximated by the effective width he 0
corresponding to the fundamental mode [14].
⎛ λ ⎞⎛ n
hev he = h + ⎜ o ⎟ ⎜ s
⎝ π ⎠ ⎜⎝ n f
Where n f and ns
⎞
⎟⎟
⎠
2σ
(n
2
f
−n
1
2 −2
s
)
(3.6)
refractive index of the MMI region and lateral
region (substrate) respectively;
he
Effective width of the multimode region;
h
Physical width of the multimode region;
λo
Free space wavelength of the input light;
σ =0
TE mode;
σ =1
TM mode.
The propagation constant can be expressed from (3.3) and (3.5).
1
2
2
2
⎛
2
v +1) π 2 ⎞ 2
v +1) π 2
v +1) πλo
(
(
(
βv ⎜ ( kon f ) −
kon f −
⎟ kon f −
2
2
⎜
⎟
h
2
k
n
h
4n f he2
e
o f e
⎝
⎠
(3.7)
24
According to Soldano et. al. (1995), the beat length of the two lowestorder modes:
Lπ =
π
β 0 − β1
4 n f h e2
3λ o
(3.8)
The propagation constants of the modes in the MMI section can be
written from (3.7) and (3.8) as:
β v kon f −
2
(v + 1) π
3 Lπ
(3.9)
Therefore, the propagation constant spacing between the first mode
and the v
th
can be written from (3.9) as
( β0 − βv ) =
v (v + 2)
3 Lπ
(3.10)
3.3.3 FIELD DISTRIBUTION INSIDE WAVEGUID
An input field profile E y ( x,0 ) at Z=0 is then given by the
superposition of all modes of that waveguide:
m−1
E ( x,0) = ∑ Av Ev ( X )
v=0
(3.11)
Where the summation should be understood as including guided as
well as radiative modes. The excitation of an optical field in an output
guide by the field in an input guide is determined by the overlap of
that field. Assuming excitation by an input field E ( x,0 ) at Z=0 that is
normalized such that [8]
∫ E( x)E ( x) dx =1
*
(3.12)
25
The excited modal amplitude, Av are then, in general, determined by
the overlap integral:
Av =
∫ E ( x,0) E ( x ) dx
∫ ( E ( x,0 ) ) dx
v
(3.13)
2
The fundamental mode field profile can be expressed from (2.14) and
[15]:
E v ( x )v = 0 = cos
πx
(3.14)
he
And the v th mode field profile in MMI section is
⎛ ( v + 1)π x ⎞
sin ⎜
⎟
he
⎝
⎠
Ev ( x ) =
⎛ ( v + 1)π ⎞
sin ⎜
⎟
2
⎝
⎠
(3.15)
Now we can rewrite the amplitude of v th mode by using the (3.14) and
(3.15) in (3.13)
Av =
⎛ π
⎜
⎝
∫ s in
⎛
∫ ⎜ s in
⎝
(v
+ 1)x ⎞
π x
dx
⎟ cos
he
he
⎠
⎛ π
⎜
⎝
(v
+ 1)x ⎞ ⎞
⎟ ⎟
he
⎠ ⎠
(3.16)
2
dx
From all previous, the multimode waveguide support the number of
modes that depend of the physical parameter, as shown in figure. 3.2
26
he
h
v =
0
Figure 3.2:
1
2
3
4
5
6 ……..
Example of amplitude-normalized lateral field
profiles Ev ( x ) . Corresponding to the first 7 guided
modes in a step-Index multimode waveguide.
The field profile at a distance Z can then written as a superposition of
all guided mode field distributions
m−1
E ( x, z ) = ∑ Av Ev ( x) exp ⎡⎣ j (ωt − βv z ) ⎤⎦
v=0
(3.17)
Taking the phase of the fundamental mode as a common factor out of
the sum, dropping it and assuming the time dependence exp(jwt)
implicit hereafter [12], the field E ( x, z ) becomes
m−1
E( x, z) = ∑AE
v v ( x) exp⎡
⎣ j ( β0 − βv ) z⎤⎦
(3.18)
v=0
A useful expression for the field at a distance z = L is then found by
(3.10) into (3.18)
⎛ ⎡ v ( v + 2) π
E ( x, L ) = ∑ Av Ev ( x ) exp ⎜ ⎢ j
⎜
3Lπ
v=0
⎝⎣
m−1
The shape of
⎤⎞
L⎥ ⎟
⎟
⎦⎠
(3.19)
E ( x.L ) , and consequently the types of images formed
[12], will be determined by the modal excitation
properties of the mode phase factor.
Av , and the
27
⎛ ⎡ v ( v + 2) π ⎤ ⎞
exp ⎜⎜ ⎢ j
L⎥ ⎟⎟
3
L
π
⎦⎠
⎝⎣
(3.20)
The field E ( x.L ) will be a reproduction (self-imaging) of the input
field E ( x.0 ) .
We
call
General
Interference
to
the
self-imaging
mechanisms which are independent of the modal excitation and
Restricted Interference to those which are obtained by exciting certain
mode alone [12].
3.3.3.1
GENERAL INTERFERENCE
The important part in (3.19) the part that expressed in (3.20),
in other meaning the E ( x, L ) will be an image of E ( x.0 ) if
⎛ ⎡ v( v + 2) π ⎤ ⎞
exp⎜⎜ ⎢ j
L⎥ ⎟⎟ =1 or ( −1)v
3Lπ
⎦⎠
⎝⎣
(3.21)
From (3.21) the single images are repeated at distance [9]
L = p( 3Lπ ) With p= 0, 1, 2, 3…
(3.22)
In addition to the single images at distance given by (3.22), multiple
images can be found as well [9].
L=
p
( 3Lπ )
2
With p = 1, 3, 5…
(3.23)
The total field at this length is found by substituting (3.23) into (3.19)
π⎞
⎛ p ⎞ m−1
⎛
E⎜ x, 3Lπ ⎟ = ∑AE
v v ( x) exp⎜ jv( v + 2) p ⎟
2⎠
⎝ 2 ⎠ v=0
⎝
(3.24)
There are two useful properties which we use them now and later
28
⎧⎪ E v ( x )
Ev (− x ) = ⎨
⎪⎩ − E v ( x )
fo r
v even
fo r
v odd
(3.25)
⎧even for v even
v ( v + 2) = ⎨
⎩odd for v odd
(3.26)
With p an odd integer, taking into account the property of (3.25) and
the mode field symmetry condition of (3.24), (3.26) can be written as
1+( −j)
1−( −j)
p
⎛ p ⎞
j
AE
x
E
x
E( −x,0)
E⎜x, 3Lπ ⎟= ∑AE
x
+
−
=
,0
+
(
)
(
)
(
)
(
)
∑
v v
v v
2
2
2
⎝
⎠ veven
vodd
,
,
p
p
(3.27)
This equation represents a pair of image of E ( x,0 ) in quadrature and
with amplitudes 1/ 2 , at distances z =
1
3
( 3Lπ ) , ( 3Lπ ) ,....... as shown in
2
2
Fig.3.3
n
ns
n
X
f
Z
h
X=0
E( x,0)
Z=
Figure 3.3:
0
1
( 3Lπ )
2
( 3Lπ )
3
( 3Lπ )
2
2 ( 3Lπ )
Multimode waveguide showing the input field E ( x,0 ) , a
mirrored single image at ( 3Lπ ) , a direct single image at
2 ( 3Lπ ) . And tow-fold images at
1
3
( 3Lπ ) and ( 3Lπ )
2
2
29
3.3.3.2
SYMMETRIC INTERFERENCE
Optical 2-way splitters can in principle is realized on the basis
of the general 2-fold imaging at lengths given by (3.23). The single
images of the input field E ( x.0 ) will now be obtained from [6].
⎛ 3L ⎞
L = p⎜ π ⎟
⎝ 4 ⎠
With p = 0, 1, 2 …
(3.28)
In general, N-fold images are obtained [10], at distances
L=
p ⎛ 3Lπ ⎞
⎟
⎜
N⎝ 4 ⎠
With p = 0, 1, 2 …
(3.29)
With N images of the input field E ( x,0 ) , symmetrically located along
the x-axis with equal spacing he / N . The comparison between (3.23)
and (3.29) 1-to-N beam splitter can be realized with multimode
waveguides four times shorter than at (3.23). In the final, we can get
the optimum and shorter length for power splitter device by using
(3.8) into (3.29) and p = 1.
L =
3.4
n
f
h e2
N λ
: N the number of output
(3.30)
o
CASCADE MULTIMODE INTERFERENCE (MMI)
MMI devices are based on the principle of self-imaging. An
input field profile in a multimode waveguide is reproduced as single
or multiple images at periodic intervals along the propagation
direction of the waveguide, as a result of beating of different modes in
the waveguide [12]. The beat length in a multimode waveguide is
proportional to the square of the width of the multimode waveguide
equation (3.8). In a standard 1X2 MMI light splitter, the width of the
multimode region is equal to twice the desired output waveguide
spacing. Therefore, in order to achieve the same output waveguide
30
spacing T (Fig3.4). The splitter was designed by [20]. That splitter
consists of two stages, which is shown in figure 3.4.
L2
W-g
W2
W1
T1
T0
W2
L1
W-g
L
Figure 3.4:
Structure of the cascaded 1X 2 splitter
CHAPTER 4
MATHEMATICAL MODEL
4.1 Introduction
In this chapter, the numeric is presented in mathematical
model to predict the dimensions and power losses of the power
splitter based on MMI. So the first function of the model is to get
accurate values of propagation constant and describe the electrical
field intensity inside the waveguide.
The optimization of 1x2, 1x4,
1x8, 1x16 and 1x32 power splitter using Matlab software will be
demonstrated. The dimensions, excess loss and imbalance have been
simulated.
Nowadays most telecommunication system takes place at the
third telecommunication window. This window has a central
wavelength of 1550 nm, and therefore while doing the calculation and
simulation the wavelength was fixed at 1550nm.
After understanding the operation principle of MMI devices,
(depending on width, wavelength, and different materials); and design
requirements, such as splitter, it is possible to apply some related
analytical and numerical solvers to design MMI-based devices. From
the previous analysis in chapter 3, we know that the MMI device that
is based on symmetrical self-imaging is the shortest one, and shorter
32
devices have better tolerance. For this reason, the device should
generally be designed to be as short as possible. However, the final
device configuration will be determined by device functions. For
example, for the power splitter in which the phase of the output is
not important, the MMI device based on symmetrical self-imaging
could be used as it is shown in chapter 3.
4.2
Model Propagation Analysis
The main feature of the MMI waveguide is that it supports
plural eigenmodes, and they interfere along the propagation to show
peculiar
lateral
field
distribution
variation
depending
on
the
propagated distances. As a result, the total field distribution is selfimaged at periodic distances, which are determined by the smallest
common multiples of the propagation constants of eigenmodes. Here
the guided-mode propagation method analysis (MPA), one of the
analytical methods, is used to illustrate the self-imaging effect in the
MMI device. In this approach, the propagation constants β v (v= 0, 1,
2, 3, …, N, where N is the number of guided modes) of multi-modes in
the MMI area . The fundamental theories were discussed in chapter 2
and 3 will be used to form the mathematical model.
4.2.1 Guided Mode
In according to the equations in chapter 3 which relate among
the width, refractive indices and propagation constant β
propagation constants of each individual mode,
, the
β v , can be
numerically determined. Each waveguide support the certain number
of modes, depend of the width and refractive indices. Soldano et. al.
33
1995
derived
approximate
formula
(3.7)
which
approximate values of the propagation constant
gives
us
the
β v without any
numerical analysis but they are inaccurate values.
Calculating the propagation constants of each mode β v
requires the use of an accurate numerical technique such as the
Newton-Raphson method [19]. With this method, the right-hand side
of equation 2.15 is subtracted from the left-hand side (separately)
and
the
method
iteratively
approximates
the
zero-point
with
increasing accuracy through each iteration. Using up to a hundred
iterations in our simulations to determine the propagation constant.
Table 4.2 shows (more accurately) the number of modes supported by
a waveguide with respect to the core width.
Table 4.1: The maximum waveguide width before another mode
becomes supported for a waveguide with nf =
3.45189744 and nc = 3.36755329.
Modes
Maximum Waveguide
Supported
Width( μ m )
14
14.3
28
28.61
48
49.05
96
100.145
192
202.335
The table above contains of the number of waveguides which each
one supports the certain number of modes, depend of the width of
waveguide. Let us take the first waveguide with width WM=14.3 μ m .
The modes diagram are shown in figure 4.1 and it is obvious to justify
the odd and even symmetry modes. A rule of thumb stated that even
number of modes will have even symmetry and likewise, the odd
modes will have odd symmetry.
34
Mode 0
Mode 1
1
The field intensity
The field intensity
1
0.5
0
-0.5
-2
-1
0
1
Width of waveguide
x 10
0.5
0
-0.5
0.5
0
-0.5
-1
-2
2
-5
x 10
Mode 4
The field intensity
The field intensity
0
-0.5
-1
0
1
Width of waveguide
0
-0.5
-5
x 10
-1
0
1
2
-5
Width of waveguide
x 10
Mode 7
1
The field intensity
1
0.5
0
-0.5
-1
0
1
Width of waveguide
0.5
-1
-2
2
Mode 6
The field intensity
2
-5
x 10
1
0.5
-1
-2
-1
0
1
Width of waveguide
Mode 5
1
-1
-2
-1
0
1
2
-5
Width of waveguide
x 10
Mode 3
1
The field intensity
The field intensity
-0.5
-5
1
-1
0
1
Width of waveguide
0
-1
-2
2
Mode 2
-1
-2
0.5
2
-5
x 10
0.5
0
-0.5
-1
-2
-1
0
1
Width of waveguide
2
-5
x 10
35
Mode 8
Mode 9
1
The field intensity
The field intensity
1
0.5
0
-0.5
-1
-2
-1
0
1
Width of waveguide
0.5
0
-0.5
-1
-2
2
-5
x 10
Mode 10
1
The field intensity
The field intensity
1
0.5
0
-0.5
-1
-2
-1
0
1
Width of waveguide
0.5
0
-0.5
-1
-2
2
-5
x 10
Mode 12
The field intensity
The field intensity
2
-5
x 10
1
0.5
0
-0.5
Figure 4.1:
-1
0
1
Width of waveguide
Mode 13
1
-1
-2
-1
0
1
2
-5
Width of waveguide
x 10
Mode 11
-1
0
1
Width of waveguide
2
-5
x 10
0.5
0
-0.5
-1
-2
-1
0
1
Width of waveguide
2
-5
x 10
Electrical field profiles ( E y ) for symmetric TE modes for a
given structure of WM =14.3 μ m , λ =1.55 μ m ,
n f =3.4519, nc = 3.3675. E y has been normalized to 1 and
According to the equation (2.14), the electric field decreases
exponentially in the clad while its dependence is sinusoidal in the
core, as was expected for the behavior of a confined mode. Figure 4.1
shows the electric field profiles for the fourteen confined modes
individual m = 0, 1, 2,…., and 13, supported by that waveguide. In
36
addition that, the figure 4.2 shows the electric field profiles for all
modes together at Z =0.
Propgation Modes Together
1
M-0
0.8
M-1
0.6
M-2
M-3
FieldIntensity
0.4
M-4
M-5
0.2
M-6
0
M-7
-0.2
M-8
M-9
-0.4
M-10
-0.6
M-11
M-12
-0.8
M-13
-1
-1.5
Figure 4.2
-1
-0.5
0
0.5
W idth of Waveguide
1
1.5
-5
x 10
Electrical field profiles ( E y ) for symmetric TE modes
together for a given structure of WM =14.3 μ m , λ =1.55 μ m ,
n f =3.4518, nc = 3.3675. E y has been normalized to 1 and
x-axis in m .
4.2.2 Fields in the MMI Waveguide
Fields in the MMI waveguide are originally excited by an input
field, and propagate with each propagation constant given by
Equations (2.14) and (2.6). The field distribution at an input end of
the MMI waveguide is given by Equation (3.17). The field distribution
at an output end of the MMI waveguide can be reconstructed as
single self-images of the input field distribution depends of the length
of device on Z direction which manage of the same image or multi
images of electrical field at the end of device. In our case the multi
image will be used to create the splitting signals from the main
signal. The width and length of 1x2, 1x4, 1x8, 1x16 and 1x32 power
splitters are calculated by mathematical model and plotted the field
intensity at the input and output ports for each device. Figure 4.3
shows the general shape of the power splitter 1xN, N may be 2, 4, 8,
16, or 32.
37
1
Output
Input
width
MMI
region
N
length
Figure 4.3 General shape of 1xN power splitter based MMI
The results are given by mathematical model as following below:
•
Power Splitter 1x2 with these parameters:
=3.4518,
ncore
nclad =3.3675, λo =1.55 μ m with respect to the width of the
output access ports Waccess =4 μ m . The optimum width and length
are 14.3
μm
LM = 248 μ m
and the length of structure
respectively. Figure 4.4 shows the normalize field intensity at
Normalize Field Intensity V/m
Normalize Field Intensity V/m
the Z= 0 and Z= LM .
The distance LM=0 micrometer
1
Width of core
0.5
0
-8
-6
-4
-2
0
2
Width of Waveguide m
4
6
8
-6
x 10
The distance LM= 248 micrometer
0.8
0.6
0.4
0.2
0
-8
-6
-4
-2
0
2
Width of Waveguide m
4
6
8
-6
x 10
Figure 4.4: Normalize field intensity at the beginning and ending of
1x2 MMI coupler
38
•
Power Splitter 1x4 with these parameters:
ncore
=3.4518,
nclad =3.3675, λo =1.55 μ m with respect to the width of the
output access ports Waccess =4 μ m . The optimum width and length
are
28.6 μ m
and
the
length
of
structure
LM =
477 μ m
respectively. Figure 4.5 shows the normalize field intensity at
NormalizeFieldIntensity V/m
NormalizeFieldIntensity V/m
the Z= 0 and Z= LM .
The distance LM=0 micrometer
1
Width of core
0.5
0
-1.5
-1
-0.5
0
0.5
Width of Waveguide m
1
1.5
-5
x 10
The distance LM=477 micrometer
0.4
0.3
0.2
0.1
0
-1.5
-1
-0.5
0
0.5
Width of Waveguide m
1
1.5
-5
x 10
Figure 4.5: Normalize field intensity at the beginning and ending of
1x4 MMI coupler
•
Power Splitter 1x8 with these parameters:
ncore
=3.4518,
nclad =3.3675, λo =1.55 μ m with respect to the width of the
output access ports Waccess =4 μ m . The optimum width and length
are 49.05 μ m
and the length of structure
LM = 687.6 μ m
respectively. Figure 4.6 shows the normalize field intensity at
the Z= 0 and Z= LM .
Normalize Field Intensity V/m
Normalize Field Intensity V/m
39
The distance LM=0 micrometer
1
Width of core
0.5
0
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
Width of Waveguide m
1.5
2
2.5
-5
x 10
The distance LM=687.6 micrometer
0.2
0.15
0.1
0.05
0
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
Width of Waveguide m
1.5
2
2.5
-5
x 10
Figure 4.6: Normalize field intensity at the beginning and ending of
1x8 MMI coupler
•
Power Splitter 1x16 with these parameters: ncore =3.4518,
nclad =3.3675, λo =1.55 μ m with respect to the width of the
output access ports Waccess =4 μ m . The optimum width and length
are 100.145 μ m and the length of structure LM = 1380 μ m
respectively. Figure 4.7 shows the normalize field intensity at
the Z= 0 and Z= LM .
Normalize Field Intensity V/m
Normalize Field Intensity V/m
40
The distance LM=0 micrometer
1
Width of core
0.5
0
-5
-4
-3
-2
-1
0
1
2
Width of Waveguide m
3
4
5
-5
x 10
The distance LM=1417 micrometer
0.1
0.05
0
-5
-4
-3
-2
-1
0
1
2
Width of Waveguide m
3
4
5
-5
x 10
Figure 4.7: Normalize field intensity at the beginning and ending of
1x16 MMI coupler
•
Power Splitter 1x32 with these parameters: ncore =3.4518,
nclad =3.3675, λo =1.55 μ m with respect to the width of the
output access ports Waccess =4 μ m . The optimum width and length
are 202.335 μ m and the length of structure LM = 2870 μ m
respectively. Figure 4.8 shows the normalize field intensity at
the Z= 0 and Z= LM .
Normalize Field Intensity V/m
Normalize Field Intensity V/m
41
The distance LM=0 micrometer
1
Width of core
0.5
0
-1
0
Width of Waveguide m
1
The distance LM=2930 micrometer
0.05
0.04
0.03
0.02
0.01
0
-1
0
Width of Waveguide m
1
-4
x 10
Figure 4.8: Normalize field intensity at the beginning and ending of
1x32 MMI coupler
4.3 Summary
In the final of this chapter, the Table 4.3 details the width,
length, excess loss and imbalance of the 1x2, 1x4, 1x8, 1x16 and
1x32 power splitter based on MMI.
Table 4.2: Results of mathematical model for 1x2, 1x4, 1x8, 1x16
and 1x32 power splitter based on MMI.
Device
Width( μ m )
Length( μ m )
Excess
Imbalance(in
loss(in dB)
dB)
1x2
14.3
248
-0.0303
0
1x4
28.61
477
-0.0349
-0.0115
1x8
49.05
689
-0.0379384
-0.068854
1x16
100.145
1417
-0.0728412
-0.0889275
1x32
202.335
2930
-0.4125477
-0.35604
42
All the values inside the table 4.2 especially the width and
length will be used in the next chapter.
CHAPTER 5
SIMULATION BY BPM-CAD
5.1 Introduction
In this chapter the simulation is presented by BPM-CAD. The
first part in this chapter is waveguide coupler which gives us the
suitable gap that avoids the coupling power among output ports of
the power splitter. After that the guide mode will be verified by BPMCAD and compared with mathematical model in chapter 4.The 1X2,
1X4, 1X8, 1X16 and 1X32 power splitter based on MMI will be
simulated by BPM-CAD. The comparison among three ways
Soldano’s formula, our mathematical model in chapter 4 and the
optimum length by BPM will come in analysis part. Finally, the
cascade MMI will be presented in last part.
5.2 WAVEGUIDE COUPLER
The beam propagation method is often used to evaluate the
performance of a coupled waveguide. We know that the field of a
confined mode extends out beyond the core region. These evanescent
tails can transfer energy from one waveguide to another if the
dielectric structure is suitable. Here we want use BPM to experiment
44
with a coupled waveguide structure. As an example, we will examine
the propagation of the mode on a waveguide which located adjacent
to an identical guide. The index profile for this structure is plotted in
figure 5.1. The couplers consist of two identical step-index
waveguide with core width of 4 μ m situated d from each other.
Index n(x)
3.4518
1
gap
2
3.3675
4
d
4
Position ( μ m )
Figure 5.1: The index profile of two slab waveguides separated by d
To begin the analysis, the materials are used in waveguide
that has strong guide, this property makes the gap distance is
smaller than the weak guide materials. Actually, we used BPM to
experiment amount of coupling power with different gap. If the light
lunch into the waveguide 1, after 100 μ m length of waveguides 1 and
2 , the output power from the waveguide 2 is called coupling power.
The table 5.1 shows the gap and amount of coupling power from
input power.
45
Table 5.1: Gap between two waveguides vs. amount of coupling
power.
Gap between two waveguides ( μ m )
Amount of coupling power (%)
1
8
2
2.5
3
0
From previous analysis we can get relation among width of
Multimode waveguide, width of output port waveguide (4 μ m ),
distance gap and the number of output ports [8].
WM = N (4 + d )
(5.1)
WM : Width of multimode waveguide
N:
d:
Number of output ports
Distance gap, It is better when we chose d ≥ 2μ m
So the width of multimode waveguide should be chosen according to
equation (5.2) with respect the number of modes that propagate
inside waveguide.
WM ≥ 6 N
(5.2)
5.3 Guide Mode
In order to verify the results were gotten by mathematical
model in chapter 4, the modes that propagate inside waveguide will
be shown by BPM in this part. If we use the same parameters that
used in chapter 4, width WM=14.3 μ m . The modes diagram are shown
in figure 5.2 each one alone but in Figure 5.3 show all modes
46
together at beginning of the waveguide.
Mode 0
Mode 1
Mode 2
C:\Documents and Settings\Administrator\Desktop\aziz\mode1x2.M2D
Mode 3
BPM_CAD: Optiwave Corporation
C:\Documents and Settings\Administrator\Desktop\aziz\mode1x2.M2D
Mode 4
C:\Documents and Settings\Administrator\Desktop\aziz\mode1x2.M2D
Mode 5
C:\Documents and Settings\Administrator\Desktop\aziz\mode1x2.M2D
BPM_CAD: Optiwave Corporation
Mode 6
C:\Documents and Settings\Administrator\Desktop\aziz\mode1x2.M2D
BPM_CAD: Optiwave Corporation
BPM_CAD: Optiwave Corporation
BPM_CAD: Optiwave Corporation
Mode 7
C:\Documents and Settings\Administrator\Desktop\aziz\mode1x2.M2D
Mode 8
C:\Documents and Settings\Administrator\Desktop\aziz\mode1x2.M2D
BPM_CAD: Optiwave Corporation
BPM_CAD: Optiwave Corporation
Mode 9
C:\Documents and Settings\Administrator\Desktop\aziz\mode1x2.M2D
BPM_CAD: Optiwave Corporation
47
Mode 10
Mode 11
Mode 12
C:\Documents and Settings\Administrator\Desktop\aziz\mode1x2.M2D
Figure 5.2:
Mode 13
BPM
C:\Documents and Settings\Administrator\Desktop\aziz\mode1x2.M2D
BPM
Electrical field profiles( E y ) for symmetric TE modes for a
given structure of WM =14.3 μ m , λ =1.55 μ m , n f =3.4519, nc
= 3.3675. E y has been normalized to 1 and x-axis in μ m .
C:\Documents and Settings\Administrator\Desktop\aziz\mode1x2.M2D
Figure 5.3
BPM_CAD: Optiwave Corporation
E y E field profiles for symmetric TE modes together for a
given structure of WM =14.3 μ m , λ =1.55 μ m , n f =3.4518, nc =
3.3675. E y has been normalized to 1 and x-axis in m .
48
According to the symmetric interference mechanism, the single
images of the input field will be obtained at the multimode section
length of (Soldano and Pennings, 1995) [12]:
⎛ 3L ⎞
L(sin gle − image ) = p ⎜ π ⎟ With p = 0, 1, 2 …
⎝ 4 ⎠
(5.3)
while the N fold images are obtained at distance:
LM =
p
N
⎛ 3L π ⎞
⎜
⎟
⎝ 4 ⎠
(5.4)
where Lπ, known as the beat length is defined by:
Lπ =
π
β 0 − β1
≈
4nW e
3λ0
(5.5)
β0 and β1 are propagation constants of the fundamental and the first
order lateral modes respectively, λ0 is the free space wavelength, n is
the effective index and We is the effective width of the multimode
waveguide. So in the first thing, the length of device will be
calculated by use equation (5.4) to compare with other methods
which shown in Table 5.2
5.4 Simulation
MMI optical splitters work on the principle of symmetric
interference, a type of self-imaging mechanism in multimode
waveguide structure [1]. As described in Chapter 3, this interference
effect can be achieved by centre-feeding the multimode waveguide.
The length and width of waveguide were calculated in chapter 4.
Here they will be used in BPM-CAD and verified. The input and
49
output access waveguides which are single mode structures are
placed at the beginning and at the end of multimode waveguide,
respectively. Five types of MMI splitters will be simulated namely;
1×2, 1×4, 1×8, 1×16 and 1×32 splitters. In order to accommodate
for separation distance between output access waveguides, the
selected multimode waveguide width of the 1×2, 1×4, 1×8, 1×16 and
1×32 splitters are set to 14.3 µm, 28.61 µm, 49.05 µm, 100.145 µm
and 202.335 µm, respectively. Figure 5.4 shows the schematic
layouts of MMI splitter.
4 µm
7.15 µm
14.3 µm
7.15 µm
4 µm
4 µm
248 µm
(a)
14.305 µm
4 µm
7.1525µm
28.61µm
4 µm
4 µm
4 µm
4 µm
477 µm
(b)
50
6.13µm
6.13µm
6.13µm
6.13µm
6.13µm
6.13µm
6.13µm
24.525µm
49.05µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
688 µm
( c)
6.26µm
6.26µm
6.26µm
6.26µm
6.26µm
6.26µm
6.26µm
6.26µm
6.26µm
6.26µm
6.26µm
6.26µm
6.26µm
6.26µm
50.0725µm
100.145µm
4µm
1420µm
[d]
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
51
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
6.33µm
101.1675µm
202.335µm
2925µm
Figure 5.4
Schematic(e)
layouts of MMI splitters:
(a) 1×2, (b) 1×4, (c) 1×8, (d) 1×16, (e) 1 x 32
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
4µm
52
The designs are then simulated on BPM-CAD™ platform. The
simulation results for designed MMI splitters are shown in Figure
5.5. The right figures show the normalized E-field and effective index
distributions at the outputs. From the figures, it clearly shows that
the output access waveguides emit uniform light distribution, which
indicates the accuracy and stability of the designed splitters
structures
(a)
(b)
53
( c)
(d)
(e)
Figure 5.5
BPM-CAD analyses of MMI optical splitters:
(a) 1×2, (b) 1×4, (c) 1×8, (d) 1×16, (e) 1×32
54
The MMI splitter is mainly characterized by an imbalance
and an excess loss. The length of device governs of the imbalance
and excess loss.
The power imbalance of the optical splitter is defined as
following (Ma Huilian et.al, 2000):
Imbalance =
10 log
Po min
( dB )
Po max
(5.6)
Where Po min and Po max are minimum and maximum output power
in the N waveguides respectively. The second property is the
Excess loss which defined as following (Chuang et. al, 1999):
⎛ N
⎜ ∑ po −i
i =1
Excess Loss = 10 log10 ⎜
⎜ pin
⎜
⎝
⎞
⎟
⎟
⎟
⎟
⎠
(5.7)
N
Where
∑p
i =1
o−i
is the carried total power in the output guides and
pin is the total power carried in the input guide. So the efficiency
of the splitter is ultimately determined by the length of MMI
section. The optimum length is the length which occur the
minimum values of imbalance and excess loss.
To evaluate the optimum length, the imbalance and the
excess loss were studied with varying length of MMI ( LMMI ), the
numerical results are shown in Fig 5.6 for 1x2, 1x4, 1x8, 1x16
and 1x32.
55
Optimum Length 248E-6 m For 1x2
0
0
-0.05
-0.1
-0.1
-0.15
-0.2
-0.3
-0.25
-0.3
-0.4
-0.35
-0.4
-0.5
-0.45
-0.6
-0.5
-0.55
Imbalance in dB
Excess Loss in dB
-0.2
-0.7
-0.6
248E-6 m
-0.65
-0.8
-0.7
-0.9
-0.75
-0.8
225
230
235
240
245
250
255
260
265
270
-1
Length * 1E-6 m
(a)
optimum Length 477e-6 m for 1x4
Access Loss in dB
-0.2
-0.02
-0.4
-0.04
-0.6
-0.06
-0.8
-0.08
-0.1
-1
-1.2
-0.12
-1.4
-0.14
-1.6
-0.16
477e-6
-0.18
-1.8
-2
450
460
470
480
Length * 1E-6 m
( b)
490
500
-0.2
Imbalance in dB
0
0
56
optimum length 688E-6 m for 1x8
0
-0.1
-0.05
-0.2
-0.1
-0.3
-0.15
-0.4
-0.2
-0.5
-0.25
-0.6
-0.3
-0.7
-0.35
-0.8
-0.4
688E-6
-0.9
-1
Imbalance in dB
Access Loss in dB
0
-0.45
665
670
675
680
685
690
695
700
705
710
-0.5
Length * 1E-6 m
(c)
-0.2
-0.4
Excess Loss in dB
-0.6
-0.8
-1
-1.2
-1.4
-1.6
-1.8
-2
-2.2
-2.4
-2.6
1420E-6 m
-2.8
-3
1360
1370
1380
1390
1400
1410
Length *1E-6 m
(d)
1420
1430
1440
0
-0.2
-0.4
-0.6
-0.8
-1
-1.2
-1.4
-1.6
-1.8
-2
-2.2
-2.4
-2.6
-2.8
-3
-3.2
-3.4
-3.6
-3.8
-4
-4.2
-4.4
-4.6
-4.8
-5
-5.2
-5.4
-5.6
-5.8
-6
1450
Imbalance in dB
Optimum Length 1420E-6 For 1x16
0
57
-0.2
-0.4
Excess Loss in dB
-0.6
-0.8
-1
-1.2
-1.4
-1.6
-1.8
-2
-2.2
-2.4
2925 E-6 m
-2.6
-2.8
-3
2850
2860
2870
2880
2890
2900
2910
2920
2930
2940
2950
0
-0.2
-0.4
-0.6
-0.8
-1
-1.2
-1.4
-1.6
-1.8
-2
-2.2
-2.4
-2.6
-2.8
-3
-3.2
-3.4
-3.6
-3.8
-4
-4.2
-4.4
-4.6
-4.8
-5
-5.2
-5.4
-5.6
-5.8
-6
Imbalance in dB
Optimum length 2925E-6 m for 1x32
0
Length *1E-6 m
(e)
Figure 5.6:
Excess loss and imbalance versus length of MMI section with
λ =1.55 μ m for (a) 1X2 at width = 14.3 WMMI , (b) 1X4 at width=
28.61 μ m , (c) 1X8 at width=49.05 μ m , (d) 1X16
width=100.145 μ m and (e) 1X32 at width= 202.335 μ m .
5.5 Analysis
Comparison
among
(Soldano
and
Pennings,
1995)’s
formula equation (5.4), our mathematical model in chapter 4 and
the optimum length by BPM-CAD was tabulated in table 5.2
at
58
Table 5.2:
Comparison among (Soldano and Pennings, 1995)’s
formula, our mathematical model and the optimum
length by BPM-CAD
The operation in MMI is more sensitive of length. Considering,
the length of MMI section mainly depends on the lowest two
propagation constant Equation (5.5), they depend on the width of
MMI section. The aim of approximate formula (Soldano and
Pennings, 1995) in the first case to calculate the propagation
constant, this formula actually accumulate small deviation from the
59
actual values of propagation constant. So the deviation of length
increases by increase the number of modes which essentially depend
on the width of MMI section.
The results of mathematical model were shown in the second
part of the previous table. It has used numerical analysis ( Newton
Raphson’s method ) to get the propagation constant of the
waveguide, the deviation of length by using this case is smaller than
the length in first case (analytical solutions) especially when increase
the number of iterations in Newton Raphson method, as it was
shown in previous table.
BPM_CAD is a powerful, user-friendly system that allows
computer-aided design of a variety of integrated and fiber optics
guided wave problems. The beam propagation method, or BPM, is a
step-by-step method of simulating the passage of light through any
waveguiding medium. In integrated and fiber optics, an optical field
can be tracked at any point as it propagates along a guiding
structure. BPM allows computer-simulated observation of the light
field distribution. The radiation and the guided field can be
examined simultaneously. The properties of 1x2, 1x4, 1x8, 1x16 and
1x32 (excess loss and imbalance) are sensitive of the length and the
width of MMI section. We have found the optimum length of these
devices by using BPM-CAD (excess loss and imbalance versus length
of MMI).
60
5.6
CASCADE MULTIMODE INTERFERENCE (MMI)
MMI devices are based on the principle of self-imaging. An
input field profile in a multimode waveguide is reproduced in single
or multiple images at periodic intervals along the propagation
direction of the waveguide, as a result of beating of different modes
in the waveguide [12]. Since the beat length in a multimode
waveguide is proportional to the square of the width of the
waveguide Equation (3.8). In a standard 1X2 MMI light splitter, the
width of the multimode region is equal to twice the desired output
waveguide spacing. Therefore, in order to achieve the output
waveguide spacing T1 in Figure 5.7. The splitter was designed by
(Milan L. at., 2003). That splitter consists of two stages, which is
shown in Figure 5.7. So the purpose of this part is to simulate the
cascade MMI and obtain the certain space between the output ports.
The relation among T1, width of MMI splitter W1, width of second
stage MMI W2 and width of output port W-g were formulated by
(Milan L. at., 2003) [20] as following:
T 1 =
W 1
+ 2
2
(W
2 − X
)
(5.8)
The first stage already have designed which has length (L1) =248 μ m and width
(W1) =14.3 μ m . The width of second stage W2 is known from Equation (5.8) by
use T1=125 μ m . Finally, the length of second stage L2 will be calculated by
Equation (5.3). With these information, the cascade MMI was simulated by BPMCAD. The Figure 5.8 shows the BPM – CAD analysis of cascade MMI.
61
L2
W-g
W2
W1
T0
T1
W2
L1
W-g
L
Figure5.7:
Figure 5.8:
Structure of the cascaded 1X 2 splitter
BPM-CAD analyses of cascade MMI
The length of stage two (L2) is 36.3mm and the excess loss is -1.8dB.
In the final, we note that length and losses of this device by use this
technique but it is easy to fabricate it and has other properties as
large bandwidth and low independence of polarization [20].
CHAPTER 6
CONCLUSIONS AND FUTURE WORK
6.1
Conclusions
In accordance with the main goal of this project, the previous
chapters have concisely explained the optical waveguide power
splitter based on Multimode Interference (MMI). The followings are
considered as a short review and overall conclusion drawn from the
work presented in this project.
The work starts with the introduction to the background that
induced the area of interest in this work. Based on the main objective
which is the studying and modelling of optical power splitter device
based on Multimode Interference.
In the first step, the theories of optical waveguides are
elaborated
in
chapter
2.
The
mathematical
foundation
and
propagation light inside waveguide have been considered. Starting
from Maxwell’s equations, the wave equation for planar waveguide is
obtained, and then TE polarized is considered to solve it. The formula
that calculates the number of modes in waveguide is shown and
described the electrical field inside the waveguide. ALGaAs is chosen
to form the waveguide because it is easy to manage the refractive
index for the core and clad. Therefore, we can obtain the waveguide
63
which has a strong guide property. Finally in that chapter, the
effective index method (EIM) is explained.
In the third chapter, self-image principle [7] which is originated
from the modes interference in a multimode waveguide structure has
been briefly stated prior to further discussions on MMI effect. In
analyzing the MMI effect, two dimension (2D) mathematical model
based on effective index method (EIM) approximation of its three
dimension counterpart is employed. Following this, the mechanism is
to form the single and multi image of electrical field profile along of
propagation length. The mathematical formulations that differentiate
the properties of these interference mechanisms are briefly described
for application in optical device design.
Mathematical model is presented in chapter four. The main
idea in this model, uses Newton Raphson’s method for numerical
analysis to get the eigenvalues of the propagation constant. So the
first function of the model is to get accurate values of the propagation
constant and describe the electrical field intensity inside the
waveguide. Thus, the main function of the model is to predict the
dimension and power losses of the power splitter based on MMI.
The simulation is presented by BPM-CAD in the chapter five.
During simulation, the length of MMI section is calculated using
three ways: Soldano and Pennings, 1995’s formula, our mathematical
model which uses Newton Raphson method in numerical analysis
and the optimum length is verified by BPM-CAD. The Excess loss and
Imbalance are calculated to compare between the first and second
ways. In the end of this chapter, cascaded MMI is presented to
increase the space between the output ports.
Overall, the operation in MMI is more sensitive to length and
width. By considering, the length of MMI section mainly depends on
64
the lowest two propagation constants, they depend on the width of
MMI section. The aim of approximation formula in the first case is to
calculate the propagation constant. So the deviation of the length
increases as the number of modes increase which essentially depend
on the width of MMI section. The mathematical model has used
numerical analysis to get the propagation constant of the waveguide;
the deviation of length by using this case is smaller than the length in
the first case. The properties of these devices (Excess loss and
Imbalance) are sensitive of the length and the width.
The modelling results agree well with simulation results.
6.2
FUTURE WORK
To study and describe the design and measurement of the
MMI device (splitter, combiner, Multiplexer and Demultiplexer )
which provide wavelength and polarization insensitive.
65
References
[1] Ma Huilian; Yang Jianyi; Jiang Xiaoqing; Wang Minghua;”
Compact and economical MMI optical power splitter for optical
communication” IEEE J. Light. Tech. Volume 2, 21-25 Aug. 2000
Page(s):1561 - 1564 vol.2
[2] Gin´es Lifante (2002).Integrated Photonics: Fundamentals. John
Wiley & Sons Ltd.
24-25 & 60-85.
[3] H. Kogelnik, (1982). Integrated Optics, 2nd ed. Berlin: SpringerVerlag, vol. 7.
[4] Boyd, J.T (1994). Photonics Inegrated Circuit. In Photonics
Devices and System. New York: Hunsperger, R. G. Ed. Marcel
Dekker.
[5] Clifford R. Pollock, (1995). Fundamental Of Optoelectronic, Irwin
Inc. pp. 62-63.
[6] G. B. Hocker and W. K. Burns, (1977). Applied Optics, 16, p. 113
[7] O. Bryngdahl (1973). Image Formation Using Self-Imaging
Technique. Journal of the Optical Society of America. 63(4): 416-419.
[8] P. A. Besse, M. Bachmann, H. Melchior, L.B. Soldano, and M.K.
Smit, (1994) Optical bandwidth and fabrication tolerances of
66
multimode interference couplers, J. Lightwave Technol. 12(6): 10041009.
[9] L.B. Soldano, F.B. Veerman, M.K. Smit, B.H. Verbeek, A.H.
Dubost and E.C.M. Pennings (1992). Planar Monomode Optical
Couplers Based on Multimode Interference Effects. IEEE Journal of
Lightwave Technology. 10(12): 1843-1850.
[10] P.A. Besse, E. Gini, M. Bachmann, and H. Melchior,(1994). New
1x2 multimode interference couplers with free selection of power
splitting ratios. In Proc. ECOC’94, Firenze, Italy,(12): 669-672.
[11] J. Leuthold, J. Eckner, E. Gamper, P.A. Besse, and H.
Melchior,(1998). Multimode interference couplers for the conversion
and combining of zero and first- order modes. J. Lightwave
Technol.16(7):1228-1239.
[12] L.B. Soldano and E.C.M. Pennings (1995). Optical Multi-Mode
Interference Devices Based on Self-Imaging: Principles and
Applications. IEEE Journal of Lightwave Technology. 13(4): 615-627.
[13] R. Ulrich. (1975). Image formation by phase coincidences in
optical waveguides.
Optics Commun. 13(3):259-264.
[14] N. S. Kapany and J. J. Burke. (1972), Optical Waveguides. New
York: Academic.
[15] Hongzhen Wei, Jinzhong Yu, Zhongli Liu, Xiaofeng Zhang, Wei
Shi, and Changshui Fang, 2001. Signal Bandwidth of general N x N
Multimode Interference Couplers. IEEE Journal of Lightwave
Technology.19(5)
67
[16] R. M. Jenkins, R. W. J. Deveraux, and J. M. Heaton. (1992),
Waveguide beam splitters and recombiners based on multimode
propagation phenomena. Opt. Lett., vol. 17(14): 991-993
[17] Bienstman, P., Six, E., Roelens, M., Vanwolleghem, M. and
Baets, R. (2002). Calculation of bending losses in dielectric waveguide
using eigenmode expansion and perfectly matched layers. IEEE
Photonics Technology Letters. 14(2): 164-166.
[18] Veldhuis, G. J. (1998). Bent waveguide devices and mechanooptical switch. University of Twente. Enschede, the Netherlands: Phd
Thesis.
[19] K.F. Riley, M.P. Hobson, and S.J. Bence, Mathematical methods
for physics and engineering (Cambridge University, Cambridge,
2003).
[20] Milan L., etc. (2003). Multimode Interference-Based Two-Stage 1
2 Light Splitter for Compact Photonic Integrated Circuits. IEEE
PHOTONICS TECHNOLOGY LETTERS, VOL. 15, NO. 5:706-708
68
Appendix A
MATLAB Code: Solve for Guided Modes
The MATLAB function below solves for the longitudinal propagation
constants, β, of the guided modes of a waveguide using the NewtonRaphson method.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%
%%%% function GuidedModesSolve
%%%% Input:
%%%% nf,nc – Core and cladding effective indices, respectively
%%%% lambda – Free-space wavelength of light
%%%% h_mmi – Full-width of multimode waveguide
%%%%
%%%% Output:
%%%% BetaGuideS – Longitudinal propagation
%%%% constants of the symmetric guided modes of the multimode
%%%% waveguide.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [BetaGuideS] = GuidedModesSolve0(nf, nc, lambda, h_mmi)
%%%%%% Multimode waveguide GUIDED Modes %%%%%%
% Symmetric Mode(s) %
k0 = (2*pi)/lambda;
k1 = k0*n1;
k2 = nc*k0;
Beta = k2:(k1-k2)/1e5:k1;
Gamma = sqrt(k1^2 - Beta.^2);
Alpha = sqrt(Beta.^2 - k2^2);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
69
%%%%%%%%% Begin Newton-Raphson Method %%%%%%%%%
F = ((Gamma./Alpha).*tan(Gamma.*(h_mmi/2))) - 1;
guess=[ ];
guess_count = 1;
%%%%%% Look for change in sign of F (rough estimate) %%%%%%
for counter = 1:length(Beta)-1
if(and(sign(F(counter)) > sign(F(counter+1)), isfinite(F(counter)) == 1))
guess(guess_count,1) = counter;
guess(guess_count,2) = Beta(counter);
guess(guess_count,3) = Beta(counter+1);
guess(guess_count,4) = F(counter);
guess_count = guess_count+1;
end
end
clear Beta;
clear Gamma;
clear Alpha;
clear F;
p_save0 = [];
%%%%%%%%% Refine rough estimates %%%%%%%%%
for counter = 1:guess_count-1
p0 = guess(counter,2);
p1 = guess(counter,3);
for iter = 1:20
Gamma = sqrt(k1^2 - p0^2);
Alpha = sqrt(p0^2 - k2^2);
F = ((Gamma./Alpha).*tan(Gamma.*(h_mmi/2))) - 1;
Gamma1 = sqrt(k1^2 - p1^2);
Alpha1 = sqrt(p1^2 - k2^2);
F1 = ((Gamma1./Alpha1).*tan(Gamma1.*(h_mmi/2))) - 1;
if(p1 == p0)
p_save0(counter) = p0;
break;
70
end
F_prime = (F1 - F)/(p1-p0);
p = p0 - (F/F_prime);
p1 = p0;
p0 = p;
end
end
GuideS0 = p_save0;
clear Beta;
clear Gamma;
clear Alpha;
clear F;
clear p0;
clear p1;
clear p;
clear p_save0;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%Beta = k2:(k1-k2)/1e5:k1;
Gamma = sqrt(k1^2 - Beta.^2);
Alpha = sqrt(Beta.^2 - k2^2);
F = ((Gamma./Alpha).*tan(Gamma.*(h_mmi/2))) + 1;
guess=[ ];
guess_count = 1;
for counter = 1:length(Beta)-1
if(and(sign(F(counter)) > sign(F(counter+1)), isfinite(F(counter)) == 1))
guess(guess_count,1) = counter;
guess(guess_count,2) = Beta(counter);
guess(guess_count,3) = Beta(counter+1);
guess(guess_count,4) = F(counter);
guess_count = guess_count+1;
end
end
clear Beta;
71
clear Gamma;
clear Alpha;
clear F;
p_save1 = [];
for counter = 1:guess_count-1
p0 = guess(counter,2);
p1 = guess(counter,3);
for iter = 1:20
Gamma = sqrt(k1^2 - p0^2);
Alpha = sqrt(p0^2 - k2^2);
F = ((Gamma./Alpha).*tan(Gamma.*(h_mmi/2))) + 1;
Gamma1 = sqrt(k1^2 - p1^2);
Alpha1 = sqrt(p1^2 - k2^2);
F1 = ((Gamma1./Alpha1).*tan(Gamma1.*(h_mmi/2)))+ 1;
if(p1 == p0)
p_save1(counter) = p0;
break;
end
F_prime = (F1 - F)/(p1-p0);
p = p0 - (F/F_prime);
p1 = p0;
p0 = p;
end
end
GuideS1 = p_save1;
clear Beta;
clear Gamma;
clear Alpha;
clear F;
clear p0;
clear p1;
clear p;
clear p_save1;
72
k=1;
for i=1:length(GuideS0)
BetaGuideS(k)=GuideS0(i);
k=k+2;
end
k=2;
for i=1:length(GuideS1)
BetaGuideS(k)=GuideS1(i);
k=k+2;
end
k=1;
for i=length(BetaGuideS):-1:1
KK(k)=BetaGuideS(i);
k=k+1;
end
BetaGuideS=KK;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
73
Appendix B
MATLAB Code: Describe the electrical field intensity for each mode.
The MATLAB program below uses the Guided Modes function which
be showed in appendix A and electric field function which will be
shown below to describe the electrical field intensity (TE-mode) for
each mode.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%This program defines the electrical field intensity %%%%
%%%% for each mode which propagate inside
waveguide.%%%%%%%%
%%%%%%%%%%%%
%%%% Input:
%%%% n1,n2 – Core and cladding effective indices, respectively
%%%% lambda – Free-space wavelength of light
%%%% h_mmi – Full-width of multimode waveguide
%%%% x
- Coordinates of waveguide in x direction (matrix)
%%%% Output:
%%%% plote elctrical field for each mode vs x
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%clear;
clear beta;
clear atten;
um=1e-6;
nf=3.45189744 ; % core refractive index (GaAs)
nc=3.36755329 ; %cladding refractive index (GaAs with 5% AL)
h_mmi=14.3*um; %Full-width of multimode waveguide
74
lambda=1.55*um; % Free-space wavelength of light
hef=h_mmi+(lambda/pi)/sqrt(nf^2-nc^2);
k0=2*pi/lambda;
%%% call the function to calculate the beta
[beta]=GuidedModesSolve0(nf, nc, lambda, h_mmi);
nef=betas/k0
%%======================================================
==
v=length(beta);
atten=sqrt(beta.^2-(k0*nc)^2);
pic=approx(v,4)
if pic==1
num=v;
else
num=4;
end
s1=1;
for r=1:pic
if(r>1)
r3=figure;
end
for a=s1:num
su1=num/(r*2);
su2=a-(r-1)*4;
subplot(su1,2,su2),
plot(x,field(a, atten(a), hef, h_mmi, x));
xlabel('Width of waveguide' )
ylabel('The field intensity')
mode=a-1;
title( mode )
end
s1=num+1;
num=num+4;
75
if r==pic-1
r1=pic-v/4;
r2=r1*4;
num=num-r2;
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%%%%%%% function Field
%%%%%%% This function defines the boundary condition
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%% Input:
%%%%%%% a1 - the mode
%%%%%%% a2 - the attenuation at this mode
%%%%%%% a3 - the effictive width of MMI section
%%%%%%% a4 - the normal width of MMI section
%%%%%%% x - A coordinate of x
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%% Output:
%%%%%%% the magntuide of electrical field at this point of x for this
mode
%%%%%%% (a1)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
function E=Field(a1, a2, a3, a4, x)
E2=sin(a1*pi*(x-a4/2)/a3);
E1=exp(-a2*(x-a4/2)).*E2;
E3=exp(a2*(x+a4/2)).*E2;
E4=E1.*(x >= a4/2) + E2.*( -a4/2< x & x<a4/2) + E3.*(x<= -a4/2);
E=(E4)*-1;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
76
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function ap =approx(o,o1)
if o==o1
ap=1;
elseif (mod(o,o1)==0)
ap=o/o1;
else
ap=o/o1-rem(o,o1)/o1+1;
end
77
Appendix C
MATLAB Code: Describe the electrical field intensity for the input and
the each output ports after Lmm from the input port.
The MATLAB program below uses the Guided Modes function which
be showed in appendix A and electric field412 function which will be
shown below to describe the electrical field intensity (TE-mode) for all
modes .
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%This program defines the electrical field intensity %%%%
%%%% for all modes which propagate inside
waveguide.%%%%%%%%
%%%%%%%%%%%%
%%%% Input:
%%%% n1,n2 – Core and cladding effective indices, respectively
%%%% lambda – Free-space wavelength of light
%%%% h_mmi – Full-width of multimode waveguide
%%%% x
- Coordinates of waveguide in x direction (matrix)
%%%% Output:
%%%% plote elctrical field for each mode vs x
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
clear;
clear beta;
clear atten;
um=1e-6;
nf=3.45189744; % core refractive index (GaAs)
78
ns=3.36755329 ; %cladding refractive index (GaAs with 5% AL)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%%% The width of waveguide and the number of outout ports are
variables.
%%% They depend on the the N ports
h_mmi=14.3*um; %%Full-width of multimode waveguide
N=2; % the number of output ports
lambda=1.55*um; % Free-space wavelength of light
hef=h_mmi+(lambda/pi)/sqrt(nf^2-ns^2);
k0=2*pi/lambda;
n=1;
%%% call the function to calculate the values of beta
[beta]=GuidedModesSolve0(nf, nc, lambda, h_mmi);
%%======================================================
==
a=length( beta);% the value of a shows the number of modes
atten=sqrt(beta.^2-(k0*ns)^2);
x=((-h_mmi/2-5*um):0.01*um:h_mmi/2+5*um);
mu=4*pi*1e-7; % Permeablity
omega=2*pi*3e8/lambda;
ampl=beta/(2*omega*mu);
length_beta= pi/(beta(1)-beta(2));
%%% the length of MMI section by use Newton Raphson Method
L=0:.75*length_beta/N: .75*length_beta/N %%%%
%%% the length of MMI section by use Soldano's formula
Lmm=nf*hef^2/(lambda*N)
b=length_beta;
for(q=1:length(L))
f(q)=figure;
plot(x,field412(a, atten, hef, h, ampl,b, x,L(q)));
79
title('The distance LM= micrometer' );
xlabel(' Width of Waveguide m');
ylabel(' Normalize Field Intensity V/m');
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% The function of calculate the total electrical field for all
modes
%%%%% in the same point Cv
function E=Field412(a1, a2, a3, a4,a5,b, x,a6)
l=0;
for m=1:2:a1
E2=sin(m*pi*(abs(x)-a4/2)/a3)/sin(m*pi*a4/(2*a3));
E1=exp(-a2(m)*(x-a4/2)).*E2;
E3=exp(a2(m)*(x+a4/2)).*E2;
E4=E1.*(x >= a4/2) + E2.*( -a4/2< x & x<a4/2) + E3.*(x<= -a4/2);
%E4=E2.*( -a3/2<= x & x<=a3/2);
I=sum(E4.^2);
%=1/sqrt(a5(m)*I);
amplt1(m)=(exp(i*(m-1)*(m+1)*pi*a6/(3*b)))*1/sqrt(a5(m)*I);
l=(E4).*amplt1(m)+l;
end
E=abs(l);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%