Introduction to Optical Waveguide Analysis 2001-Kenji Kawano, Tsutomu Kitoh

Introduction to Optical Waveguide Analysis: Solving Maxwell's Equations
and the SchroÈdinger Equation. Kenji Kawano, Tsutomu Kitoh
Copyright # 2001 John Wiley & Sons, Inc.
ISBNs: 0-471-40634-1 (Hardback); 0-471-22160-0 (Electronic)

INTRODUCTION TO
OPTICAL WAVEGUIDE
ANALYSIS


INTRODUCTION TO
OPTICAL WAVEGUIDE
ANALYSIS

Solving Maxwell's Equations and
the SchroÈdinger Equation

KENJI KAWANO and TSUTOMU KITOH

A Wiley-Interscience Publication
JOHN WILEY & SONS, INC.
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To our wives,
Mariko and Kumiko


CONTENTS
Preface = xi

1

Fundamental Equations

1

1.1 Maxwell's Equations = 1
1.2 Wave Equations = 3
1.3 Poynting Vectors = 7

1.4 Boundary Conditions for Electromagnetic Fields = 9
Problems = 10
Reference = 12

2

Analytical Methods

13

2.1 Method for a Three-Layer Slab Optical Waveguide = 13
2.2 Effective Index Method = 20
2.3 Marcatili's Method = 23
2.4 Method for an Optical Fiber = 36
Problems = 55
References = 57
vii


viii

3

CONTENTS

Finite-Element Methods

59

3.1 Variational Method = 59

3.2 Galerkin Method = 68
3.3 Area Coordinates and Triangular Elements = 72
3.4 Derivation of Eigenvalue Matrix Equations = 84
3.5 Matrix Elements = 89
3.6 Programming = 105
3.7 Boundary Conditions = 110
Problems = 113
References = 115
4

Finite-Difference Methods

117

4.1 Finite-Difference Approximations = 118
4.2 Wave Equations = 120
4.3 Finite-Difference Expressions of Wave Equations = 127
4.4 Programming = 150
4.5 Boundary Conditions = 153
4.6 Numerical Example = 160
Problems = 161
References = 164
5

Beam Propagation Methods

165

5.1 Fast Fourier Transform Beam Propagation Method = 165
5.2 Finite-Difference Beam Propagation Method = 180

5.3 Wide-Angle Analysis Using Pade Approximant
Operators = 204
5.4 Three-Dimensional Semivectorial Analysis = 216
5.5 Three-Dimensional Fully Vectorial Analysis = 222
Problems = 227
References = 230
6

Finite-Difference Time-Domain Method
6.1 Discretization of Electromagnetic Fields = 233
6.2 Stability Condition = 239
6.3 Absorbing Boundary Conditions = 241

233


ix

CONTENTS

Problems = 245
References = 249
7

SchroÈdinger Equation

251

7.1 Time-Dependent State = 251
7.2 Finite-Difference Analysis of Time-Independent State = 253

7.3 Finite-Element Analysis of Time-Independent State = 254
References = 263
Appendix A Vectorial Formulas

265

Appendix B Integration Formula for Area Coordinates

267

Index

273


PREFACE
This book was originally published in Japanese in October 1998 with the
intention of providing a straightforward presentation of the sophisticated
techniques used in optical waveguide analyses. Apparently, we were
successful because the Japanese version has been well accepted by
students in undergraduate, postgraduate, and Ph.D. courses as well as
by researchers at universities and colleges and by researchers and
engineers in the private sector of the optoelectronics ®eld. Since we did
not want to change the fundamental presentation of the original, this
English version is, except for the newly added optical ®ber analyses and
problems, essentially a direct translation of the Japanese version.
Optical waveguide devices already play important roles in telecommunications systems, and their importance will certainly grow in the future.
People considering which computer programs to use when designing
optical waveguide devices have two choices: develop their own or use
those available on the market. A thorough understanding of optical

waveguide analysis is, of course, indispensable if we are to develop our
own programs. And computer-aided design (CAD) software for optical
waveguides is available on the market. The CAD software can be used
more effectively by designers who understand the features of each analysis
method. Furthermore, an understanding of the wave equations and how
they are solved helps us understand the optical waveguides themselves.
Since each analysis method has its own features, different methods are
required for different targets. Thus, several kinds of analysis methods have
xi


xii

PREFACE

to be mastered. Writing formal programs based on equations is risky
unless one knows the approximations used in deriving those equations, the
errors due to those approximations, and the stability of the solutions.
Mastering several kinds of analysis techniques in a short time is
dif®cult not only for beginners but also for busy researchers and
engineers. Indeed, it was when we found ourselves devoting substantial
effort to mastering various analysis techniques while at the same time
designing, fabricating, and measuring optical waveguide devices that we
saw the need for an easy-to-understand presentation of analysis techniques.
This book is intended to guide the reader to a comprehensive understanding of optical waveguide analyses through self-study. It is important
to note that the intermediate processes in the mathematical manipulations
have not been omitted. The manipulations presented here are very detailed
so that they can be easily understood by readers who are not familiar with
them. Furthermore, the errors and stabilities of the solutions are discussed
as clearly and concisely as possible. Someone using this book as a

reference should be able to understand the papers in the ®eld, develop
programs, and even improve the conventional optical waveguide theories.
Which optical waveguide analyses should be mastered is also an
important consideration. Methods touted as superior have sometimes
proven to be inadequate with regard to their accuracy, the stability of
their solutions, and central processing unit (CPU) time they require. The
methods discussed in this book are ones widely accepted around the
world. Using them, we have developed programs we use on a daily basis
in our laboratories and con®rmed their accuracy, stability, and effectiveness in terms of CPU time.
This book treats both analytical methods and numerical methods.
Chapter 1 summarizes Maxwell's equations, vectorial wave equations,
and the boundary conditions for electromagnetic ®elds. Chapter 2
discusses the analysis of a three-layer slab optical waveguide, the effective
index method, Marcatili's method, and the analysis of an optical ®ber.
Chapter 3 explains the widely utilized scalar ®nite-element method. It ®rst
discusses its basic theory and then derives the matrix elements in the
eigenvalue equation and explains how their calculation can be
programmed. Chapter 4 discusses the semivectorial ®nite-difference
method. It derives the fully vectorial and semivectorial wave equations,
discusses their relations, and then derives explicit expressions for the
quasi-TE and quasi-TM modes. It shows formulations of Ex and Hy
expressions for the quasi-TE (transverse electric) mode and Ey and Hx
expressions for the quasi-TM (transverse magnetic) mode. The none-


PREFACE

xiii

quidistant discretization scheme used in this chapter is more versatile than

the equidistant discretization reported by Stern. The discretization errors
due to these formulations are also discussed. Chapter 5 discusses beam
propagation methods for the design of two- and three-dimensional (2D,
3D) optical waveguides. Discussed here are the fast Fourier transform
beam propagation method (FFT-BPM), the ®nite-difference beam propagation method (FD-BPM), the transparent boundary conditions, the wideangle FD-BPM using the Pade approximant operators, the 3D semivectorial analysis based on the alternate-direction implicit method, and
the fully vectorial analysis. The concepts of these methods are discussed
in detail and their equations are derived. Also discussed are the error
factors of the FFT-BPM, the physical meaning of the Fresnel equation,
the problems with the wide-angle FFT-BPM, and the stability of the
FD-BPM. Chapter 6 discusses the ®nite-difference time-domain method
(FD-TDM). The FD-TDM is a little dif®cult to apply to 3D optical
waveguides from the viewpoint of computer memory and CPU time, but
it is an important analysis method and is applicable to 2D structures.
Covered in this chapter are the Yee lattice, explicit 3D difference
formulation, and absorbing boundary conditions. Quantum wells, which
are indispensable in semiconductor optoelectronic devices, cannot be
designed without solving the SchroÈdinger equation. Chapter 7 discusses
how to solve the SchroÈdinger equation with the effective mass approximation. Since the structure of the SchroÈdinger equation is the same as that
of the optical wave equation, the techniques to solve the optical wave
equation can be used to solve the SchroÈdinger equation.
Space is saved by including only a few examples in this book. The
quasi-TEM and hybrid-mode analyses for the electrodes of microwave
integrated circuits and optical devices have also been omitted because of
space limitations. Finally, we should mention that readers are able to get
information on the vendors that provide CAD software for the numerical
methods discussed in this book from the Internet.
We hope this book will help people who want to master optical
waveguide analyses and will facilitate optoelectronics research and development.
Kanagawa, Japan
March 2001

KENJI KAWANO and TSUTOMU KITOH


INTRODUCTION TO
OPTICAL WAVEGUIDE
ANALYSIS


Introduction to Optical Waveguide Analysis: Solving Maxwell's Equations
and the SchroÈdinger Equation. Kenji Kawano, Tsutomu Kitoh
Copyright # 2001 John Wiley & Sons, Inc.
ISBNs: 0-471-40634-1 (Hardback); 0-471-22160-0 (Electronic)

CHAPTER 1

FUNDAMENTAL EQUATIONS
This chapter summarizes Maxwell's equations, vectorial wave equations,
and the boundary conditions for electromagnetic ®elds.
1.1 MAXWELL'S EQUATIONS
The electric ®eld E (in volts per meter), the magnetic ®eld H (amperes per
meter), the electric ¯ux density D (coulombs for square meters), and the
magnetic ¯ux density B (amperes per square meter) are related to each
other through the equations
D ˆ eE;

…1:1†

B ˆ mH;

…1:2†

where the permittivity e and permeability m are de®ned as
e ˆ e0 er ;

…1:3†

m ˆ m0 mr :

…1:4†

Here, e0 and m0 are the permittivity and permeability of a vacuum, and er
and mr are the relative permittivity and permeability of the material. Since
the relative permeability mr is 1 for materials other than magnetic
1


2

FUNDAMENTAL EQUATIONS

materials, it is assumed throughout this book to be 1. Denoting the
velocity of light in a vacuum as c0 , we obtain
e0 ˆ

1
c20 m0

% 8:854188 Â 10À12

m0 ˆ 4p  10À7

H=m:

F=m

…1:5†
…1:6†

The current density J (in amperes per square meter) in a conductive
material is given by
J ˆ sE:

…1:7†

The electromagnetic ®elds satisfy the following well-known Maxwell
equations [1]:
@B
;
@t

…1:8†

@D
‡ J:
@t

…1:9†

=3E ˆ À

=3H ˆ

Since the equation = ? …=3A† ˆ 0 holds for an arbitrary vector A, from
Eqs. (1.8) and (1.9), we can easily derive
= ? B ˆ 0;

…1:10†

= ? D ˆ r:

…1:11†

The current density J is related to the charge density r (in coulombs per
square meter) as follows:
=?JˆÀ

@r
:
@t

…1:12†

Equations (1.10) and (1.11) can be derived from Eqs. (1.8), (1.9), and
(1.12).


1.2 WAVE EQUATIONS

3

1.2 WAVE EQUATIONS
Let us assume that an electromagnetic ®eld oscillates at a single angular
frequency o (in radians per meter). Vector A, which designates an
electromagnetic ®eld, is expressed as
 exp… jot†g:
A…r; t† ˆ RefA…r†

…1:13†

Using this form of representation, we can write the following phasor
expressions for the electric ®eld E, the magnetic ®eld H, the electric ¯ux
density D, and the magnetic ¯ux density B:
 exp… jot†g;
E…r; t† ˆ RefE…r†

…1:14†

 exp… jot†g;
H…r; t† ˆ RefH…r†

…1:15†

 exp… jot†g;
D…r; t† ˆ RefD…r†

…1:16†

 exp… jot†g:
B…r; t† ˆ RefB…r†

…1:17†

 H,
 D,
 and B in the phasor
In what follows, for simplicity we denote E,
representation as E, H, D, and B. Using these expressions, we can write
Eqs. (1.8) to (1.11) as
=3E ˆ ÀjoB ˆ Àjom0 H;

…1:18†

=3H ˆ joD ˆ joeE;

…1:19†

= ? H ˆ 0;

…1:20†

= ? …er E† ˆ 0;

…1:21†

where it is assumed that mr ˆ 1 and r ˆ 0.

1.2.1 Wave Equation for Electric Field E
Applying a vectorial rotation operator =3 to Eq. (1.18), we get
=3…=3E† ˆ Àjom0 =3H:

…1:22†


4

FUNDAMENTAL EQUATIONS

Using the vectorial formula
=3…=3A† ˆ =…= ? A† À H2 A;

…1:23†

we can rewrite the left-hand side of Eq. (1.22) as
=…= ? E† À H2 E:

…1:24†

The symbol H2 is a Laplacian de®ned as
H2 ˆ

@2
@2
@2
‡
‡
:
@x2 @y2 @z2

…1:25†

Since Eq. (1.21) can be rewritten as
= ? …er E† ˆ =er ? E ‡ er = ? E ˆ 0;
we obtain
=?EˆÀ

=er
? E:
er

…1:26†

Thus, the left-hand side of Eq. (1.22) becomes


=er
À=
? E À H2 E:
er

…1:27†

On the other hand, using Eq. (1.19), we get for the right-hand side of
Eq. (1.22)
k02 er E;

…1:28†

where k0 is the wave number in a vacuum and is expressed as
p o
k0 ˆ o e0 m0 ˆ :

c0

…1:29†


1.2 WAVE EQUATIONS

5

Thus, for a medium with the relative permittivity er, the vectorial wave
equation for the electric ®eld E is


=er
H E‡=
? E ‡ k02 er E ˆ 0:
er
2

…1:30†

And using the wave number k in that medium, given by
p
p
p
k ˆ k0 n ˆ k0 er ˆ o e0 er m0 ˆ o em0 ;

…1:31†

we can rewrite Eq. (1.30) as



=er
H E‡=
? E ‡ k 2 E ˆ 0:
er
2

…1:32†

When the relative permittivity er is constant in the medium, this
vectorial wave equation can be reduced to the Helmholtz equation
H2 E ‡ k 2 E ˆ 0:

…1:33†

1.2.2 Wave Equation for Magnetic Field H
Applying the vectorial rotation operator =3 to Eq. (1.19), we get
=3…=3H† ˆ joe0 =3…er E†:
Thus,
=…= ? H† À H2 H ˆ joe0 …=er 3E ‡ er =3E†
ˆ joe0 …=er 3E† ‡ joe0 er …Àjom0 H†
ˆ joe0 …=er 3E† ‡ k02 er H:

…1:34†

Using
Eˆ

1

=3H
joe0 er

…1:35†


6

FUNDAMENTAL EQUATIONS

obtained from Eqs. (1.19) and (1.20), we get from Eq. (1.30) the following
vectorial wave equation for the magnetic ®eld H:
H2 H ‡

=er
3…=3H† ‡ k02 er H ˆ 0:
er

…1:36†

Using Eq. (1.31), we can rewrite Eq. (1.36) as
H2 H ‡

=er
3…=3H† ‡ k 2 H ˆ 0:
er

…1:37†

When the relative permittivity er is constant in the medium, this

vectorial wave equation can be reduced to the Helmholtz equation
H2 H ‡ k 2 H ˆ 0:

…1:38†

Now, we discuss an optical waveguide whose structure is uniform in the
z direction. The derivative of an electromagnetic ®eld with respect to the z
coordinate is constant such that
@
ˆ Àjb;
@z

…1:39†

where b is the propagation constant and is the z-directed component of the
wave number k. The ratio of the propagation constant in the medium, b, to
the wave number in a vacuum, k0 , is called the effective index:
neff ˆ

b
:
k0

…1:40†

When l0 is the wavelength in a vacuum,
bˆ

2p
2p

2p
neff ˆ
ˆ
;
l0
l0 =neff leff

…1:41†

where leff ˆ l0 =neff is the z-directed component of the wavelength in the
medium. The physical meaning of the propagation constant b is the phase
rotation per unit propagation distance. Thus, the effective index neff can be
interpreted as the ratio of a wavelength in the medium to the wavelength in
a vacuum, or as the ratio of a phase rotation in the medium to the phase
rotation in a vacuum.


1.3

POYNTING VECTORS

7

We can summarize the Helmholtz equation for the electric ®eld E as
H2c E ‡ …k 2 À b2 †E ˆ 0

…1:42†

H2c E ‡ k02 …er À n2eff †E ˆ 0:

…1:43†

or

For the magnetic ®eld H, on the other hand, we get the Helmholtz equation
H2c H ‡ …k 2 À b2 †H ˆ 0

…1:44†

H2c H ‡ k02 …er À n2eff †H ˆ 0;

…1:45†

or

where we used the de®nition H2c ˆ @2 =@x2 ‡ @2 =@y2 .

1.3 POYNTING VECTORS
In this section, the time-dependent electric and magnetic ®elds are
expressed as E…r; t† and H…r; t†, and the time-independent electric and


magnetic ®elds are expressed as E…r†
and H…r†.
Because the voltage is the
integral of an electric ®eld and because the magnetic ®eld is created by a
current, the product of the electric ®eld and the magnetic ®eld is related to
the energy of the electromagnetic ®elds. Applying a divergence operator
= ? to E3H, we get
= ? …E3H† ˆ H ? =3E ‡ E ? =3H:

Substituting Maxwell's equations (1.8) and (1.9) into this equation, we get
@H
@H
À eE ?
À sE2
= ? …E3H† ˆ ÀmH ?
@t
@t


@ 1 2 1 2
ˆÀ
eE ‡ mH À sE2 :
@t 2
2

…1:46†


8

FUNDAMENTAL EQUATIONS

When Eq. (1.46) is integrated over a volume V , we get
…

…
V

= ? …E3H† dV ˆ

…E3H†n dS

… 
…
@
1 2 1 2
eE ‡ mH dV À sE2 dV ;
ˆÀ
@t V 2
2
V
…1:47†
S

where we make use of Gauss's law and n designates a component normal
to the surface S of the volume V .
The ®rst two terms of the last equation correspond to the rate of the
reduction of the stored energy in volume V per unit time, while the third
term corresponds to the rate of reduction of the energy
„ due to Joule
heating in volume V per unit time. Thus, the term s …E3H†n dS is
considered to be the rate of energy loss through the surface.
Thus,
S ˆ E3H

…1:48†

is the energy that passes through a unit area per unit time. It is called a
Poynting vector.

For an electromagnetic wave that oscillates at a single angular
frequency o, the time-averaged Poynting vector hSi is calculated as
follows:
hSi ˆ hE3Hi
 exp… jot†g3RefH…r†
 exp… jot†gi
ˆ hRefE…r†

 exp…Àjot† H
 exp… jot† ‡ H*
 exp…Àjot†
E exp… jot† ‡ E*
3
ˆ
2
2
1 
 ‡ E*3
 H
 ‡ E3
 H
 exp… j2ot† ‡ E*3
 H*
 exp…Àj2ot†i
ˆ h…E3H*
ˆ

4
1
 

2 RefhE3H*ig:

…1:49†
Here, we used hexp… j2ot†i ˆ hexp…Àj2ot†i ˆ 0.
Thus, for an electromagnetic wave oscillating at a single angular
frequency, the quantity
 H*

S ˆ 12 E3

…1:50†


1.4

BOUNDARY CONDITIONS FOR ELECTROMAGNETIC FIELDS

9

is de®ned as a complex Poynting vector and the energy actually propagating is considered to be the real part of it.

1.4 BOUNDARY CONDITIONS FOR ELECTROMAGNETIC
FIELDS
The boundary conditions required for the electromagnetic ®elds are
summarized as follows:
(a) Tangential components of the electric ®elds are continuous such
that
E1t ˆ E2t :

…1:51†

(b) When no current ¯ows on the surface, tangential components of the
magnetic ®elds are continuous such that
H1t ˆ H2t :

…1:52†

When a current ¯ows on the surface, the magnetic ®elds are
discontinuous and are related to the current density JS as follows:
H1t À H2t ˆ JS :

…1:53†

Or, since the magnetic ®eld and the current are perpendicular to
each other, the vectorial representation is
n3…H1 À H2 † ˆ JS :

…1:54†

(c) When there is no charge on the surface, the normal components of
the electric ¯ux densities are continuous such that
D1n ˆ D2n :

…1:55†

When there are charges on the surface, the electric ¯ux densities
are discontinuous and are related to the charge density rS as
follows:
D1n À D2n ˆ rS :

…1:56†


10

FUNDAMENTAL EQUATIONS

(d) Normal components of the magnetic ¯ux densities are continuous
such that
B1n ˆ B2n :

…1:57†

Here, the vectors n and t in these equations are respectively unit normal
and tangential vectors at the boundary.

PROBLEMS
1. Use Maxwell's equations to specify the features of a plane wave
propagating in a homogeneous nonconductive medium.

ANSWER
Maxwell's equations are written as
@Ez @Ey
À
ˆ Àjom0 Hx ;
@y
@z

…P1:1†

@Ex @Ez
À
ˆ Àjom0 Hy ;
@z
@x

…P1:2†

@Ey @Ex
À
ˆ Àjom0 Hz ;
@x
@y

…P1:3†

@Hz @Hy
À
ˆ joe0 er Ex ;
@y
@z

…P1:4†

@Hx @Hz
À
ˆ joe0 er Ey ;
@z
@x

…P1:5†

@Hy @Hx
À
ˆ joe0 er Ez :
@x
@y

…P1:6†

Since the electric and magnetic ®elds of the plane wave depend not on
the x and y coordinates but on the z coordinate, the derivatives with respect
to the coordinates for directions other than the propagation direction are
zero. That is, @=@x ˆ @=@y ˆ 0.


PROBLEMS

11

From Eqs. (P1.3) and (P1.6), we get
Hz ˆ Ez ˆ 0;

…P1:7†

dEy
ˆ jom0 Hx ;
dz

…P1:8†

dEx
ˆ Àjom0 Hy ;
dz

…P1:9†

dHy
ˆ Àjoe0 er Ex ;
dz

…P1:10†

dHx
ˆ joe0 er Ey :
dz

…P1:11†

The remaining equations are

Equations (P1.8)±(P1.11) are categorized into two sets:
Set 1:

dEx
ˆ Àjom0 Hy
dz

Set 2:

dEy
ˆ jom0 Hx
dz

and
and

dHy
ˆ Àjoe0 er Ex :
dz
dHx
ˆ joe0 er Ey :
dz

…P1:12†
…P1:13†

The equations of set 1 can be reduced to
d 2 Ex
‡ k 2 Ex ˆ 0
dz2

and

d 2 Hy
‡ k 2 Hy ˆ 0;
dz2

…P1:14†

where k 2 ˆ o2 e0 m0 er ˆ k02 er . And the equations of set 2 can be reduced to
d 2 Ey
‡ k 2 Ey ˆ 0
dz2

and

d 2 Hx
‡ k 2 Hx ˆ 0:
dz2

…P1:15†

Here, we discuss a plane wave propagating in the z direction. Considering that Eq. (P1.14) implies that both the electric ®eld component Ex and
the magnetic ®eld component Hy propagate with the wave number k,
where it should be noted that the propagation constant b is equal to the
wave number k in this case and that the pure imaginary number j
[ˆ exp…12 jp†] corresponds to phase rotation by 90 , we can illustrate the
propagation of the electric ®eld component Ex and the magnetic ®eld


12

FUNDAMENTAL EQUATIONS

FIGURE P1.1. Propagation of an electromagnetic ®eld.

component Hy , as shown in Fig. P1.1. When we substitute Ex for Ey and
ÀHy for Hx, the equations of set 2 are equivalent to those of set 1. Since
the ®eld components in set 2 can be obtained by rotating the ®eld

components in set 1 by 90 , sets 1 and 2 are basically equivalent to
each other.
The features of the plane wave are summarized as follows: (1) the
electric and magnetic ®elds are uniform in directions perpendicular to the
propagation direction, that is, @=@x ˆ @=@y ˆ 0; (2) the ®elds have no
component in the propagation direction, that is, Hz ˆ Ez ˆ 0; (3) the
electric ®eld and the magnetic ®eld components are perpendicular to each
other; and (4) the propagation direction is the direction in which a screw
being turned to the right, as if the electric ®eld component were being
turned toward the magnetic ®eld component, advances.
2. Under the assumption that the relative permeability in the medium is
equal to 1 and p
that
 a plane wave propagates in the ‡z direction, prove
p
that m0 Hy ˆ eEx .
ANSWER
The derivative with respect to the z coordinate can be reduced to
p
d=dz ˆ Àjk ˆ Àjo em0 by using Eq. (P1.14). Thus, the relation follows
from Eq. (P1.12).
REFERENCE
[1] R. E. Collin, Foundations for Microwave Engineering, McGraw-Hill, New
York, 1966.


Introduction to Optical Waveguide Analysis: Solving Maxwell's Equations
and the SchroÈdinger Equation. Kenji Kawano, Tsutomu Kitoh
Copyright # 2001 John Wiley & Sons, Inc.
ISBNs: 0-471-40634-1 (Hardback); 0-471-22160-0 (Electronic)

CHAPTER 2

ANALYTICAL METHODS
Before discussing the numerical methods in Chapters 3±7, we ®rst
describe analytical methods: a method for a three-layer slab optical
waveguide, an effective index method, and Marcatili's method. For
actual optical waveguides, the analytical methods are less accurate than
the numerical methods, but they are easier to use and more transparent.
In this chapter, we also discuss a cylindrical coordinate analysis of the
step-index optical ®ber.

2.1 METHOD FOR A THREE-LAYER SLAB OPTICAL
WAVEGUIDE
In this section, we discuss an analysis for a three-layer slab optical
waveguide with a one-dimensional (1D) structure. The reader is referred
to the literature for analyses of other multilayer structures [1, 2].
Figure 2.1 shows a three-layer slab optical waveguide with refractive
indexes n1 , n2 , and n3 . Its structure is uniform in the y and z directions.
Regions 1 and 3 are cladding layers, and region 2 is a core layer that has a
refractive index higher than that of the cladding layers. Since the
tangential ®eld components are connected at the interfaces between
adjacent media, we can start with the Helmholtz equations (1.47) and
(1.49), which are for uniform media. Furthermore, since the structure is
13


14

ANALYTICAL METHODS

FIGURE 2.1. Three-layer slab optical waveguide.

uniform in the y direction, we can assume @=@y ˆ 0. Thus, the equation for
the electric ®eld E is
d2E
‡ k02 …er À n2eff †E ˆ 0:
dx2

…2:1†

Similarly, we easily get the equation for the magnetic ®eld H:
d2H
‡ k02 …er À n2eff †H ˆ 0:
dx2

…2:2†

Next, we discuss the two modes that propagate in the three-layer slab
optical waveguide: the transverse electric mode (TE mode) and the
transverse magnetic mode (TM mode). For better understanding, we
again derive the wave equation from Maxwell's equations
=3E ˆ Àjom0 H;

…2:3†

=3H ˆ joe0 er E;

…2:4†

whose component representations are
@Ez @Ey
À
ˆ Àjom0 Hx ;
@y
@z

…2:5†

@Ex @Ez
À
ˆ Àjom0 Hy ;
@z
@x

…2:6†

@Ey @Ex
À
ˆ Àjom0 Hz ;
@x
@y

…2:7†