Introduction to Optical Waveguide Analysis: Solving Maxwell's Equations and the SchroÈdinger Equation. Kenji Kawano, Tsu...
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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. New York / Chichester / Weinheim / Brisbane / Singapore / Toronto
Designations used by companies to distinguish their products are often claimed as trademarks. In all instances where John Wiley & Sons, Inc., is aware of a claim, the product names appear in initial capital or ALL CAPITAL LETTERS. Readers, however, should contact the appropriate companies for more complete information regarding trademarks and registration. Copyright # 2001 by John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic or mechanical, including uploading, downloading, printing, decompiling, recording or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the Publisher. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: PERMREQ @ WILEY.COM. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold with the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional person should be sought. ISBN 0-471-22160-0 This title is also available in print as ISBN 0-471-40634-1. For more information about Wiley products, visit our web site at www.Wiley.com.
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
INDEX Area coordinate, 74 Alternate-direction implicit (ADI) method, 216 Angular frequency, 3 Bandwidth, 109 Basis function, 62 Beam propagation method (BPM), 165 ADI-BPM, 216 ®rst Fourier transform beam propagation method (FFT-BPM), 165 ®nite-difference beam propagation method (FD-BPM), 180 Bessel function, 40 Bessel function of ®rst kind, 40 Bessel function of second kind, 40 modi®ed Bessel function of ®rst kind, 40 modi®ed Bessel function of second kind, 40 Boundary condition, 9, 27, 110, 153, 197 absorbing boundary condition (ABC), 241 analytical boundary condition, 154 transparent boundary condition (TBC), 197
Characteristic equation, 17, 20, 41, 53 Charge density, 1 Cladding, 13, 37, 125 Core, 13, 37, 125 Cramer's formula, 76 Crank-Nicolson scheme, 195 Current density, 1 Cylindrical coordinate system, 38 Derivative, 119 ®rst derivative, 119 second derivative, 119 Difference center, 131 hypothetical difference center, 131 Discretization, 130 equidistant discretization, 130 nonequidistant discretization, 130 Dirichlet condition, 66, 111, 153, 257 Dominant mode, 111 Effective index, 6 effective index method, 20 Eigenvalue, 88, 151 eigenvalue matrix equation, 68, 72, 151, 257 Eigenvector, 88, 151 Electric ®eld, 1 273
274
INDEX
Electric ¯ux density, 1 Element, 64 triangular element, 64, 72 ®rst-order triangular element, 64, 73, 91, 106 second-order triangular element, 64, 79, 95, 108 x Epq mode, 24, 113 y Epq mode, 31, 113 Even mode, 111 Expansion coef®cient, 63 Explicit scheme, 239 Finite-element method (FEM), 59 scalar ®nite-element method (SC-FEM), 59 Finite-difference method (FDM), 116 scalar ®nite-difference method (SC-FDM), 150 semivectorial ®nite-difference method (SV-FDM), 117 Finite-difference time-domain method (FD-TDM), 233 Fourier transform, 170 discrete Fourier transform, 170 inverse discrete Fourier transform, 170 Fresnel approximation, 167±168, 187 Functional, 62 Fully vectorial analysis, 222 Galerkin method, 68 Helmholtz equaiotn, 5±7 Hybrid-mode analysis, 47 Implicit scheme, 186 Interpolation function, 64 Joule heating, 8 Laplacian, 4 Line element, 257 ®rst-order line element, 257 second-order line element, 260 Local coordinate, 107, 110 LP mode, 38
Magnetic ®eld, 1 Magnetic ¯ux density, 1 Marcatili's method, 23 Maxwell's equations, 1 Matrix element, 89 Mirror-symmetrical plane, 111 Multistep method, 213 Neumann condition, 66, 111, 153, 257 Node, 64, 73, 79, 129, 257 Normalized frequency, 41 Odd mode, 111 Optical ®ber, 36 step-index optical ®ber, 37 Pade approximant operator, 204 Para-axial approximation, 167 Permeability, 1 relative permeability, 1 Permittivity, 1 relative permittivity, 1 Phase-shift lens, 170, 173 Phasor expression, 3 Plane wave, 10 Plank constant, 252 Potential, 252 Power con®nement factor (G factor), 55±56 Poynting vector, 7 Principal ®eld component, 125, 182, 184 Propagation constant, 6 Quantum well, 252 Quasi-TE mode, 125, 128 Quasi-TM mode, 125, 147 Rayleigh-Ritz method, 62 Reference index, 166, 187 Residual, 68 error residual, 68 weighted residual method, 69 SchroÈdinger equation, 251 normalized SchroÈdinger equation, 255 time-dependent SchroÈdinger equation, 251
INDEX
time-independent SchroÈdinger equation, 253±254 Shape function, 64, 78, 83 Slab optical waveguide, 13 Slowly varying envelope approximation (SVEA), 166 Stability condition, 195, 239 Taylor series expansion, 118 Tohmas method, 203 Transverse electric (TE) mode, 14, 181, 186 Transverse magnetic (TM) mode, 14, 184, 190
Variational method, 59 Variational principle, 62 Wave equation, 5, 6 scalar wave equation, 84, 127 semivectorial wave equation, 124 vectorial wave equation, 4, 120 Wave number, 5 Weak form, 69 Wide-angle formulation, 167 Wide-angle analysis, 204 Wide-angle order, 205 Yee lattice, 235
275
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 eE;
1:1
B mH;
1:2
where the permittivity e and permeability m are de®ned as e e0 er ;
1:3
m 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
1 c20 m0
8:854188 10
m0 4p 10
7
12
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 sE:
1:7
The electromagnetic ®elds satisfy the following well-known Maxwell equations [1]: =3E
@B ; @t
1:8
=3H
@D J: @t
1:9
Since the equation = ?
=3A 0 holds for an arbitrary vector A, from Eqs. (1.8) and (1.9), we can easily derive = ? B 0;
1:10
= ? D r:
1:11
The current density J is related to the charge density r (in coulombs per square meter) as follows: =?J
@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
jotg: A
r; t 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
jotg; E
r; t RefE
r
1:14
exp
jotg; H
r; t RefH
r
1:15
exp
jotg; D
r; t RefD
r
1:16
exp
jotg: B
r; t 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
joB
jom0 H;
1:18
=3H joD joeE;
1:19
= ? H 0;
1:20
= ?
er E 0;
1:21
where it is assumed that mr 1 and r 0.
1.2.1 Wave Equation for Electric Field E Applying a vectorial rotation operator =3 to Eq. (1.18), we get =3
=3E
jom0 =3H:
1:22
4
FUNDAMENTAL EQUATIONS
Using the vectorial formula =3
=3A =
= ? A
H2 A;
1:23
we can rewrite the left-hand side of Eq. (1.22) as H2 E:
=
= ? E
1:24
The symbol H2 is a Laplacian de®ned as H2
@2 @2 @2 : @x2 @y2 @z2
1:25
Since Eq. (1.21) can be rewritten as = ?
er E =er ? E er = ? E 0; we obtain =?E
=er ? E: er
1:26
Thus, the left-hand side of Eq. (1.22) becomes =er = ?E er
H2 E:
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 o e0 m0 : 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 0: er 2
1:30
And using the wave number k in that medium, given by p p p k k0 n k0 er o e0 er m0 o em0 ;
1:31
we can rewrite Eq. (1.30) as =er H E= ? E k 2 E 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 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 joe0 =3
er E: Thus, =
= ? H
H2 H joe0
=er 3E er =3E joe0
=er 3E joe0 er
jom0 H joe0
=er 3E k02 er H:
1:34
Using E
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 0: er
1:36
Using Eq. (1.31), we can rewrite Eq. (1.36) as H2 H
=er 3
=3H k 2 H 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 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 @ @z
jb;
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
b : k0
1:40
When l0 is the wavelength in a vacuum, b
2p 2p 2p neff ; l0 l0 =neff leff
1:41
where leff 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 H2? E
k 2
b2 E 0
1:42
n2eff E 0:
1:43
or H2? E k02
er
For the magnetic ®eld H, on the other hand, we get the Helmholtz equation H2? H
k 2
b2 H 0
1:44
n2eff H 0;
1:45
or H2? H k02
er
where we used the de®nition H2? @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 H ? =3E E ? =3H: Substituting Maxwell's equations (1.8) and (1.9) into this equation, we get = ?
E3H
@H @H eE ? mH ? @t @t @ 1 2 1 2 eE mH @t 2 2
sE2 sE2 :
1:46
8
FUNDAMENTAL EQUATIONS
When Eq. (1.46) is integrated over a volume V , we get
V
= ?
E3H dV
S
E3Hn dS
@ 1 2 1 2 eE mH dV @t V 2 2
V
sE2 dV ;
1:47
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
E3Hn dS is considered to be the rate of energy loss through the surface. Thus, S 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 hE3Hi exp
jotg3RefH
r exp
jotgi hRefE
r exp
jot H exp
jot H* exp
jot E exp
jot E* 3 2 2 1 E*3 H E3 H exp
j2ot E*3 H* exp
j2oti 4 h
E3H* H*ig: 1 RefhE3 2
1:49 Here, we used hexp
j2oti hexp
j2oti 0. Thus, for an electromagnetic wave oscillating at a single angular frequency, the quantity H* S 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 E2t :
1:51
(b) When no current ¯ows on the surface, tangential components of the magnetic ®elds are continuous such that H1t 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
H2t JS :
1:53
Or, since the magnetic ®eld and the current are perpendicular to each other, the vectorial representation is n3
H1
H2 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 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
D2n rS :
1:56
10
FUNDAMENTAL EQUATIONS
(d) Normal components of the magnetic ¯ux densities are continuous such that B1n 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 @y
@Ey @z
jom0 Hx ;
P1:1
@Ex @z
@Ez @x
jom0 Hy ;
P1:2
@Ey @x
@Ex @y
jom0 Hz ;
P1:3
@Hz @y
@Hy joe0 er Ex ; @z
P1:4
@Hx @z
@Hz joe0 er Ey ; @x
P1:5
@Hy @x
@Hx joe0 er Ez : @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 @=@y 0.
PROBLEMS
11
From Eqs. (P1.3) and (P1.6), we get Hz Ez 0;
P1:7
dEy jom0 Hx ; dz
P1:8
dEx dz
jom0 Hy ;
P1:9
dHy dz
joe0 er Ex ;
P1:10
The remaining equations are
dHx joe0 er Ey : dz
P1:11
Equations (P1.8)±(P1.11) are categorized into two sets: Set 1:
dEx dz
Set 2:
dEy jom0 Hx dz
jom0 Hy
and and
dHy dz
joe0 er Ex :
dHx joe0 er Ey : dz
P1:12
P1:13
The equations of set 1 can be reduced to d 2 Ex k 2 Ex 0 dz2
and
d 2 Hy k 2 Hy 0; dz2
P1:14
where k 2 o2 e0 m0 er k02 er . And the equations of set 2 can be reduced to d 2 Ey k 2 Ey 0 dz2
and
d 2 Hx k 2 Hx 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 [ 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 Hy 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 @=@y 0; (2) the ®elds have no component in the propagation direction, that is, Hz Ez 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 eEx . ANSWER The derivative with respect to the z coordinate can be reduced to p d=dz jk jo 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 0. Thus, the equation for the electric ®eld E is d2E k02
er dx2
n2eff E 0:
2:1
Similarly, we easily get the equation for the magnetic ®eld H: d2H k02
er dx2
n2eff H 0:
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
jom0 H;
=3H joe0 er E;
2:3
2:4
whose component representations are @Ez @y
@Ey @z
jom0 Hx ;
2:5
@Ex @z
@Ez @x
jom0 Hy ;
2:6
@Ey @x
@Ex @y
jom0 Hz ;
2:7
2.1
METHOD FOR A THREE-LAYER SLAB OPTICAL WAVEGUIDE
15
@Hz @y
@Hy joe0 er Ex ; @z
2:8
@Hx @z
@Hz joe0 er Ey ; @x
2:9
@Hy @x
@Hx joe0 er Ez : @y
2:10
As mentioned in Chapter 1, we assume here that the relative permeability mr 1. That is, m mr m0 m0 . Since the structure is uniform in the propagation direction, the derivative with respect to the z coordinate, @=@z, can be replaced by jb. The effective index can be expressed as neff b=k0 , where k0 is the wave number in a vacuum.
2.1.1 TE Mode In the TE mode, the electric ®eld is not in the longitudinal direction (Ez 0) but in the transverse direction (Ey 6 0). Since the structure is uniform in the y direction, @=@y 0. Substitution of these relations into Eq. (2.10) results in @Hy =@x 0. Since this means that Hy is constant, we can assume that Hy 0. Furthermore, substitution of Ez Hy 0 into Eq. (2.6) results in @Ex =@z 0, which means that Ex 0. We thus get Ex Ez Hy 0:
2:11
Substituting Hx
b=om0 Ey , derived from Eq. (2.5), and Hz
j=om0 @Ey =@x, derived from Eq. (2.7), into Eq. (2.9), we get the following wave equation for the principal electric ®eld component Ey : d 2 Ey k02
er dx2 p where k0 o e0 m0 .
n2eff Ey 0;
2:12
16
ANALYTICAL METHODS
Next, we derive the characteristic equation used to calculate the effective index neff . The principal electric ®eld component Ey in regions 1, 2, and 3 can be expressed as q as g1 k0 n2eff n21
region 1; Ey
x C1 exp
g1 x q g2 k0 n22 n2eff
region 2;
C2 cos
g2 x a
as
C3 exp g3
x
q as g3 k0 n2eff n23
W
2:13
2:14
region 3:
2:15
Here, C1 , C2 , and C3 are unknown constants. Since the number of unknowns is 4 (neff , C1 , C2 , and C3 ), four equations are needed to determine the effective index neff . To obtain the four equations, we impose boundary conditions on the tangential electric ®eld component Ey and the tangential magnetic ®eld component Hz at x 0 and x W . The tangential magnetic ®eld component Hz is Hz
1 @Ey ; jom0 @x
2:16
which for the three regions is expressed as g1 C exp
g1 x
region 1; jom0 1 g 2 C2 sin
g2 x a
region 2; jom0 g 3 C3 exp g3
x W
region 3: jom0
Hz
x
2:17
2:18
2:19
Since the boundary condition requirement is that the tangential electric ®eld components as well as the tangential magnetic ®eld components are equal at the interfaces between adjacent media, the boundary conditions on these ®eld components at x 0 are expressed as Ey1
0 Ey2
0;
2:20
Hz1
0 Hz2
0
2:21
2.1
METHOD FOR A THREE-LAYER SLAB OPTICAL WAVEGUIDE
17
and at x W are expressed as Ey2
W Ey3
W ;
2:22
Hz2
W Hz3
W :
2:23
C1 C2 cos a
from Eq:
2:20;
2:24
g1 C1 g2 C2 sin a
from Eq:
2:21;
2:25
from Eq:
2:22;
2:26
from Eq:
2:23:
2:27
The resultant equations are
C2 cos
g2 W a C3 g2 C2 sin
g2 W a
g 3 C3
Thus, dividing Eq. (2.25) by Eq. (2.24), we get a
g1 tan q1 p g2 1
q1 0; 1; 2; . . .:
2:28
On the other hand, dividing Eq. (2.27) by Eq. (2.26), we get g3 g2 W tan g2 1
a q2 p
q2 0; 1; 2; . . .:
2:29
Substitution of a in Eq. (2.28) into Eq. (2.29) results in the following characteristic equation: g1 1 g3 g2 W tan tan qp g2 g2 1
q 0; 1; 2; . . .:
2:30
Or, using tan
1
y
p x 2
x tan ; y 1
2:31
18
ANALYTICAL METHODS
we can rewrite this equation as g2 W
tan
1
g2 g1
tan
1
g2
q 1p g3
q 0; 1; 2; . . .:
2:32
2.1.2 TM Mode In the TM mode, the magnetic ®eld component is not in the longitudinal direction (Hz 0) but in the transverse direction (Hy 6 0). Since the structure is uniform in the y direction, @=@y 0. Thus, we get @Ey =@x 0 from Eq. (2.7). Since this means that Ey is constant, we can assume that Ey 0. Furthermore, substitution of Hz Ey 0 into Eq. (2.9) results in @Hx =@z 0, which means that Hx 0. We thus get Hx Hz Ey 0:
2:33
Substituting Ex
b=oe0 er Hy , derived from Eq. (2.8), and Ez
j=oe0 er @Hy =@x, derived from Eq. (2.10), into Eq. (2.6), we get the following wave equation for the principal magnetic ®eld component Hy : d 2 Hy k02
er dx2
n2eff Hy 0:
2:34
The ®eld components on which the boundary conditions should be imposed are the principal magnetic ®eld component Hy and the longitudinal electric ®eld component Ez . The principal magnetic ®eld component Hy can be expressed as q as g1 k0 n2eff n21
Hy
x C1 exp
g1 x
region 1;
2:35
C2 cos
g2 x a
as
q g2 k0 n22 n2eff
region 2;
2:36
C3 exp g3
x
W
q as g3 k0 n2eff n23
region 3:
2:37
2.1
METHOD FOR A THREE-LAYER SLAB OPTICAL WAVEGUIDE
19
The tangential electric ®eld component Ez is Ez
1 @Hy ; joe0 er @x
2:38
which is expressed as Ez
x
g1 C exp
g1 x joe0 er 1
region 1;
2:39
region 2;
2:40
W
region 3:
2:41
g2 C sin
g2 x a joe0 er 2 g3 C exp g3
x joe0 er 3
Imposing the boundary conditions on the tangential ®elds at x 0 and x W , we get C1 C2 cos a
from Hy1
0 Hy2
0;
2:42
g1 g C 1 C sin a er1 1 er2 2
from Ez1
0 Ez2
0;
2:43
from Hy2
W Hy3
W ;
2:44
from Ez2
W Ez3
W :
2:45
C2 cos
g2 W a C3 g2 C sin
g2 W a er2 2
g2 C er2 3
Dividing Eq. (2.43) by Eq. (2.42), we get a
tan
1
er2 g1 q1 p er1 g2
q1 0; 1; 2; . . .:
2:46
On the other hand, dividing Eq. (2.45) by Eq. (2.44), we get g2 W tan
1
er2 g3 er3 g2
a q2 p
q2 0; 1; 2; . . .:
2:47
20
ANALYTICAL METHODS
Substitution of the variable a in Eq. (2.46) into Eq. (2.47) results in the following characteristic equation: g2 W tan
1
er2 g1 1 er2 g3 tan qp er1 g2 er3 g2
q 0; 1; 2; . . .:
2:48
Using Eq. (2.31), we also get g2 W
tan
1
er1 g2 er2 g1
q 1p
tan
1
er3 g2 er2 g3
2:49
q 0; 1; 2; . . .:
Comparing the characteristic equations (2.30) and (2.32) for the TE mode and Eqs. (2.48) and (2.49) for the TM mode, one discovers that the characteristic equations for the TM mode contain the ratio of the relative permittivities of adjacent media.
2.2 EFFECTIVE INDEX METHOD Here, we discuss the effective index method, which allows us to analyze two-dimensional (2D) optical waveguide structures by simply repeating the slab optical waveguide analyses. Figure 2.2 shows an example of a 2D optical waveguide and illustrates the concept of the effective index method. We consider the scalar wave equation @2 f
x; y @2 f
x; y k02
er
x; y @x2 @y2
n2eff f
x; y 0;
2:50
where neff is the effective index to be obtained. We separate the wave function f
x; y into two functions: f
x; y f
x g
y:
2:51
2.2
EFFECTIVE INDEX METHOD
21
FIGURE 2.2. Concept of the effective index method.
This corresponds to the assumption that there is no interaction between the variables x and y. Substituting Eq. (2.51) into Eq. (2.50) and dividing the resultant equation by the wave function f
x; y, we get 1 d 2 f
x 1 d 2 g
y k02
er
x; y f
x dx2 g
y dy2
n2eff 0:
2:52
22
ANALYTICAL METHODS
Setting the sum of the second and third terms of Eq. (2.52) equal to k02 N 2
x, we get 1 d 2 g
y k02 er
x; y k02 N 2
x: g
y dy2 This means that the sum of the ®rst and fourth terms is equal to 1 d 2 f
x f
x dx2
k02 n2eff
2:53 k02 N 2
x:
k02 N 2
x:
2:54
Through these procedures we get the two independent equations d 2 g
y k02 er
x; y dy2
N 2
xg
y 0
2:55
d 2 f
x k02
N 2
x dx2
n2eff f
x 0:
2:56
and
The effective index calculation procedure can be summarized as follows: (a) As shown in Fig. 2.2, replace the 2D optical waveguide with a combination of 1D optical waveguides. (b) For each 1D optical waveguide, calculate the effective index along the y axis. (c) Model an optical slab waveguide by placing the effective indexes calculated in step (b) along the x axis. (d) Obtain the effective index by solving the model obtained in step (c) along the x axis. It should be noted that, for the TE mode of the 2D optical waveguide, we ®rst do the TE-mode analysis and then the TM-mode analysis. And, for
2.3
MARCATILI'S METHOD
23
the TM-mode analysis of the 2D optical waveguide, we do these analyses in the opposite order.
2.3 MARCATILI'S METHOD Here, we discuss Marcatili's method for analyzing 2D optical waveguides [3]. Proposed in 1969, it is still in wide use. Figure 2.3 shows a cross-sectional view of a buried optical waveguide. The core has a refractive index n1 , width 2a, and height 2b. It is surrounded by cladding that has a refractive index n2 . In Marcatili's method, it is assumed that the electric ®elds and magnetic ®elds are con®ned to the core and do not exist in the four hatched regions shown in Fig. 2.3. Thus, the continuity conditions for the electric ®elds and the magnetic ®elds are imposed only at the interfaces of regions of 1 and 2, 1 and 3, 1 and 4, and 1 and 5. x mode, which has Ex and Hy as principal ®eld We discuss the Epq y mode, which has Ey and Hx as principal ®eld components, and the Epq components. Here, p and q and are integers and respectively correspond to the numbers of peaks of optical power in the x and y directions. Thus, unlike ordinary mode orders, which begin from 0, they begin from 1.
FIGURE 2.3. Marcatili's method.
24
ANALYTICAL METHODS
x 2.3.1 Epq Mode x The electric ®eld of the Epq mode is assumed to be polarized in the x direction, which results in Ey 0. Since the structure of the optical waveguide is assumed to be invariant in the z direction, the derivative with respect to z is replaced by jb. The component representations shown in Eqs. (2.5)±(2.10) are reduced to
jbEx
@Ez @y
jom0 Hx ;
2:57
@Ez @x
jom0 Hy ;
2:58
@Ex @y
jom0 Hz ;
2:59
@Hz jbHy joe0 er Ex ; @y
2:60
@Hz 0: @x
2:61
jbHx Thus, we get @Hy @x
@Hx joe0 er Ez ; @y
2:62
where 1 @Ez jom0 @y 1 jbEx Hy jom0
Hx
Hz
1 @Ex jom0 @y
@Ez @x
from Eq:
2:57;
2:63
from Eq:
2:58;
2:64
from Eq:
2:59:
2:65
On the other hand, the component representation of the divergence equation = ?
er E 0
2:66
2.3
MARCATILI'S METHOD
25
is @
e E
jb
er Ez 0: @x r x
2:67
Thus, we get the longitudinal electric ®eld 1 @
e E jber @x r x 1 @er 1 @Ex Ex jber @x jb @x 1 @Ex : jb @x
Ez
2:68
Here, we assumed that 1=er @er =@x 0. That is, we ignored the dependence of the relative permittivity er on the coordinate x in each region. This assumption will also be used for each element in the ®nite-element method discussed in Chapter 3. Eliminating Ez by substituting Eq. (2.68) into Eqs. (2.63) and (2.64), we get 1 @Ez 1 @2 Ex from Eq:
2:63 jom0 @y om0 b @x @y 1 @ 1 @Ex Hy jbEx jom0 @x jb @x 1 @2 Ex b2 Ex from Eq:
2:64: om0 b @x2 Hx
2:69
2:70
The above results can be summarized as 1 @2 Ex om0 b @x @y 1 b2 Ex Hy om0 b
Hx
@2 Ex @x2
from Eq:
2:69;
2:71
from Eq:
2:70;
2:72
Hz
1 @Ex jom0 @y
from Eq:
2:65;
2:73
Ez
1 @Ex jb @x
from Eq:
2:68:
2:74
26
ANALYTICAL METHODS
Substituting Eqs. (2.72) and (2.73) into Eq. (2.60), we get a wave x mode such that equation for a principal ®eld component Ex for the Epq @ 2 Ex @ 2 Ex 2 k02
er @x2 @y
n2eff Ex 0:
2:75
The electric ®eld and the magnetic ®eld components to be connected are Ex and Hz at the boundaries y b and are Ez and Hy at the boundaries x a. Since the principal ®eld component Ex is a solution (i.e., wave function) of the wave equation (2.75), we get the following ®eld components in regions 1±5: Ex C1 cos
kx x a1 cos
ky y a2 C2 cos
kx x a1 exp
gy
y C3 exp gx
x
region 1;
2:76
b
region 2;
2:77
a cos
ky y a2
region 3;
2:78
C4 cos
kx x a1 expgy
y b
region 4;
2:79
C5 expgx
x a cos
ky y a2
region 5:
2:80
Substituting these wave functions into the wave equation (2.75), we get (after some mathematical manipulations) the following relations for the wave numbers: kx2 ky2 b2 k02 n21 ;
2:81
g2y b2 k02 n22 ;
2:82
g2x ky2 b2 k02 n22 :
2:83
kx2
Subtracting Eq. (2.81) from Eq. (2.83) and Eq. (2.81) from Eq. (2.82), we get g2x k02
n21
n22
kx2 ;
2:84
g2y k02
n21
n22
ky2 :
2:85
2.3
MARCATILI'S METHOD
27
The next step is to impose the boundary conditions speci®ed by Eqs. (1.55) and (1.56) on the electric and magnetic ®elds. A. Connection at y b: Ex and Hz Setting the electric ®eld components Ex of regions 1 and 2 equal at y b, from Eqs. (2.76) and (2.77), we get C1 cos
ky b a2 C2 :
2:86
And setting the magnetic ®eld components Hz of regions 1 and 2, obtained by substituting Eqs. (2.76) and (2.77) into Eq. (2.73), equal at y b, we get C1 ky sin
ky b a2 C2 gy :
2:87
Dividing Eq. (2.87) by Eq. (2.86), we get tan
ky b a2
gy : ky
Therefore ky b a2 tan
1
! gy q1 p ky
q1 0; 1; . . .:
2:88
On the other hand, setting the electric ®eld components Ex of regions 1 and 4 equal at y b, from Eqs. (2.76) and (2.79), we get C1 cos
ky b a2 C4 :
2:89
And setting the magnetic ®eld components Hz of regions 1 and 4, obtained by substituting Eqs. (2.76) and (2.79) into Eq. (2.73), equal at y b, we get C1 ky sin
ky b
a2 C4 gy :
2:90
28
ANALYTICAL METHODS
Dividing Eq. (2.90) by Eq. (2.89), we get
ky b
a2 tan
1
! gy q2 p ky
q2 0; 1; . . .:
2:91
And adding Eq. (2.88) to Eq. (2.91), we get
ky b tan
1
! gy 12
q ky
1p
q 1; 2; . . .:
2:92
B. Connection at x a: Ez and Hy Setting the electric ®eld components Ez of regions 1 and 3, obtained by substituting Eqs. (2.76) and (2.78) into Eq. (2.74), equal at x a, we get C1 kx sin
kx a a1 C3 gx :
2:93
And setting the magnetic ®eld components Hy of regions 1 and 3, obtained by substituting Eqs. (2.76) and (2.78) into Eq. (2.72), equal at x a, we get
b2 kx2 C1 cos
kx a a1
b2
g2x C3 :
2:94
Dividing Eq. (2.93) by Eq. (2.94), we get tan
kx a a1
b2 kx2 gx :
b2 g2x kx
2:95
Substituting the relations b2 kx2 k02 n21
ky2 ;
2:96
b2
ky2 ;
2:97
g2x k02 n22
2.3
MARCATILI'S METHOD
29
which are obtained from Eqs. (2.81) and (2.83), into Eq. (2.95), we get
tan
kx a a1
k02 n21
k02 n22
ky2 gx : ky2 kx
Therefore
kx a a1 tan
1
k02 n21
k02 n22
! ky2 gx p1 p ky2 kx
p1 0; 1; . . .:
2:98
On the other hand, setting the electric ®eld components Ez of regions 1 and 5, obtained by substituting Eqs. (2.76) and (2.78) into Eq. (2.74), equal at x a, we get C1 kx sin
kx a
a1 C5 gx :
2:99
And setting the magnetic ®eld components Hy of regions 1 and 5, obtained by substituting Eqs. (2.76) and (2.78) into Eq. (2.72), equal at x a, we get
b2 kx2 C1 cos
kx a a1
b2
g2x C5 :
2:100
Substituting Eqs. (2.96) and (2.97) into the equation derived by dividing Eq. (2.99) by Eq. (2.100), we get
tan
kx a
a1
b2 kx2 gx :
b2 g2x kx
Therefore
tan
kx a
a1
k02 n21
k02 n22
ky2 gx ; ky2 kx
30
ANALYTICAL METHODS
and
kx a
a1 tan
1
k02 n21
k02 n22
! ky2 gx p2 p ky2 kx
p2 0; 1; . . .:
2:101
Now, adding Eq. (2.98) to Eq. (2.101), we get
kx a tan
k02 er1
k02 er2
1
! ky2 gx 12
p ky2 kx
1p
p 1; 2; . . .;
2:102
where er1 n21 and er2 n22 . Or, since k0 n1;2 ky for most cases, we get er1 gx kx a tan 12
p er2 kx 1
1p
p 1; 2; . . .:
2:103
The propagation constant b (or the effective index neff ) can be obtained as follows: 1. Obtain kx by using a numerical technique such as a successive bisection method to solve Eq. (2.103) [or Eq. (2.102)]. (It should be noted that in each process of the successive bisection routine Eq. (2.84) can be used to obtain the gx corresponding to kx .) 2. Similarly, obtain ky by making use of Eqs. (2.85) and (2.92). 3. Finally, obtain the propagation constant b from Eq. (2.81). Interesting points are as follows: Equation (2.92) corresponds to the characteristic equation of the TE mode for a three-layer slab optical waveguide parallel to the x axis. Equation (2.103), on the other hand, corresponds to the characteristic equation of the TM mode for a threelayer slab optical waveguide parallel to the y axis. The correspondence is the same as that described in the last part of Section 2.2. The x and y axes are related to each other through Eq. (2.81).
2.3
MARCATILI'S METHOD
31
y 2.3.2 Epq Mode x The magnetic ®eld of the Epq mode is assumed to be polarized in the x direction, which results in Hy 0. The component representations shown in Eqs. (2.5)±(2.10) are reduced to
jbHx
@Hz joe0 er Ex ; @y
2:104
@Hz joe0 er Ey ; @x
2:105
@Hx joe0 er Ez ; @y
2:106
@Ez jbEy @y jbEx @Ey @x
jom0 Hx ;
2:107
@Ez 0; @x
2:108
@Ex @y
2:109
jom0 Hz ;
1 @Hz joe0 er @y 1 Ey jbHx joe0 er
Ex
Ez
@Hz @x
from Eq:
2:104;
2:110
from Eq:
2:105;
2:11
from Eq:
2:106:
2:112
1 @Hx joe0 er @y
On the other hand, the component representation of the magnetic divergence equation =?H0
2:113
is expressed as @Hx @x
jbHz 0:
2:114
32
ANALYTICAL METHODS
Thus, we get the longitudinal magnetic ®eld component Hz
1 @Hx : jb @x
2:115
Eliminating Hz by substituting Eq. (2.115) into Eqs. (2.110) and (2.111), we get 1 @ 1 @Hx 1 @2 Hx from Eq:
2:110;
2:116 Ex joe0 er @y jb @x oe0 er b @x @y 1 @ 1 @Hx Ey jbHx joe0 er @x jb @x 1 @ 2 Hx 2 b Hx from Eq:
2:111:
2:117 oe0 er b @x2 The above results can be summarized as 1 @2 Hx oe0 er b @x @y 1 Ey b 2 Hx oe0 er b Ex
@ 2 Hx @x2
from Eq:
2:116;
2:118
from Eq:
2:117;
2:119
Ez
1 @Hx joe0 er @y
from Eq:
2:112;
2:120
Hz
1 @Hx jb @x
from Eq:
2:115:
2:121
Substituting Eqs. (2.119) and (2.120) into Eq. (2.107), we get the y following wave equation for a principal ®eld component Hx for the Epq mode: @2 Hx @2 Hx 2 k02
er @x2 @y
n2eff Hx 0:
2:122
The electric ®eld and the magnetic ®eld components to be connected are Hx and Ez at the boundaries y b and are Hz and Ey at the boundaries x a.
2.3
MARCATILI'S METHOD
33
Since the principal ®eld component Hx is a solution (i.e., a wave function) of the wave equation (2.122), we get Hx C1 cos
kx x a1 cos
ky y a2
region 1;
2:123
C2 cos
kx x a1 exp gy
y
region 2;
2:124
a cos
ky y a2
region 3;
2:125
C3 exp gx
x
b
C4 cos
kx x a1 expgy
y b
region 4;
2:126
C5 expgx
x a cos
ky y a2
region 5;
2:127
Substituting these wave functions into the wave equation (2.122), we get (after some mathematical manipulations) for the wave numbers the x mode. same equations speci®ed by Eqs. (2.81)±(2.85) for the Epq Equations (2.96) and (2.97) also hold. The next step is to impose the boundary conditions on the tangential electric ®elds and the magnetic ®eld components at the interfaces between different media. A. Connection at y b: Ez and Hx Setting the electric ®eld components Ez of regions 1 and 2, obtained by substituting Eqs. (2.123) and (2.124) into Eq. (2.120), equal at y b, we get C1
gy ky sin
ky b a2 C2 : er1 er2
2:128
And setting the electric ®elds Hx of regions 1 and 2 equal at y b, from Eqs. (2.123) and (2.124), we get C1 cos
ky b a2 C2 : Dividing Eq. (2.128) by Eq. (2.129), we get tan
ky b a2
er1 gy : er2 ky
2:129
34
ANALYTICAL METHODS
Therefore ky b a2 tan
1
! er1 gy q1 p er2 ky
q1 0; 1; . . .:
2:130
On the other hand, setting the electric ®eld components Ez of regions 1 and 4, which are obtained by substituting Eqs. (2.123) and (2.126) into Eq. (2.120), equal at y b, we get C1
ky sin
ky b a2 er1
C4
gy : er2
2:131
Setting the magnetic ®eld components Hx of regions 1 and 4 equal at y b, from Eqs. (2.123) and (2.126), we get C1 cos
ky b a2 C4 :
2:132
Dividing Eq. (2.131) by Eq. (2.132), we get tan
ky b
a2
er1 gy : er2 ky
Therefore ky b
a2 tan
1
! er1 gy q2 p er2 ky
q2 0; 1; . . .:
2:133
Adding Eq. (2.130) to Eq. (2.133), we get ky b tan
1
! er1 gy 12
q er2 ky
1p
q 1; 2; . . .
2:134
B. Connection at x a: Ey and Hz Setting the electric ®eld components Hz of regions 1 and 3, obtained by substituting Eqs. (2.123) and (2.125) into Eq. (2.119), equal at x a, we get C1
b2 kx2 b2 g2x cos
kx a a1 C3 : er1 er2
2:135
2.3
MARCATILI'S METHOD
35
Setting the electric ®eld components Hz of regions 1 and 3, obtained by substituting Eqs. (2.123) and (2.125) into Eq. (2.121), equal at x a, we get C1 kx sin
kx a a1 C3 gx :
2:136
Dividing Eq. (2.136) by Eq. (2.135), we get tan
kx a a1
er2
b2 kx2 gx : er1
b2 g2x kx
2:137
Here, making use of Eqs. (2.96) and (2.97), we get tan
kx a a1
er2
k02 n21 er1
k02 n22
ky2 gx : ky2 kx
Therefore kx a a1 tan
1
er2
k02 n21 er1
k02 n22
! ky2 gx p1 p ky2 kx
p1 0; 1; . . .:
2:138
On the other hand, setting the electric ®eld components Ey of regions 1 and 5, obtained by substituting Eqs. (2.123) and (2.127) into Eq. (2.119), equal at x a, we get C1
b2 kx2 b2 g2x cos
kx a a1 C5 : er1 er2
2:139
And setting the magnetic ®eld components Hz of regions 1 and 5, obtained by substituting Eqs. (2.123) and (2.127) into Eq. (2.121), equal at x a, we get C1 kx sin
kx a
a1 C3 gx :
Dividing Eq. (2.140) by Eq. (2.139), we get tan
kx a
a1
er2
b2 kx2 gx : er1
b2 g2x kx
2:140
36
ANALYTICAL METHODS
And substituting Eqs. (2.96) and (2.97) into this equation, we get tan
kx a
a1
er2
k02 n21 er1
k02 n22
ky2 gx : ky2 kx
Therefore kx a
a1 tan
1
er2
k02 n21 er1
k02 n22
! ky2 gx p2 p ky2 kx
p2 0; 1; . . .:
2:141
Here, adding Eq. (2.138) to Eq. (2.141), we get kx a tan
1
er2
k02 n21 er1
k02 n22
! ky2 gx 12
p ky2 kx
1p
p 1; 2; . . .:
2:142
Or, since k0 n1;2 ky for most cases, we get gx kx a tan 12
p kx 1
1p
p 1; 2; . . .:
2:143
The propagation constant b (or the effective index neff ) is obtained in x mode. just the same way as for the Epq Equation (2.134) corresponds to the characteristic equation of the TM mode for a three-layer slab optical waveguide parallel to the x axis. Equation (2.143), on the other hand, corresponds to the characteristic equation of the TE mode for a three-layer slab optical waveguide parallel to the y axis. The correspondence is the same as that described in the last part of Section 2.2.
2.4 METHOD FOR AN OPTICAL FIBER In this section, we discuss an analysis of a step-index optical ®ber. An optical ®ber consists of a core and a cladding and is axially symmetric. Since the refractive index of the core is slightly higher than that of the cladding, the optical ®eld is largely con®ned to the core. A single-mode
2.4
METHOD FOR AN OPTICAL FIBER
37
FIGURE 2.4. Step-index optical ®ber.
®ber, which has only one guided mode, plays a key role in telecommunications systems. Figure 2.4 shows a cross-sectional view of a step-index optical ®ber. The core of a radius a has a uniform refractive index n1 , slightly higher than the refractive index of the cladding, n2 . Thus, the relative permittivities of the core and cladding are respectively er1 n21 and er2 n22 . The exact vectorial wave equations for the electric ®eld and the magnetic ®eld were shown in Chapter 1 as Eqs. (1.30) and (1.36). Since the structure of the optical ®ber is uniform in the propagation direction, we can, as shown in Eq. (1.39), substitute jb for the derivatives of the electric and magnetic ®elds with respect to z. Thus, we can write H2? E
=er = ? E k02
er Er
n2eff E 0;
2:144
=er 3
=3H k02
er er
n2eff H 0;
2:145
H2? H
where E and H are respectively the electric ®elds and the magnetic ®elds and neff is the effective index to be obtained. Although the structure shown in Fig. 2.4 can be analyzed exactly by using the hybrid-mode analysis [4], the analysis procedure is a little complicated. Fortunately, however, the weakly guiding approximation can be used because the refractive index difference between the core and the
38
ANALYTICAL METHODS
cladding is very small, about 1%. This approximation simpli®es the analysis signi®cantly, and the modes obtained are called linearly polarized modes [5].
2.4.1 Linearly Polarized Modes (LP Modes) A. Field Expressions Since the derivative of the relative permittivity er is small, neglecting the derivatives in the vectorial wave equations (2.144) and (2.145) gives a good approximation. Using this approximation, we can reduce the vectorial wave equations to the scalar Helmholtz equations for the tangential electric ®elds and the magnetic ®elds: H2? E? k02
er
n2eff E? 0;
2:146
H2? H? k02
er
n2eff H? 0:
2:147
Since the optical ®ber is axially symmetric, the Laplacian H2 is rewritten for a cylindrical coordinate system as follows: @2 H2 H2? 2 @z 1 @ @ 1 @2 @2 r 2 2 2 r @r @r r @y @z
@2 1 @ 1 @2 @2 : @r2 r @r r2 @y2 @z2
To solve Eq. (2.146), we assume that the tangential electric ®eld component E? (i.e., Ex or Ey ) is given by Ei
r; y R
rY
y;
2:148
where the subscript i x; y. Substituting the electric ®eld component (2.148) into Eq. (2.146) and dividing the resultant equation by R
rY
y, we get 1 @2 R
r 1 @R
r 1 1 @2 Y
y k02
er R
r @r2 r @r r2 Y
y @y2
n2eff 0:
2.4
METHOD FOR AN OPTICAL FIBER
39
Therefore r2 @2 R
r 1 @R
r r2 k02
er R
r @r2 r @r
1 @2 Y
y :
2:149 Y
y @y2
n2eff
Since the left-hand side of Eq. (2.149) is a function of only the variable r and the right-hand side is a function of only the variable y, both sides have to be constant. Thus, we get r2 d 2 R
r 1 dR
r r2 k02
er r dr R
r dr2
n2eff l 2
2:150
and 1 d 2 Y
y Y
y dy2
l2:
2:151
Equations (2.149) and (2.151) are summarized as d 2 R
r 1 dR
r 2 k0 er dr2 r dr
n2eff
l2 R
r 0 r2
2:152
and d 2 Y
y l 2 Y
y 0: dy2
2:153
The solution of Eq. (2.153) is an oscillation with a single frequency and is expressed as Y
y sin
ly f;
2:154
where l and f are respectively an integer and an arbitrary constant phase. The next step is to solve Eq. (2.152). Using the variable transformations u~ 2 k02
er
n2eff
2:155
and x u~ r;
2:156
40
ANALYTICAL METHODS
we write the ®rst and second derivatives with respect to r as d d dx d u~ dr dx dr dx
and
2 d2 d 2 dx 2 d ~ ~ u u : dr2 dx2 dr dx2
2:157
Equation (2.152) is then rewritten as u~
2d
R
r 1 dR
r 2 k0 u~ 2 u~ x=~u d u~ dx2
l2 R
r 0:
x=~u2
2
Therefore d 2 R
r 1 dR
r 2 k0 1 x dx dx2 as
l2 R
r 0: x2
2:158
Solutions for Eq. (2.158) are lth-order Bessel functions and are written ur 8 ur > BNl < AJl a a R
r wr wr > : CK DIl l a a
for r a; for r a:
2:159
Here, Jl
ur=a and Nl
ur=a are the lth-order Bessel functions of the ®rst and second kinds, and Kl
wr=a and Il
wr=a are the lth-order modi®ed Bessel functions of the ®rst and second kinds. The parameters u2 and w2 are de®ned as u2 k02 a2
er1
n2eff ;
2:160
w2 k02 a2
n2eff
er2 :
2:161
Thus, we get the following very important relation between u and w: u2 w2 v2 ;
2:162
p v k0 a er1 er2
2:163
where
2.4
METHOD FOR AN OPTICAL FIBER
41
is the normalized frequency. The parameters u and w are respectively considered to be the normalized lateral propagation constant in the core and the normalized lateral decay constant in the cladding. The effective index neff has to satisfy the relation n1 neff n2 :
2:164
Since the Bessel function of the second kind, Nl
ur=a, diverges at r 0 and the modi®ed Bessel function of the second kind, Il
wr=a, diverges at r 1, the coef®cients B and D of those functions have to be zero. Thus, from Eq. (2.159), we get 8 ur > < AJl a R
r wr > : CK l a
for r a; for r a:
2:165
B. Characteristic Equation The boundary conditions to be satis®ed by the radial wave function R
r are R
a
0 R
a 0
2:166
and dR
r dr a
0
dR
r : dr a0
2:167
From Eqs. (2.166) and (2.167), we get AJl
u AuJl0
u
CKl
w 0;
2:168
CwKl0
w 0:
2:169
Equations (2.166) and (2.167) can be rewritten as the matrix equation
Jl
u uJl0
u
Kl
w wKl0
w
A C
0:
2:170
42
ANALYTICAL METHODS
When the coef®cients A and C are nontrivial, the determinant of their coef®cient matrix has to be zero such that Jl
u 0 uJ
u l
Kl
w 0: wKl0
w
2:171
Equation (2.171) can be rewritten as wJl
uKl0
w uJl0 Kl
w 0; where the prime denotes the derivative with respect to r. Thus, we get the well-known characteristic equation uJl0
u wKl0
w : Jl
u Kl
w
2:172
The effective index neff can be obtained by solving a combination of Eqs. (2.162) and (2.172). In other words, since u and w are functions of the effective index neff , the characteristic equation (2.172) is a function of neff . Solutions of this equation are called linearly polarized modes (LP modes). 1. Explicit Forms of the Characteristic Equation LP0m MODES (l 0 AND m 1)
Equation (2.172) is rewritten for l 0 as
uJ00
u wK00
w : J0
u K0
w
2:173
Making use of the Bessel function formulas J00
z
J1
z;
2:174
K00
z
K1
z;
2:175
we rewrite Eq. (2.173) as uJ1
u wK1
w : J0
u K0
w
2.4
METHOD FOR AN OPTICAL FIBER
43
Therefore J0
u K
w 0 : uJ1
u wK1
w
2:176
Here, we consider the LPlm mode. Equation (2.172) is rewritten for l 1 as
LPlm MODES (l 1 AND m 1)
uJ10
u wK10
w : J1
u K1
w
2:177
Making use of the Bessel function formulas J10
z J0
z zK10
z
z 1 J1
z;
zK0
z
K1
z;
2:178
2:179
we rewrite Eq. (2.177) as u
J0
u
u 1 J1
u J1
u
wK0
w K1
w : K1
w
Therefore J1
u uJ0
u
K1
w : wK0
w
2:180
In general, making use of the Bessel function formulas Jn0
z Jn 1
z zKn0
z
nz 1 Jn
z;
zKn 1
z
nKn
z;
one can rewrite Eq. (2.177) as u
Jl 1
u lu 1 Jl
u Jl
u
wKl 1
w lKl
w : Kl
w
2:181
2:182
44
ANALYTICAL METHODS
Therefore Jl
u uJl 1
u
Kl
w : wKl 1
w
2:183
2. Value Range of u The characteristic equations for LPlm modes have solutions only within limited ranges of the parameter u, and we need to know what these ranges are. These ranges can be determined by investigating the limits w ! 0 and w ! 1, where the former corresponds to u ! v . Through this process, the single-mode condition and cutoff conditions for the higher order LPlm modes will also be clari®ed. LP0m MODES (l 0 AND m 1) First, we investigate the limit w ! 0 (i.e., u ! v). Since the zeroth-order and lth-order modi®ed Bessel functions of the ®rst kind can, for the limit of z ! 0, be respectively expressed asymptotically as
K0
z
ln z;
2:184
Kl
z 12 G
l
12 z
1
for l > 0:
2:185
The right-hand side of Eq. (2.176) can be rewritten as K0
w ln w wK1
w w
1=2G
1
1=2w
1
ln w ! 1 for w ! 0:
The left-hand side of Eq. (2.176) also has to go to 1. That is, J0
v ! 1: vJ1
v
2:186
The possible solutions for Eq. (2.186) are v ! 0 and J1
v ! 0. Since we get J0
v ! 1 and J1
v ! 0 for v ! 0, Eq. (2.186) holds. This implies that the cutoff value vc for the normalized frequency v is zero. In other words, the LP01 mode has no cutoff condition. Next, we discuss the cutoff conditions for higher order modes: LP0m modes where m 2. Here, we assume that j1;m 1 is the
m 1th zero of the Bessel function of the ®rst kind. That is, J1
j1;m 1 0. Since the signs of J0
v and J1
v are the
2.4
METHOD FOR AN OPTICAL FIBER
45
same for the limit v ! j1;m 1 0, Eq. (2.186) holds. This means the cutoff value vc of the LP0m mode is given by vc j1;m 1 :
2:187
Thus, we can summarize the cutoff conditions for LP0m modes as follows: LP01 mode:
vc 0;
LP0m mode:
vc j1;m
1
for m 2:
2:188
On the other hand, when w 1, the asymptotic expansion of the lthorder modi®ed Bessel function of the ®rst kind is r p e Kl 2 r p e 2 r p e 2
w
w
w
1
l; n P n n0
2w
l; 0
2w0 :
2:189
Since for w 1 the right-hand side of Eq. (2.176) can be rewritten as p w p=2e K0
w 1 p w ! 0; wK1
w w p=2e w
2:190
the left-hand side of Eq. (2.176) also has to go to zero. That is, J0
u ! 0: uJ1
u
2:191
This implies that the asymptotic value of u is given by u j0;m :
2:192
46
ANALYTICAL METHODS
Thus, we can summarize the asymptotic values of u for LP0m modes as follows: LP01 mode:
u j0;1 ;
LP0m mode:
u j0;m
for m 2:
2:193
As in the above discussion, we ®rst investigate the limit w ! 0 (i.e., u ! v). For the limit z ! 0, the lthorder modi®ed Bessel functions of the ®rst kind were shown in Eqs. (2.185) and (2.185). The
l 1th-order modi®ed Bessel function of the ®rst kind for the limit z ! 0 is LPlm MODES (l 1 AND m 1)
Kl 1
z 12 G
l
1
12 z
l1
for z ! 0:
2:194
Making use of this approximation, we express the right-hand side of Eq. (2.183) as Kl
w wKl 1
w
1
1=2G
l
1=2w l w
1=2G
l 1
1=2w l1
1
l 1!
1=2w l w
l 2!
1=2w l1
1
l w
2
l 1 ! w2
1
2 w 1 for w ! 0:
The left-hand side of Eq. (2.183) also has go to Jl
v ! vJl 1
v
1:
2:195
1. That is,
2:196
The possible solutions of Eq. (2.196) are v ! 0 and Jl 1
v ! 0. Since the left-hand side of Eq. (2.183) diverges to 1 for the limit v ! 0, the limit v ! 0 cannot be a solution. On the other hand, since the signs of Jl
v and Jl 1
v are opposite for v ! jl 1;m 0, jl 1;m can be a solution for Eq. (2.196). Here, jl 1;m is the zero of the
l 1th-order Bessel function of the ®rst kind, Jl 1
v.
2.4
METHOD FOR AN OPTICAL FIBER
47
The cutoff condition for LPlm modes is LPlm mode:
vc jl
1;m :
2:197
On the other hand, since, according to the asymptotic expression (2.189), the right-hand side of Eq. (2.183) is 0 for w 1, the left-hand side of Eq. (2.176) also has to go to zero. That is, Jl
u ! 0: uJl 1
u
2:198
This implies that the asymptotic value of u is given by u jl;m :
2:199
We can thus summarize the possible value ranges of the parameter u as LP01 mode:
0 u < j0;1 ;
LP0m mode:
j1;m
LPlm mode:
jl
1
1;m
u < j0;m
for m 2;
u < jl;m
for l 1; m 1:
2:200
It should be noted that since the minimum value of the zeros of the Bessel functions is j0;1 (2.404826), the single-mode condition for an optical ®ber with a step index is given by the cutoff value for the LP11 mode: vc j0;1 :
2:201
Figure 2.5 shows examples of u±o curves. The crossing points of the curve u2 w2 v2 and the other curves give the effective indexes and con®rm the value ranges of u speci®ed above. Figure 2.6 shows examples of calculated ®eld distributions. It should be noted that l and m of LPlm respectively correspond to the number of dark lines in the azimuthal direction and the number of bright peaks in the radial direction. 2.4.2 Hybrid-Mode Analysis This section discusses a more exact analysis for the step-index optical ®ber. Since the above LP-mode analysis can meet the ordinary demands for ®ber analyses and the hybrid-mode analyses are very specialized, some readers may wish to skip this section.
48
ANALYTICAL METHODS
FIGURE 2.5. Relation between u and w (v 4).
FIGURE 2.6. Field distributions for the LP mode (v 20): (a) LP0;1 ; (b) LP1,1; (c) LP2,2; (d) LP5,2.
A. Field Expressions The cylindrical electric ®eld E and the cylindrical magnetic ®eld H are expressed as E
r; y; z E
r; y exp j
ot
bt;
2:202
H
r; y; z H
r; y exp j
ot
bt;
2:203
2.4
METHOD FOR AN OPTICAL FIBER
49
where E0 E r r Ey u E z z;
2:204
H0 H r r H y u H z z:
2:205
Here, r, u, and z are respectively unit vectors in the radial, azimuthal, and longitudinal directions. Applying the rotation formula 1 @Az =3A r @y
@Ay @Ar r @z @z
@Az 1 @
rAy u r @r @r
1 @Ar z r @y
2:206
for a vector A Ar r Ay u Az z to the Maxwell equations =3E
jom0 H;
=3H
joe0 er E
2:207 joeE;
2:208
1 @Ez jbEy r @y
jom0 Hr ;
2:209
@Ez @r
jom0 Hy ;
2:210
1 @Er r @y
jom0 Hz ;
2:211
we get
jbEr 1 @
rEy r @r
1 @Hz jbHy joeEr ; r @y jbHr 1 @
rHy r @r
2:212
@Hz @r
joeEy ;
2:213
1 @Hr r @y
joeEz :
2:214
50
ANALYTICAL METHODS
Expressing the tangential ®eld components (Er , Ey , Hr , and Hy ) as functions of the longitudinal ®eld components (Ez and Hz ), we get 1 @Ez ; r @y
2:215
@Ez ; @r
2:216
jbHy
1 @Hz ; r @y
2:217
joeEy jbHr
@Hz : @r
2:218
jom0 Hr
jbEy
jbEr jom0 Hy joeEr
The radial and azimuthal ®eld components are obtained as follows: Equation
2:216 b Eq.
2:217 om0 : Er
@Ez 1 @Hz om0 : b r @y @r b2
j o2 em
2:219
Equation
2:215 b Eq.
2:218 om0 : Ey
j b2
o2 em
@Hz om0 : @r
2:220
1 @Ez oe : r @y
2:221
1 @Hz @Ez oe : b r @y @r
2:222
1 @Ez b r @y
Equation
2:215 oe Eq.
2:218 b: Hr
j b2
o2 em
@H b z @r
Equation
2:216 oe Eq.
2:217 b: Hy
j o2 em
b2
Substituting the Hr of Eq. (2.221) and the Hy of Eq. (2.222) into Eq. (2.214), we get @2 Ez 1 @Ez 1 @2 Ez
o2 em r @r r2 @y2 @r2
b2 Ez 0:
2:223
2.4
METHOD FOR AN OPTICAL FIBER
51
And substituting the Er of Eq. (2.219) and the Ey of Eq. (2.220) into Eq. (2.211), we get @2 Ez 1 @Ez 1 @2 Ez
o2 em r @r r2 @y2 @r2
b2 Ez 0:
2:224
To solve Eqs. (2.223), we assume that the longitudinal ®eld components Ez and Hz are given by Ez
r; yor Hz
r; y Rz
rYz
y:
2:225
Thus, we get the following wave equations for Rz
r and Yz
y: d 2 Rz
r 1 dRz
r 2 k0 er dr2 r dr
n2eff
n2 R
r 0 r2 z
2:226
and d 2 Yz
y n2 Yz
y 0: 2 dy
2:227
The solution of Eq. (2.227) is an oscillation with a single frequency and is expressed as Yz
y sin
ny f;
2:228
where n and f are respectively an integer and an arbitrary constant phase. Through the procedure shown for the LP mode, we obtain the radial wave function Rz
r in the core as the Bessel function of the ®rst kind, Jn
ur=a, and obtain the radial wave function in the cladding as the
52
ANALYTICAL METHODS
modi®ed Bessel function of the ®rst kind, Kn
ur=a. Thus, we ®nally get the following ®eld components: 8 ur > > AJ < n a sin
ny f for r a; Ez wr > > : CKn sin
ny f for r a; a 8 ur > > for r a; BJ < n a cos
ny f Hz wr > > : DKn cos
ny f for r a: a
2:229
2:230
Substituting Eqs. (2.229) and (2.230) into Eqs. (2.219)±(2.222), we get the wave functions as follows: 1. In the core (r a): ur sin
ny f;
2:231 a jb 0 ur jom0 n ur J Jn A Er B sin
ny f;
2:232 u=a n a a
u=a2 r jb n ur jom0 0 ur Ey A J Jn B cos
ny f; u=a n a a
u=a2 r Ez AJn
2:233 ur Hz BJn cos
ny f;
2:234 a joe1 n ur jb 0 ur Hr A Jn B J cos
ny f;
2:235 a u=a n a
u=a2 r joe1 0 ur jb n ur Jn sin
ny f: J Hy A B a u=a n a
u=a2 r
2:236
2.4
METHOD FOR AN OPTICAL FIBER
53
2. In the cladding (r a): wr Ez CKn sin
ny f;
2:237 a jb 0 wr jom0 n wr Er C Kn D Kn sin
ny f; w=a a a
w=a2 r Ey C
wr
jb n Kn a
w=a2 r
D
wr
jom0 0 K w=a n a
2:238 cos
ny f;
2:239
wr
Hz DKn cos
ny f;
2:240 a joe2 n ur jb 0 wr K K cos
ny f; Hr C D n a w=a n a
u=a2 r
2:241 joe2 0 wr jb n wr Hy C Kn sin
ny f: Kn D 2 a a w=a
w=a r
2:242 The normalized lateral propagation constant u in the core and the normalized lateral decay constant w in the cladding were respectively de®ned in Eqs. (2.160) and (2.161).
B. Characteristic Equation The boundary conditions to be satis®ed are that each of the tangential ®eld components (Ez , Ey , Hz , and Hy ) is continuous at r a. They are expressed as Ez
a
0; y Ez
a 0; y;
2:243
Ey
a
0; y Ey
a 0; y;
2:244
Hz
a
0; y Hz
a 0; y;
2:245
Hy
a
0; y Hy
a 0; y;
2:246
54
ANALYTICAL METHODS
Substituting the equations for the ®eld components [i.e., Eqs. (2.231)± (2.242)], into the equations for the boundary conditions [i.e., Eqs. (2.243)±(2.246)], we get the matrix equation 0
Jn
B B B B B B B B @
jb n Jn
u=a2 a 0
0
Kn
0
jom0 0 J u=a n
jb n Kn
w=a2 a
jom0 0 K w=a n
Jn
joe1 0 J u=a n
0
jb n Jn
u=a2 a
joe2 0 K w=a n
Kn jb n Kn
w=a2 a
1
0 1 C A C CB C CB B C CB C 0: CB C C@ C A C A D
2:247
When the coef®cients A, B, C, and D have nontrivial solutions, the determinant of the coef®cients of Eq. (2.247) has to be zero. After some mathematical manipulations, we get the characteristic equation
Jn0 K0 n uJn wJn
e1 Jn0 Kn0 1 e1 1 1 2 1 n :
2:248 u2 w2 e2 u2 w2 e2 uJn wJn
This is the characteristic equation for the hybrid mode (i.e., Ez 6 0 and Hz 6 0). Since u and w are functions of the effective index neff , this characteristic equation is also a function of neff . For n 0, Eq. (2.248) is reduced to
Jn0 K0 n 0 uJn wJn
2:249
e1 Jn0 Kn0 0: e2 uJn wJn
2:250
or
It can be shown that Eqs. (2.249) and (2.250) respectively correspond to the TE mode (Ez 0) and the TM mode (Hz 0).
PROBLEMS
55
PROBLEMS 1. Derive the expressions for the power con®nement factors (G factors) in the core for the TE mode and the TM mode of the three-layer slab optical waveguide shown in Fig. 2.1.
ANSWER a. TE mode. The principal electric ®eld component is Ey and the other ®eld components are Ex Ez Hy 0; b E; Hx om0 y j @Ey Hz : om0 @x
P2:1
P2:2
P2:3
Since the complex Poynting vector S is de®ned as S 12 E3H*;
P2:4
the power propagating in the z direction, Sz , is given by Sz 12
E3H*z
1 2 Ey H*x dx
b jEy j2 dx: 2om0
P2:5
The power con®nement factor in the core is thus
W
GTE
1=2om0 jEy j2 dx 0 :
1 2
1=2om0 jEy j dx 1
P2:6
56
ANALYTICAL METHODS
b. TM mode. The principal electric ®eld component is Hy is and the other ®eld components are Hx Hz Ey 0; b H; Ex oe0 er y b @Hx Ez : joe0 er @x
P2:7
P2:8
P2:9
The power propagating in the z direction, Sz , is given by Sz
1 2
Ex H*y dx
b 1 jH j2 dx: 2oe0 er y
P2:10
The power con®nement factor in the core is thus
GTM
W 1
1=2oe0 jHy j2 dx e 0 r :
1 1 2
1=2oe0 jHy j dx 1 er
P2:11
2. The characteristic equations for the LP0m mode, Eq. (2.176), and for the LPlm mode, Eq. (2.183), were derived separately. Show that, when l 0, Eq. (2.176) is included in Eq. (2.183).
ANSWER Substituting l into Eq. (2.183), we get J0
u K0
w : uJ 1
u wK 1
w
P2:12
REFERENCES
57
Since the Bessel functions here have the formulas J n
z
1 n Jn
z;
P2:13
K n
z Kn
z;
P2:14
J 1
z
P2:15
assuming n 1, we get J1
z;
K 1
z K1
z:
P2:16
The characteristic equation (2.176) can be derived by substituting Eqs. (P2.15) and (P2.16) into Eq. (P2.12). REFERENCES [1] T. Tanaka and Y. Suematsu, ``An exact analysis of cylindrical ®ber with index distribution by matrix method and its application to focusing ®ber,'' Trans. IECE Jpn., vol. E-59, no. 11, pp. 1±8, 1976. [2] K. Kawano, Introduction and Application of Optical Coupling Systems to Optical Devices, 2nd ed., Gendai Kohgakusha, Tokyo, 1998 (in Japanese). [3] E. A. Marcatili, ``Dielectric rectangular waveguide and directional coupler for integrated optics,'' Bell Syst. Tech. J., vol. 48, pp. 2071±2102, 1969. [4] E. Snitzer, ``Cylindrical dielectric waveguide modes,'' J. Opt. Soc. Am., vol. 51, pp. 491±498, 1961. [5] D. Gloge, ``Weakly guiding ®bers,'' Appl. Opt., vol. 10, pp. 2252±2258, 1971.
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 3
FINITE-ELEMENT METHODS Like the ®nite-difference methods (FDMs) that will be discussed in Chapter 4, the ®nite-element methods (FEMs) [1±3] are widely utilized numerical methods. The scalar (SC) FEM has several advantages over the fully vectorial (V) FEM. The main ones are that the SC-FEM has no spurious problem and the matrixes in the eigenvalue equation are small and symmetrical These contribute to numerical ef®ciency. Here, we discuss the use of the SC-FEM in the 2D cross-sectional analysis of optical waveguides.
3.1 VARIATIONAL METHOD As shown in Fig. 3.1, there are two kinds of ways that can be used to solve optical waveguide problems [2±4]: the variational method and weighted residual method, of which the Galerkin method is representative. Both the variational and the weighted residual methods eventually require that the same matrix eigenvalue equations be solved. This section will focus on a variational method; the next section will discuss a weighted residual method. In the variational method, the wave equation is not directly solved. Instead, the analysis region is divided into many segments and the variational principle is applied to the sum of the discretized functionals for all segments. 59
60
FINITE-ELEMENT METHODS
FIGURE 3.1. Analysis based on the ®nite-element method.
Figure 3.2 shows an analysis region O surrounded by a boundary G. Here, n is the outward-directed unit vector normal to the surface of the analysis region O. A variational method for obtaining the effective index neff is ®rst discussed here by using the scalar wave equation [2,3] @2 f @2 f k02
er @x2 @y2
n2eff f 0:
3:1
Multiplying this equation by variation df of the function f and integrating the product over the whole analysis region O, we get
2
@ f @2 f df df k02
er dx dy @x2 @y2 O O
n2eff fdx dy 0:
3:2
Here, applying the partial integration with respect to variables x and y, we get
@f @f dy df dx df @x @y G G
@ df @f @ df @f dx dy df k02
er @x @x @y @y O O
FIGURE 3.2. Analysis region.
n2eff f dx dy 0;
3.1
61
VARIATIONAL METHOD
where the terms inside the brackets are those whose integration orders with respect to x and y were reduced from 2 to 1 because of the partial integration. Summarizing the second and the third terms, we get
@f @f dy df dx df @x @y G G
@ df @f @ df @f @x @x @y @y O
df
k02
er
n2eff f
dx dy 0:
In addition, substituting the relations ( ) 2 @ df @f @f @ df @f @f 1 @f d d ; @x @x @x @x @x @x 2 @x ( ) 2 @ df @f 1 @f d ; @y @y 2 @y 1 2 f df d f 2 into the above equation, we get
@f @f dy f dx d f @x @y G G "
( 2 2 1 @f @f d k02
er 2 @x @y O
) n2eff f2
dx dy 0:
3:3
Here, we introduce the following function I : 1 I 2
"
(
2 2 @f @f k02
er @x @y O
@f @f f dy f dx : @x @y G G
) n2eff f
2
# dx dy
3:4
62
FINITE-ELEMENT METHODS
Since the line integral calculus term can be rewritten as
@f f dy @x G
@f @f f dy dx @x @y G
@f dG; f @n G
@f f dx @y G
3:5
I can be rewritten as 1 I 2
"
( 2 2 @f @f k02
er @x @y O
) 2
n2eff f
# dx dy
@f f dG ; @n G
3:6
where @=@n is the derivative with respect to the normal vector n. Here, I is a function of f, which is a function of x and y. A function of a function is generally called a functional. Using the functional I , we can rewrite Eq. (3.3) as dI 0;
3:7
which means that the stationary condition is imposed on the functional I . We can instead ®rst de®ne the functional I given by Eq. (3.4) or (3.6). Then, imposing the stationary condition on the functional I , we get
df
O
@2 f @2 f k02
er @x2 @y2
n2eff f dx dy 0:
3:8
When Eq. (3.8) holds for an arbitrary variation df of the function f, the wave equation (3.1) has to hold. Thus, the wave equation (3.1) is obtained by imposing the stationary condition on the functional I . This means that solving the wave equation (3.1) is equivalent to setting the variation dI of the functional I , which can be obtained by Eq. (3.4) or (3.6), to zero, that is, to imposing the stationary condition on the functional I . This is called the variational principle, and a method for solving problems by using the variational principle is called a variational method. In the Rayleigh±Ritz method, the unknown function is formed by a linear combination of known basis functions that satisfy the boundary conditions, and the variational principle is applied to the functional.
3.1
VARIATIONAL METHOD
63
The calculation procedure for the application of the variational method to the FEMs is summarized as follows: The analysis region is ®rst divided into segments, which are called elements, and the functional Ie is calculated for each element e. Then, the total functional I for the whole analysis region is obtained by summing up the functional Ie for all elements: P
3:9 I Ie : e
The ®nal eigenvalue matrix equation is obtained by imposing the stationary condition on the functional I . It should be noted that the difference between an ordinary variational method and an FEM is that the former treats the analysis region as one area and the latter treats the region as the sum of elements. Since the total functional is a linear combination of the functionals for the elements, the variation dI of the whole system is a sum of variation dIe of each element e. Thus, Eq. (3.7) can be rewritten as P
3:10 dI dIe 0: e
The functional Ie of an element e surrounded by the boundary Ge is given by "
( ) # 2 2 1 @fe @fe k02
er n2eff f2e dx dy Ie 2 @x @y e "
# @fe dG :
3:11 fe @n Ge Next, the wave function fe is expanded as fe
Me P i1
Ni fei Ne T ffe g
3:12
by using the basis function Ne in element e, where Me is the number of nodes in element e and T is the transposing operator for a matrix. Then the basis function Ne and the expansion coef®cient ffe g are expressed as Ne N1 ffe g
f1
N2 f2
N3 f3
NMe T ; T
fMe :
3:13
3:14
64
FINITE-ELEMENT METHODS
In the FEMs, a basis function Ni is called a shape function or an interpolation function. As discussed later, the expansion coef®cient fi corresponds to a ®eld component at each node. Two-dimensional cross-sectional analyses of optical waveguides often use ®rst-order triangular elements having three nodes or second-order triangular elements having six nodes. We make use of the equation T 2 T @fe @fe @fe @Ne T ffe g @Ne T ffe g @x @x @x @x @x
@ffe gT Ne @Ne T ffe g @Ne @Ne T ffe gT ffe g; @x @x @x @x
3:15
where @fe =@x is a scalar quantity and is rewritten as
@fe =@xT . Similarly, we can rewrite
@fe @y
2
ffe gT
@Ne @Ne T ffe g @y @y
3:16
and f2e
fe T fe ffe gT Ne Ne T ffe g;
3:17
where we made use of the fact that fe is also a scalar quantity. The functional Ie given by Eq. (3.11) can thus be rewritten as
@Ne @Ne T @Ne @Ne T @x @y @x @y e "
# @f fe e dG k02
er n2eff Ne Ne T dx dyffe g @n Ge "
# 1 @fe 2 T ffe g
Ae l Be ffe g fe dG :
3:18 2 @n Ge
1 Ie ffe gT 2
3.1
VARIATIONAL METHOD
65
Here, matrixes Ae and Be and the quantity l2 are given by
Ae
Be
e
e
@Ne @Ne T @Ne @Ne T @x @y @x @y
dx dy;
Ne Ne T dx dy;
3:19
3:20
and l2 k02
er
n2eff :
3:21
Since the functional Ie given by Eq. (3.18) is obtained for only element e, we have to sum up all the elements to obtain the functional I for the whole analysis region. According to Eq. (3.9), 1P ff gT
Ae I 2 e e 1 ffgT
A 2
"
P
@f l Be ffe g fe e dG @n e Ge "
# P @f l2 Bffg fe e dG ; @n e Ge 2
#
3:22
where ffg A B
P e
P e
P e
ffe g;
3:23
Ae ;
3:24
Be :
3:25
Now, consider the second term of Eq. (3.22), which is the line integral calculus term and can be expressed as follows: "
P e
# @fe fe dG @n Ge
P e
@f fe1 e1 dG @n Ge1
! @fe fe dG : @n Ge
Here, we assume that the wave function fe and its derivative with respect to the normal to the surface of element e, @fe =@n, are continuous at the boundaries with neighboring elements. Since, under this assumption, the line integral calculus terms inside the analysis region are canceled out,
66
FINITE-ELEMENT METHODS
the line integral calculus term of the periphery of the whole analysis region remains, which will be discussed in Section 3.4. Thus, the second term of Eq. (3.22) can be reduced to
@f f dG : @n G
3:26
Finally, the functional I for the whole analysis region given by Eq. (3.22) is obtained as I
T 1 2 ffg
A
2
l Bffg
@f f dG : @n G
3:27
Next, we impose on the functional I at the boundaries the Dirichlet condition f0
3:28
@f 0: @n
3:29
or the Neumann condition
Using these boundary conditions, we can reduce the functional I to the simple form 1P ff gT
Ae l2 Be ffe g 2 e e 1 ffgT
A l2 Bffg: 2
I
3:30
Now that we have obtained the functional I , the next step is to impose the variational principle on it. Although having a total of only two nodes is impossible for 2D cross-sectional optical waveguides, this case will be used here for ease of understanding. The wave function vector ffg and matrixes S, A, and B for the eigenvalue l2 are expressed as ffg
f1 f2
3:31
3.1
VARIATIONAL METHOD
67
and S A
2
l B
S12 : S22
S11 S12
3:32
Here, we make use of the symmetry for S. [The symmetry of the matrix, S ST , will be discussed later.] Thus, we get I 12 ffgT Sffg 12
f1 f2 12
f1 f2
S11
S12
S12
S22
!
f1
!
3:33
f2 ! S11 f1 S12 f2
S12 f1 S22 f2
12
S11 f21 2S12 f1 f2 S22 f22 :
3:34
The derivatives of Eq. (3.34) with respect to f1 and f2 are @I S11 f1 S12 f2 ; @f1
3:35
@I S12 f1 S22 f2 : @f2
3:36
And the matrix expression for these derivatives is @I f @ 1 f2
S11 S12
S12 S22
f1 f2
3:37
or @I
A @ffg
l2 Bffg:
3:38
68
FINITE-ELEMENT METHODS
This is the same equation we get for general cases. Since solving the wave equation by using the FEM with the variational principle is the imposition of the stationary condition on the functional I as @I 0; @ffg
3:39
it is equivalent to solving the eigenvalue matrix equation
A
l2 Bffg 0:
3:40
3.2 GALERKIN METHOD Since the functional is used to solve the problems, we have to ®nd it when we use the FEM with the variational principle. On the other hand, the partial differential equations governing almost all the physical phenomena that we encounter are already known. This is certainly true with regard to the wave equations for the electromagnetic ®elds and for the SchroÈdinger equation, solutions of which are the targets of this book. The weighted residual methods, especially the Galerkin method, is quite powerful for solving them. The Galerkin method is widely used not only in the FEM but also in methods of microwave analyses, such as the spectral domain approach (SDA) [5, 6]. Since the wave function f is a true solution for the wave equation (3.1), the right-hand term of Eq. (3.1) is de®nitely zero. The true wave function f, however, cannot actually be known; we can obtain only an approximate When the true wave function f in Eq. (3.1) is replaced wave function f. by the approximate one, the right-hand term does not become zero but generates the error R, which is called the error residual: @2 f @2 f k02
er @x2 @y2
n2eff f R:
3:41
It is natural to think that the difference between f and f can be decreased by averagely setting the error residual R equal to zero in the whole analysis region. As is well known, the electromagnetic ®elds concentrate mostly in the core where the refractive index is higher than in the cladding. Thus, some weighting should be used when setting R equal to zero. Introducing the weight function c, we get
cR dx dy 0:
3:42
3.2 GALERKIN METHOD
69
Rewriting the error residual R explicitly, we get
( 2 @ f @2 f k02
er c @x2 @y2
) n2eff f
dx dy 0:
3:43
The procedure discussed above is the weighted residual method. Partially integrating Eq. (3.43) with respect to x and y, we get "
#
!
@f @f @c @f @c @f dy c dx dx dy c @x @y @x @x @y @y
ck02
er n2eff f dx dy 0;
3:44
which can be rewritten as "
#
! @f @c @f @c @f dG c dx dy @n @x @x @y @y G
ck02
er n2eff f dx dy 0:
3:45
Here, the relation given by Eq. (3.5) was used, and G dG and @=@n are respectively the line integral calculus at the boundary G and the derivative with respect to the normal vector n. The rank of the integral calculus of the ®rst term inside the square brackets in Eq. (3.45) is decreased by 1 as a result of the partial integration. The rank of the derivatives of the second term is also decreased from 2 to 1, and these derivatives are called the weak forms. The weighted residual method, in which both the approximate wave function f and the weight function c are expanded by the same basis functions, is called the Galerkin method. In using the FEM, we ®rst divide the analysis region into many elements, then apply the Galerkin method to each element, then sum up the contributions of all the elements. The expansion coef®cients obtained as an eigenvector correspond to the ®elds at nodes in the analysis region.
70
FINITE-ELEMENT METHODS
Since the calculation procedure will be discussed in detail later, here we simply summarize it. The equation for element e in the divided elements is expressed from Eq. (3.45) as "
#
! @f e @ce @f e @ce @f e dG ce dx dy @n @y @y Ge e @x @x
ce k02
er n2eff f e dx dy 0:
3:46 e
Here, it should be noted that f e and ce in element e are expanded by using the same basis functions: P f e fei Ni Ne T ffe g; Me i
ce
Me P i
fei Ni Ne T ffe g;
3:47
3:48
where Me is the number of nodes in e. The Ne and ffe g were de®ned in Eqs. (3.13) and (3.14). The basis function Ni and the expansion coef®cient fei correspond to the shape function and the ®eld component. Substituting Eqs. (3.47) and (3.48) into Eq. (3.46), we get
( @Ne @Ne T @Ne @Ne T T ffe g @x @y @x @y e ) "
# @ f f e e dG 0 k02
er n2eff Ne Ne T dx dyffe g @n Ge
3:49 for element e. Applying the de®nitions for Ae , Be , and l2 shown in Eqs. (3.19) to (3.21), we can rewrite the above equation as "
# @ f 2 e T f e ffe g
Ae l Be ffe g dG 0 @n Ge or T
ffe g
Ae
2
l Be ffe g
"
# @ f e f e dG 0: @n Ge
3:50
3.2 GALERKIN METHOD
71
Since Eq. (3.50) gives the contribution of only element e, it is necessary to sum up the contributions of all the elements in the analysis region. Thus, we get P ffe gT
Ae e
"
P
2
l Be ffe g
e
# @ f e dG 0 f e @n Ge
or T
ffg
A
"
P
2
l Bffg
# f @fe dG 0; e @n Ge
e
3:51
where the de®nitions ffg
P ffe g; e
A
P e
Ae ;
B
P e
Be
3:52
in Eqs. (3.23)±(3.25) were used. With respect to the second term in Eq. (3.51), we assume that the wave function fe and its normal derivative @fe =@n are continuous at the boundaries between elements. This assumption cancels out the line integral calculus terms, so the second term in Eq. (3.51) can be reduced to ! f @f dG : @n G
3:53
Substituting Eq. (3.53) into (3.51), we get ffgT
A
l2 Bffg
! @ f dG 0: f @n G
3:54
When the Dirichlet condition or the Neumann condition is the boundary condition, the second term in Eq. (3.51) becomes zero and Eq. (3.54) is simpli®ed to ffgT
A
l2 Bffg 0:
3:55
72
FINITE-ELEMENT METHODS
We ®nally get the following eigenvalue matrix equation to be solved:
A
l2 Bffg 0:
3:56
Comparing Eq. (3.56) with Eq. (3.40), one can see that the eigenvalue matrix equation for the Galerkin method is identical to that for a variational method. 3.3 AREA COORDINATES AND TRIANGULAR ELEMENTS We have roughly discussed the calculation procedures for the variational method and Galerkin method. Before describing the detailed formulations of the global matrixes for the eigenvalue matrix equations, we have to investigate the elements, which are indispensable when we divide the analysis region into segments. In the analysis of 2D cross-sectional structures, triangular elements using polynomials are generally used to approximate ®eld distributions. The concept of polynomial approximations is illustrated in Fig. 3.3. In Fig. 3.3a, the true wave function is approximated by a linear function, and, in Fig. 3.3b, it is approximated by a quadratic function. Although the higher order polynomial approximations can bring about more accurate results, they result in a larger number of nodes at which optical ®elds are de®ned. In addition, when there are more nodes, we need to use a more complicated mathematical analysis and more computer memory. Discussions in this book are therefore limited to two widely utilized triangular elements: the ®rst-order triangular element, which requires three nodes, and the second-order triangular element, which requires six nodes.
FIGURE 3.3. Approximations by polynomial functions: (a) linear function; (b) quadratic function.
3.3 AREA COORDINATES AND TRIANGULAR ELEMENTS
73
FIGURE 3.4. First-order triangular element.
3.3.1 First-Order Triangular Elements Figure 3.4 shows a ®rst-order triangular element. In this element, the wave function f
x; y at an arbitrary coordinate
x; y inside the element is expanded using shape functions N1 , N2 , and N3 (with ®elds f1 , f2 , and f3 ) at the vertexes on which nodes are placed: f
x; y N1 f1 N2 f2 N3 f3 NT ffg;
3:57
where N N1 N2 N3 T and ffg
f1 f2 f3 T . The shape functions N and the ®eld vectors ffg respectively correspond to the basis functions and the expansion coef®cients. The coordinates of nodes 1, 2, and 3 are respectively
x1 ; y1 ,
x2 ; y2 and
x3 ; y3 . In determining the explicit forms of shape functions N1 , N2 , and N3, the use of area coordinates is convenient. Figure 3.5 shows a triangle we use here for discussing the area coordinates. An arbitrary coordinate in this triangle is denoted by p
x; y. Figure 3.6 shows another triangle formed by nodes denoted 1, 2, and 3. As is well known, the area S123 of the triangle is given by S123 12 jA3Bj:
3:58
When we assume that i, j, and k are unit vectors in the x, y, and z directions, the vector A from node 1 to node 2 and the vector B from node 1 to node 3 are expressed as A
x2
x1 i
y2
y1 j;
B
x3
x1 i
y3
y1 j:
3:59
74
FINITE-ELEMENT METHODS
FIGURE 3.5. Area coordinates.
Using these expressions, we get i A3B x2 x1 x x1 3 x2 x1 x3 x1
j y2
y1
y3
y1
y2 y3
k 0 0
y1 k: y 1
3:60
Thus, we can obtain the area S123 of the triangle 123 as 1 x S123 2 2 x3
x1 x1
y2 y3
y1 : y1
FIGURE 3.6. Area of a triangle.
3:61
3.3 AREA COORDINATES AND TRIANGULAR ELEMENTS
75
We similarly get area S23p of triangle 23p formed by nodes 2 and 3 and point p; area S1p3 of triangle 1p3 formed by node 1, point p, and node 3; and area S12p of triangle 12p formed by nodes 1 and 2 and point p: 1 x2 x y2 y S23p ;
3:62 2 x3 x y3 y 1 x3 x y3 y S1p3 ;
3:63 2 x1 x y1 y 1 x2 x y2 y :
3:64 S12p 2 x1 x y1 y Area coordinate Li can be de®ned as the ratio of the area of the triangle formed by point p and the side opposite node i to the whole area of the triangle. In Fig. 3.5, the area coordinate L1 , which is related to node 1, is de®ned as the ratio of the area of triangle 23p to the area of triangle 123. Similarly, the area coordinate L2 , related to node 2, is de®ned as the ratio of the area of triangle 1p3 to the area of triangle 123. And area coordinate L3 , related to node 3, is de®ned as the ratio of the area of triangle 12p to the area of triangle 123. The explicit expressions for L1, L2 , and L3 are x2 x y2 y x S23p x y3 y 3 ; L1
3:65 S123 x2 x1 y2 y1 x x y y 3
L2
L3
S1p3 S123
S12p S123
x3 x 1 x2 x 3 x2 x 1 x2 x 3
1
3
x
y3
x x1
y1 y2
x1
y3
x
y2
x x1 x1
y1 y2 y3
1
y y
; y1 y
3:66
; y1 y1
3:67
1
y y
Generally, since the value of the determinant for a matrix is invariant under transposition, we get the relations x2 x1 y2 y1 x1 y1 1 x1 x2 x3 x2 y2 1 y1 y2 y3
2S123 :
3:68 x x1 y3 y1 3 x3 y3 1 1 1 1
76
FINITE-ELEMENT METHODS
We similarly get the following relations for the other determinants: x2 x 3
x
y2
x
y3
x3 x 1
x
y3
x
y1
x2 x 1
x
y2
x
y1
x y x2 y x3 x y x3 y x1 x y x2 y x1
y y2 y3 y y3 y1 y y2 y1
1 x 1 y 1 1 1 x 1 y 1 1 1 x 1 y 1 1
x2 y2 1 x3 y3 1 x2 y2 1
x3 y3 ; 1 x1 x1 y1 y1 1 1 x1 x1 y1 y1 1 1
x3 y y3 ; 1 1 x2 x y2 y : 1 1 x
Rewriting Eqs. (3.65)±(3.67) by using these relations, we get x x2 L1 y y2 1 1 x1 x L2 y1 y 1 1 x1 x2 L3 y1 y2 1 1
x3 y3 1 x3 y3 1 x y 1
x1 y1 1 x1 y1 1 x1 y1 1
x2 y2 1 x2 y2 1 x2 y2 1
x3 y3 ; 1 x3 y3 ; 1 x3 y3 ; 1
3:69
3:70
3:71
As shown in these equations, area coordinates L1 , L2 , and L3 are linear functions of x and y. According to Cramer's formula, Eqs. (3.69)±(3.71) mean that L1 , L2 , and L3 are roots of the following algebraic matrix equations: 0 1 0 x1 x @ y A @ y1 1 1
x2 y2 1
10 1 L1 x3 y 3 A@ L 2 A; 1 L3
3:72
where it can be found from the bottom row that the sum of L1 , L2 , and L3 is 1.
3.3 AREA COORDINATES AND TRIANGULAR ELEMENTS
77
Now, we return our attention to the shape functions in Eq. (3.57). Since here we are interested in a ®rst-order triangular element, we approximate shape functions N1 , N2 , and N3 by the following ®rst-order plane functions: N1 a1 x b1 y c1 ;
3:73
N2 a2 x b2 y c2 ;
3:74
N3 a3 x b3 y c3 ;
3:75
for which the matrix expression is 0
1 0 a1 N1 @ N2 A @ a 2 N3 a3
b1 b2 b3
10 1 x c1 c2 A@ y A: 1 c3
3:76
The conditions to be imposed on shape function N1 are the following: When p
x; y coincides with node 1
x1 ; y1 : N1 a1 x1 b1 y1 c1 1;
3:77 When p
x; y coincides with node 2
x2 ; y2 : N1 a1 x2 b1 y2 c1 0;
3:78 When p
x; y coincides with node 3
x3 ; y3 : N1 a1 x3 b1 y3 c1 0:
3:79 The matrix expression for Eqs. (3.77)±(3.79) is 0
x1 @ x2 x3
y1 y2 y3
10 1 0 1 1 1 a1 1 A@ b 1 A @ 0 A: 0 1 c1
3:80
Imposing the same conditions on N2 and N3, we get 0
x1
B @ x2 x3 0 x1 B @ x2 x3
y1 y2
0 1 0 CB C B C 1 A@ b 2 A @ 1 A; 0 1 c3 10 1 0 1 a3 1 0 CB C B C 1 A@ b 3 A @ 0 A:
y3
1
y1 y2 y3
1
10
a2
c3
1
1
3:81
3:82
78
FINITE-ELEMENT METHODS
Equations (3.80)±(3.82) can be rewritten as 0
x1 @ x2 x3
y1 y2 y3
10 1 a1 1 A@ b 1 1 c1
a2 b2 c2
1 0 1 a3 b3 A @ 0 0 c3
1 0 0 1 0 A: 0 1
3:83
Using the de®nitions 0
x1 @ X x2 x3
y1 y2 y3
1 1 1 A; 1
0
a1 @ A b1 c1
1 a3 b3 A; c3
a2 b2 c2
0
1 @ I 0 0
1 0 0 1 0 A; 0 1
3:84
we can rewrite Eq. (3.83) as X A I :
3:85
Taking the transposition T of Eq. (3.85), we get AT X T I :
3:86
Thus, matrix X T is found to be an inverse matrix of AT . Multiplying Eq. (3.76) by matrix X T , we get 0
1 0 1 0 1 0 1 N1 x x x X T @ N2 A X T AT @ y A I @ y A @ y A: N3 1 1 1 This results in 0 1 0 0 1 x1 N1 x @ y A X T @ N2 A @ y1 N3 1 1
x2 y2 1
10 1 x3 N1 @ A y3 N 2 A: 1 N3
3:87
Comparing Eq. (3.72) with Eq. (3.87), we ®nd the following important relations between shape functions N1 , N2 , and N3 and area coordinates L1 , L2 , and L3 : N1 L1 ;
N2 L2 ;
N3 L3 :
3:88
79
3.3 AREA COORDINATES AND TRIANGULAR ELEMENTS
FIGURE 3.7. Shape function N1 for the ®rst-order triangular element.
Figure 3.7 shows N1 as an example. As shown in this ®gure, N1 is a plane function whose value is 1 at node 1. Similarly, N2 and N3 are respectively plane functions whose values are 1 at nodes 2 and 3. 3.3.2 Second-Order Triangular Elements Figure 3.8 shows a second-order triangular element. As in the ®rst-order triangular element shown in Fig. 3.4, nodes 1, 2, and 3 are positioned at the vertexes of the triangle. The second-order triangular element has three additional nodes set midway between the pairs of vertexes. The wave function f
x; y at an arbitrary coordinate
x; y is expanded using six shape functions and the values of wave functions f1 ; . . . ; f6 at the nodes as f
x; y
6 P i1
Ni fi N T ffg;
FIGURE 3.8. Second-order triangular element.
3:89
80
FINITE-ELEMENT METHODS
where N N1 N2 N3 N4 N5 N6 T and ffg
f1 f2 f3 f4 f5 f6 T . The coordinates of nodes 1±6 are respectively
x1 ; y1 ; . . . ;
x6 ; y6 . As mentioned before, the ®rst-order shape function is a linear function of x and y. The second-order shape functions should be quadratic polynomial functions of x and y. The area coordinates L1 , L2 , and L3 are also linear functions of x and y, so the second-order shape functions must be quadratic functions of the area coordinates. A. Shape Function N1 Shape function N1 is 1 at node 1 and is 0 at all other nodes. Since L1 1 and L2 L3 0 at node 1, N1 should be a quadratic function of only area coordinate L1 . Thus, N1 aL21 bL1 c:
3:90
The following relations should hold: At node 1
L1 1:
N1 a b c 1:
3:91
At node 6
L1 0:5:
N1 0:25a 0:5b c 0:
3:92
At node 3
L1 0:
N1 c 0:
3:93
From Eqs. (3.91)±(3.93), we obtain a 2;
b
1;
c 0:
3:94
1:
3:95
Thus, we get N1 2L21
L1 L1
2L1
B. Shape Function N2 Shape function N2 is 1 at node 2 and is 0 at all other nodes. Since L2 1 and L1 L3 0 at node 2, N2 should be a quadratic function of only area coordinate L2 . Thus, N2 aL22 bL2 c:
3:96
3.3 AREA COORDINATES AND TRIANGULAR ELEMENTS
81
The following relations should hold: At node 2
L2 1:
N2 a b c 1:
3:97
At node 4
L2 0:5:
N2 0:25a 0:5b c 0:
3:98
At node 1
L2 0:
N2 c 0:
3:99
From Eqs. (3.97)±(3.99), we obtain a 2;
b
1;
c 0:
3:100
Thus, we get N2 L2
2L2
1:
3:101
C. Shape Function N3 Shape function N3 is 1 at node 3 and is 0 at all other nodes. Since L3 1 and L1 L2 0 at node 3, N3 should be a quadratic function of only area coordinate L3 . Thus, N3 aL23 bL3 c:
3:102
The following relations should hold: At node 3
L3 1:
N3 a b c 1:
3:103
At node 6
L3 0:5:
N3 0:25a 0:5b c 0:
3:104
At node 1
L3 0:
N3 c 0:
3:105
From Eqs. (3.103)±(3.105), we obtain a 2;
b
1;
c 0:
3:106
Thus, we get N3 L3
2L3
1:
3:107
82
FINITE-ELEMENT METHODS
D. Shape Function N4 Shape function N4 is 1 at node 4 and is 0 at all other nodes. Since L1 L2 0:5 and L3 0 at node 4, N4 should be a quadratic function of area coordinates L1 and L2 . Thus, N4 aL21 bL22 cL1 L2 dL1 eL2 f :
3:108
The following relations should hold: At node 4
L1 L2 0:5:
N4 0:25a 0:25b 0:25c
At node 1
L1 1; L2 0:
0:5d 0:5e f 1:
3:109 N4 a d f 0:
3:110
At node 2
L1 0; L2 1:
N4 b e f 0:
At node 5
L1 0; L2 0:5:
N4 0:25b 0:5e f 0:
3:112
At node 6
L1 0:5; L2 0:
N4 0:25a 0:5d f 0:
3:113
At node 3
L1 L2 0:
N4 f 0:
3:111
3:114
From Eqs. (3.109)±(3.114), we obtain c 4 and a b d e f 0:
3:115
N4 4L1 L2 :
3:116
Thus, we get
E. Shape Function N5 Shape function N5 is 1 at node 5 and is 0 at all other nodes. Since L1 0 and L2 L3 0:5 at node 5, N5 should be a quadratic function of area coordinates L2 and L3 . Thus, N5 aL22 bL23 cL2 L3 dL2 eL3 f :
3:117
The following relations should hold: At node 5
L2 L3 0:5:
N5 0:25a 0:25b 0:25c 0:5d 0:5e f 1:
3:118
At node 2
L2 1; L3 0:
N5 a d f 0:
3:119
At node 3
L2 0; L3 1:
N5 b e f 0:
3:120
At node 4
L2 0:5; L3 0:
N5 0:25a 0:5d f 0:
3:121
At node 6
L2 0; L3 0:5:
N5 0:25b 0:5e f 0:
3:122
At node 1
L2 L3 0:
N5 f 0:
3:123
From Eqs. (3.118)±(3.123), we obtain c4
and a b c d e f 0:
3:124
3.3 AREA COORDINATES AND TRIANGULAR ELEMENTS
83
Thus, we get N5 4L2 L3 :
3:125
F. Shape Function N6 The shape function N6 is 1 at node 6 and is 0 at all other nodes. Since L1 L3 0:5 and L2 0 at node 6, N6 should be a quadratic function of the area coordinates L1 and L3 . Thus, N6 aL21 bL23 cL1 L3 dL1 eL3 f :
3:126
The following relations should hold: At node 6
L1 L3 0:5:
N6 0:25a 0:25b 0:25c 0:5d 0:5e f 1:
3:127
At node 1
L1 1; L3 0:
N6 a d f 0:
3:128
At node 3
L1 0; L3 1:
N6 b e f 0:
3:129
At node 4
L1 0:5; L3 0:
N6 0:25a 0:5d f 0:
3:130
At node 5
L1 0; L3 0:5:
N6 0:25b 0:5e f 0:
3:131
At node 2
L1 L3 0:
N6 f 0:
3:132
From Eqs. (3.127)±(3.132), we obtain c 4 and a b d e f 0:
3:133
N6 4L3 L1 :
3:134
Thus, we get
The above results can be summarized as N1 L1
2L1
1;
3:135
N2 L2
2L2
1;
3:136
N3 L3
2L3
1;
3:137
N4 4L1 L2 ;
3:138
N5 4L2 L3 ;
3:139
N6 4L3 L1 :
3:140
84
FINITE-ELEMENT METHODS
3.4 DERIVATION OF EIGENVALUE MATRIX EQUATIONS x Next, we move to the derivation of eigenvalue matrix equations for the Epq y and Epq modes. As shown in Eqs. (2.75) and (2.76) in Marcatili's method , the wave equations are
x mode: Epq y mode: Epq
@2 Ex @2 Ex 2
k02 er @x2 @y 2 @ Hx @ 2 Hx 2
k02 er @x2 @y
b2 Ex 0;
3:141
b2 Hx 0:
3:142
Taking into consideration the continuity conditions at neighboring elements, which will be discussed later, we can rewrite the wave equation y mode as for the Epq ! b2 Hx 0: er
1 @2 Hx @2 Hx k02 er @x2 @y2
3:143
Using Eqs. (3.141) and (3.142), we get the scalar wave equations [2, 3] Z
2
@2 f @2 f
k02 x2 @x2 @y2
Z2 b2 f 0;
3:144
x2 er n2r ;
3:145
x mode where for the Epq
f Ex ;
Z2 1;
y mode and for the Epq
f Hx ;
Z2
1 1 ; er n2r
x2 1:
3:146
To derive the eigenvalue matrix equation for these modes, we use the Galerkin method discussed in Section 3.2. After the analysis region is divided into many elements, the wave function fe at the nodes in e is
3.4 DERIVATION OF EIGENVALUE MATRIX EQUATIONS
85
expressed by shape functions N1 and wave functions fei . In other words, the wave function is expanded by the shape functions as fe
Me P i1
Nei fei Ne T ffe g;
3:147
where Me is the number of nodes in e and T is a transposing operator for a matrix. We also used the following de®nitions: Ne N1
N2
N3 NMe T ;
3:148
ffe g
f1
f2
f3 fMe T ;
3:149
where the numbers of the nodes Me in the ®rst-order and second-order triangular elements are respectively 3 and 6. Substituting Eq. (3.147) into the wave equation (3.144), we get Z2e
@2 @2 Ne T ffe g
k02 x2e @x2 @y2
Z2e b2 Ne T ffe g 0:
Multiplying the left-hand side of this equation by the shape function Ne and integrating it in element e, we get
e
Ne Z2e
@2 @2 2 Ne T dx dyffe g 2 @x @y
e
k02 x2e
Z2e b2 Ne Ne T dx dyffe g f0g:
3:150
Partially integrating the ®rst term of Eq. (3.150) with respect to x and y, we get "
#
T T @N @N e e Z2e Ne Z2e Ne dy dx ffe g @x @y Ge Ge
T @Ne @Ne T 2 @Ne @Ne Ze dx dyffe g @x @y @x @y e
k02 x2e Z2e b2 Ne Ne T dx dyffe g f0g: e
86
FINITE-ELEMENT METHODS
Making use of Eq. (3.5), we get "
Ge
# @Ne T dG ffe g @n
Z2e Ne T
e
e
Z2e
@Ne @Ne T @Ne @Ne T @x @y @x @y
k02 x2e
dx dyffe g
Z2e b2 Ne Ne T dx dyffe g f0g;
3:151
R where @=@n is the derivative with respect to the outside normal and G dG e is the line integration at boundary Ge . Since the method we are discussing is an FEM, it is necessary to sum up the contributions from all the elements: "
P e
# T @N e Z2e Ne dG ffe g @n Ge P e
P
e
e
e
Z2e
@Ne @Ne T @Ne @Ne T dx dyffe g @x @y @x @y
k02 x2e
Z2e b2 Ne Ne T dx dyffe g f0g:
3:152
Here, we focus on the ®rst term of the above equation. As mentioned before, G dG is a line integration at the boundary of element e. To e simplify the argument here, we assume that the left-hand side of Eq. (3.152) is multiplied by the wave function vector ffe gT . Thus, we get the relation "
P e
Ge
@Ne T Z2e Ne @n
# dG ffe g
2
P6 6 ! 4 Z2e ffe gT Ne |{z} e Ge
Wave function
3 @Ne T ffe g @n |{z}
Derivative of wave function
7 dG7 5:
3.4 DERIVATION OF EIGENVALUE MATRIX EQUATIONS
87
This means that the ®rst term in Eq. (3.152) can be rewritten as "
P e
Ge
P e
Z2e fe
Ge1
@fe dG @n
#
Z2e1 fe1
@fe1 dG @n
Ge
Z2e fe
! @fe dG : @n
Here, we assume that the wave function fe and its normal derivative Z2e @fe =@n with constant Z2e are continuous at the boundaries with neighboring elements. Through this assumption, the line integral calculus terms inside the analysis region are canceled out, since, as shown in Fig. 3.9, the directions of the line integral calculus terms are opposite for each pair of neighboring elements. As a result, only the line integral calculus term of the periphery of the whole analysis region remains. Although this assumption is one of the limitations of the SC-FEM, it is a relatively good approximation. Thus, the ®rst term in Eq. (3.152) can be rewritten as G
Z2 f
@f dG: @n
FIGURE 3.9. Canceling out of line integral calculus terms.
3:153
88
FINITE-ELEMENT METHODS
Substituting the line integral calculus term (3.153) into Eq. (3.152), we can reduce Eq. (3.152) to 2
K
b M ffg
G
Z2 f
@f dG f0g: @n
3:154
When we impose the Dirichlet or Neumann conditions on ®elds or their derivatives given by Eqs. (3.28) and (3.29), we can neglect the last term on the left-hand side of Eq. (3.154). Then Eq. (3.154) can be simpli®ed to the eigenvalue matrix equation
K
b2 M ffg f0g;
3:155
where the square of the propagation constant b is an eigenvalue and ffg is an eigenvector. Here, we used the de®nitions P
@Ne @Ne T @Ne @Ne T K @x @y @x @y e e
k02 x2e Ne Ne T dx dy ; M ffg
P e
Z2e
Z2e
e
e
Ne Ne T dx dy;
P ffe g:
dx dy
3:156
3:157
3:158
e
Since the relative permittivity ere is assumed to be constant in an element, Z2e and x2e are also constant in the element. The variable transformations x xk0 ;
3:159
y yk0 ;
3:160
which are useful for suppressing the round-off errors in the calculation, enable the wave equation (3.155) to be reduced to
K
ffg f0g; n2eff M
3:161
3.5
MATRIX ELEMENTS
89
where K
P
@Ne @Ne T @Ne @Ne T @x @y @x @y e e
x2e Ne Ne T d x d y ; Z2e
d x d y
3:162
e
M Zeff
P e
Z2e
e
Ne Ne T d x d y ;
3:163
b : k0
3:164
3.5 MATRIX ELEMENTS In this section, we discuss the matrix elements for the ®rst- and secondorder triangular elements. A way to form the global matrixes using matrix elements will be shown in the next section. The eigenvalue equation given in Eqs. (3.155)±(3.158) are
K
b2 M ffg f0g;
3:165
where P
@Ne @Ne T @Ne @Ne T K @x @y @x @y e e
k02 x2e Ne Ne T dx dy ; Z2e
e
M ffg
dx dy
3:166
P f Z2e
Ae Be k02 x2e Ce g; e
P e
P e
Z2e
Ne Ne T dx dy
3:167
Z2e Ce ;
P ffe g: e
e
3:168
90
FINITE-ELEMENT METHODS
As mentioned before, since the parameters Z2e and x2e are constant in each element, only the following terms shown in Eqs. (3.165)±(3.167) have to be calculated to obtain explicit expressions for matrixes K and M :
Ae
Be
Ce
@Ne @Ne T dx dy; @x e @x
3:169
@Ne @Ne T dx dy; @y e @y
3:170
e
Ne Ne T dx dy:
3:171
Since the shape function Ne is expressed by area coordinates L1 , L2 , and L3, they are useful for performing the integral calculus in Eqs. (3.169)±(3.171). For later convenience, these coordinates are rewritten as L1 L2 L3
Q1
x
x2 R1
y 2Se
y2
Q2
x
x3 R2
y 2Se
y3
Q3
x
x1 R3
y 2Se
y1
;
3:172
;
3:173
;
3:174
where Q1 y 2
y3 ;
3:175
Q2 y 3
y1 ;
3:176
Q3 y 1
y2 ;
3:177
R1 x 3
x2 ;
3:178
R2 x 1
x3 ;
3:179
R3 x 2
x1 ;
3:180
and Se 12
y3
y1
x2
x1
x3
x1
y2
y1 :
3:181
3.5
MATRIX ELEMENTS
91
Here, Se is the area of the triangle 123 shown in Fig. 3.4. The derivatives of area coordinates L1 , L2 , and L3 with respect to coordinates x and y are @L1 Q1 ; @x 2Se @L1 R 1; @y 2Se
@L2 Q2 ; @x 2Se @L2 R 2; @y 2Se
@L3 Q3 ; @x 2Se @L3 R 3: @y 2Se
3:182
3:183
The following calculations will use the following convenient integration formula for the area coordinates:
e
Li1 Lj2 Lk3 dx dy
i!j!k! 2S
i j k 2! e
i; j; k 0; 1; 2; 3; . . .:
3:184
The derivation of this formula is shown in Appendix B. 3.5.1 First-Order Triangular Elements Figure 3.4 shows a ®rst-order triangular element. Since it has three nodes, its shape function Ne has three components: Ne N1
N2
N3 T :
3:185
As shown in Eq. (3.88), we have the following important relations between the shape functions N1 , N2 , and N3 and the area coordinates L1 , L2 , and L3 : N1 L1 ;
N2 L2 ;
N3 L3 :
3:186
The next step is to derive the explicit expressions of Eqs. (3.169) to (3.171). A. e
@Ne =@x
@Ne T =@x dx dy The component representation of the matrix is
Ae
0
a11 @Ne @Ne @ dx dy a21 @x e @x a31 T
a12 a22 a32
1 a13 a23 A; a33
3:187
92
FINITE-ELEMENT METHODS
and the integrand is 1 @N1 @N2 @N3 C @x @x @x @Ne @Ne T B @N2 =@x C B @ A @x @x @N3 =@x 1 2 0 @N1 @N1 @N2 @N1 @N3 B @x @x @x @x @x C C B C B 2 C B @N2 @N2 @N3 C B @N2 @N1 B C: C B @x @x @x @x @x C B B 2 C @ @N @N @N @N @N3 A 3 1 3 2 @x @x @x @x @x 0
@N1 =@x
3:188
Using the integration formula shown in Eq. (3.184), we obtain the following matrix elements:
a11
a12
a13
a22
2
2 @N1 @L1 dx dy dx dy e @x e @x 2
Q1 Q2 dx dy 1 ; 2Se 4Se e
@N1 @N2 @L1 @L2 dx dy dx dy @x @x e @x e @x
Q1 Q2 QQ dx dy 1 2 a21 ; 2Se 2Se 4Se e
@N1 @N3 @L1 @L3 dx dy dx dy @x @x e @x e @x
Q1 Q3 QQ dx dy 1 3 a31 ; 2Se 2Se 4Se e 2
2 @N2 @L2 dx dy dx dy e @x e @x 2
Q2 Q2 dx dy 2 ; 2Se 4Se e
3:189
3:190
3:191
3:192
3.5
a23
e
e
a33
e
e
@N2 @x Q2 2Se
93
@N3 @L2 @L3 dx dy dx dy @x @x e @x
Q3 QQ dx dy 2 3 a32 ; 2Se 4Se
@N3 @x Q3 2Se
MATRIX ELEMENTS
2
2
dx dy
dx dy
e
@L3 @x
2
3:193
dx dy
Q23 : 4Se
3:194
B. e
@Ne =@y
@Ne T =@y dx dy The component representation of the matrix is 0
Be
T
b11
@Ne @Ne B dx dy @ b21 @y @y e b31
b12
b13
1
b22
C b23 A;
b32
b33
3:195
and the integrand is 1 @N3 B @y C C B C B B 2 C T C B @Ne @Ne @N2 @N2 @N3 C B @N2 @N1 B C: B @y @y @y @y @y @y @y C C B C B B 2 C A @ @N3 @N1 @N3 @N2 @N3 @y @y @y @y @y
3:196 0
@N1 @y
2
@N1 @y
@N2 @y
@N1 @y
Comparing Eq. (3.187) with Eq. (3.195), we ®nd that the differences between them are the variables for the derivatives. Thus, the elements for
94
FINITE-ELEMENT METHODS
the matrix in Eq. (3.195) can be obtained by simply substituting Ri for Qi in Eqs. (3.189)±(3.194):
b11 b12
2 @L1 R2 dx dy dx dy 1 ; 4Se e e @y
@N1 @N2 @L1 @L2 dx dy dx dy @y @y e @y e @y @N1 @y
2
R1 R2 b21 ; 4Se
@N1 @N3 @L1 @L3 dx dy dx dy @y @y e @y e @y
b13
R1 R3 b31 ; 4Se 2
2 @N2 @L2 R2 dx dy dx dy 2 ; 4Se e @y e @y
@N2 @N3 @L2 @L3 dx dy dx dy @y @y e @y e @y
b22 b23
R2 R3 b32 ; 4Se 2
2 @N3 @L3 R2 dx dy dx dy 3 : 4Se e @y e @y
b33
C.
e Ne Ne
T
3:198
3:199
3:200
3:201
3:202
dx dy The component representation of the matrix is
Ce
3:197
0
c11 Ne Ne T dx dy @ c21 e c31
c12 c22 c32
1 c13 c23 A; c33
3:203
and the integrand is 0
N12 T Ne Ne @ N2 N1 N3 N1
N1 N2 N22 N3 N2
1 N1 N3 N 2 N 3 A: N32
3:204
3.5
MATRIX ELEMENTS
95
Using the integration formula shown in Eq. (3.184), we obtain the following matrix elements:
S c11 N12 dx dy L21 dx dy e ;
3:205 6 e e
S N1 N2 dx dy L1 L2 dx dy e c21 ;
3:206 c12 12 e e
S N1 N3 dx dy L1 L3 dx dy e c31 ;
3:207 c13 12 e e
S N22 dx dy L22 dx dy e ; c22
3:208 6 e e
S N2 N3 dx dy L2 L3 dx dy e c32 ;
3:209 c23 12 e e
S
3:210 N32 dx dy L23 dx dy e : c33 6 e e We have so far obtained matrixes Ae , Be , and Ce for the ®rst-order triangular element e. We can form the global matrixes K and M by substituting matrixes Ae , Be , and Ce into Eqs. (3.166) and (3.167) and summing them up. Because the matrixes Ae , Be , and Ce are symmetrical 3 3 matrixes, the global matrixes are also symmetrical. 3.5.2 Second-Order Triangular Elements Since the second-order triangular element (Fig. 3.8) has six nodes, its shape function Ne has six components: Ne N1
N2
N3
N4
N5
N6 T :
3:211
As shown in Eqs. (3.135)±(3.140), the important relations between shape functions N1 ; . . . ; N6 and area coordinates L1 , L2 , and L3 are N1 L1
2L1
1;
3:212
N2 L2
2L2
1;
3:213
N3 L3
2L3
1;
3:214
N4 4L1 L2 ;
3:215
N5 4L2 L3 ;
3:216
N6 4L3 L1 :
3:217
96
FINITE-ELEMENT METHODS
We derive the explicit expressions of Eqs. (3.169)±(3.171) for the second-order triangular element in a way similar to that in which we derived the corresponding expressions for the ®rst-order triangular element. A. e
@Ne =@x
@Ne T =@x dx dy The component representation of the matrix is 0
a11 B a21 B
B a31 @Ne @Ne T dx dy B Ae B a41 @x e @x B @ a51 a61
a12 a22 a32 a42 a52 a62
a13 a23 a33 a43 a53 a63
a14 a24 a34 a44 a54 a64
a15 a25 a35 a45 a55 a65
1 a16 a26 C C a36 C C; a46 C C a56 A a66
3:218
and the integrand is @Ne @Ne T @x @x T @N1 @N2 @N3 @N4 @N5 @N6 @N1 @N2 @N3 @N4 @N5 @N6 @x @x @x @x @x @x @x @x @x @x @x @x 1 0 2 @N1 @N1 @N2 @N1 @N3 @N1 @N4 @N1 @N5 @N1 @N6 C B B @x @x @x @x @x @x @x @x @x @x @x C C B 2 C B @N @N @N @N @N @N @N @N @N @N @N B 2 1 2 2 3 2 4 2 5 2 6C C B B @x @x @x @x @x @x @x @x @x @x @x C C B 2 C B @N3 @N1 @N4 @N1 @N5 @N1 @N6 C B @N3 @N1 @N3 @N2 C B C B @x @x @x @x @x @x @x @x @x @x @x C: B 2 C B @N4 @N4 @N5 @N4 @N6 C B @N4 @N1 @N4 @N2 @N4 @N3 C B C B @x @x @x @x @x @x @x @x @x @x @x C B 2 C B @N5 @N5 @N6 C B @N5 @N1 @N5 @N2 @N5 @N3 @N5 @N4 C B B @x @x @x @x @x @x @x @x @x @x @x C B 2 C C B @ @N6 @N1 @N6 @N2 @N6 @N3 @N6 @N4 @N6 @N5 @N6 A @x @x @x @x @x @x @x @x @x @x @x
3:219
3.5
MATRIX ELEMENTS
97
The derivatives of the shape functions are @Ni @L Q
i 1; 2; 3;
4Li 1 i i
4Li 1 @x @x 2Se @N4 @L2 @Li 2 4 Li L2
Q2 L1 Q2 L2 ; Se @x @x @x @N5 @L @L 2 4 L2 3 L3 2
Q3 L2 Q2 L3 ; Se @x @x @x @N6 @L @L 2 4 L3 1 L1 3
Q1 L3 Q3 L1 : Se @x @x @x
3:220
3:221
3:222
3:223
Using the integration formula shown in Eq. (3.184) and the relations (
e
4Li
e
4Li
1Lj dx dy
0
i 6 j
1 3 Se
i j
1 dx dy 13 Se ;
3:224
3:225
we obtain the following matrix elements for Eq. (3.218):
a11
2 2
@N1 Q1 dx dy
4L1 2Se e @x e 2
Q1
16L21 8L1 1 dx dy 2Se e 2 Q1 2! 1 1 8 2Se 16 4! 3! 2! 2Se 2 Q1 1 1 1 16 8 2Ae 43 32 2 2Se 2 Q1 4 4 1 Q2 2Ae 1; 3 3 2 2Se 4Se
12 dx dy
3:226
98
a12
a13
FINITE-ELEMENT METHODS
@N1 @N2 dx dy @x e @x
Q1 Q2
4L1 1
4L2 1 dx dy 2Se 2Se e
Q1 Q2
16L1 L2 4L1 4L2 1 dx dy 4Se e Q1 Q2 1 1 1 1 4 4 2Se 16 4! 3! 3! 2! 4Se2 Q1 Q2 2 4 1 Q1 Q2 a21 ; 2Se 2 3 3 2 4Se 12Se
@N1 @N2 dx dy @x e @x
Q1 Q3
4L1 1
4L3 1 dx dy 2Se 2Se e Q1 Q3 a31 ; 12Se
@N1 @N4 dx dy @x e @x
Q1 2
4L1 1
Q2 L1 Q1 L2 dx dy 2Se Se e
a14
3:227
3:228
Q1 Q2 a41 ;
3:229 3Se
@N1 @N5 dx dy @x e @x
Q1 2
4L1 1
Q3 L2 Q2 L3 dx dy 0 a51 ;
3:230 2Se Se e
@N1 @N6 dx dy @x e @x
Q1 2
4L1 1
Q1 L3 Q3 L1 dx dy 2Se Se e
a15
a16
Q1 Q3 a61 ; 3Se
3:231
3.5
a22 a23
a24
a25
a26
a33 a34
a35
MATRIX ELEMENTS
99
2 2
@N2 Q2 Q2 dx dy
4L2 12 dx dy 2 ;
3:232 2Ae 4Se e @x e
@N2 @N3 dx dy @x e @x
Q2 Q3
4L2 1
4L3 1 dx dy 2Se 2Se e Q2 Q3 a32 ;
3:233 12Se
@N2 @N4 dx dy @x e @x
Q2 2
4L2 1
Q2 L1 Q1 L2 dx dy 2Se Se e QQ 1 2 a42 ;
3:234 3Se
@N2 @N5 dx dy @x e @x
Q2 2 QQ
4L2 1
Q3 L2 Q2 L3 dx dy 2 3 2Se Se 3Se e a52 ;
@N2 @N6 dx dy @x e @x
Q2 2
4L2 1
Q1 L3 Q3 L1 dx dy 0 a62 ; 2Se Se e 2 2
@N3 Q3 Q2 dx dy
4L3 12 dx dy 3 ; 2Se 4Se e @x e
@N3 @N4 dx dy @x e @x
Q3 2
4L3 1
Q2 L1 Q1 L2 dx dy 0 a43 ; 2Se Se e
@N3 @N5 dx dy @x e @x
Q3 2
4L3 1
Q3 L2 Q2 L3 dx dy 2Se Se e QQ 2 3 a53 ; 3Se
3:235
3:236
3:237
3:238
3:239
100
FINITE-ELEMENT METHODS
a36
@N3 @N6 dx dy @x e @x
Q3 2
4L3 1
Q1 L3 Q3 L1 dx dy 2Se Se e
a44
Q3 Q1 a63 ; 3Se 2 2
@N4 2 dx dy
Q2 L1 Q1 L2 2 dx dy @x A e e e 2
2
Q22 L21 2Q1 Q2 L1 L2 Q21 L22 dx dy Se e
a45
2
Q2 Q1 Q2 Q22 ; 3Se 1
@N4 @N5 dx dy @x e @x 2
2
Q2 L1 Q1 L2
Q3 L2 Q2 L3 dx dy Ae e
a46
1
Q Q Q22 2Q1 Q3 Q1 Q2 a54 ; 3Se 2 3
@N4 @N6 dx dy @x e @x 2
2
Q2 L1 Q1 L2
Q1 L3 Q3 L1 dx dy Se e
a55
1
Q Q 2Q2 Q3 Q21 Q1 Q3 a64 ; 3Se 1 2 2 2
@N5 2 dx dy
Q3 L2 Q2 L3 2 dx dy Se e @x e 2
2
Q23 L22 2Q2 Q3 L2 L3 Q22 L23 dx dy Se e
2
Q2 Q2 Q3 Q22 ; 3Se 3
3:240
3:241
3:242
3:243
3:244
3.5
MATRIX ELEMENTS
a56
a66
@N5 @N6 dx dy @x e @x 2
2
Q3 L2 Q2 L3
Q1 L3 Q3 L1 dx dy Se e 1
Q Q Q23 2Q1 Q2 Q2 Q3 a65 ; 3Se 1 3 2 2
@N6 2 dx dy
Q1 L3 Q3 L1 2 dx dy S @x e e e 2
2
Q21 L3 2Q1 Q3 L1 L3 Q23 L21 dx dy Se e 2
Q2 Q1 Q3 Q23 : 3Se 1
101
3:245
3:246
B. e
@Ne =@y
@Ne T =@y dx dy The component representation of the matrix is 0
b11 B b21 B
B b31 @Ne @Ne T dx dy B Be B b41 @y e @y B @ b51 b61
b12 b22 b32 b42 b52 b62
b13 b23 b33 b43 b53 b63
b14 b24 b34 b44 b54 b64
b15 b25 b35 b45 b55 b65
1 b16 b26 C C b36 C C: b46 C C b56 A b66
3:247
Comparing Eq. (3.219) with Eq. (3.247), we ®nd that the differences between them are the variables for the derivatives. Thus, the elements for the matrix in Eq. (3.247) can be obtained by simply substituting Ri for Qi in Eqs. (3.226)±(3.246):
b11
e
b12 b13
@N1 @y
2
R21 ; 4Se
@N2 dx dy @y e
@N1 @N3 dx dy @y e @y @N1 @y
dx dy
3:248 R1 R2 b21 ; 12Se
3:249
R1 R3 b31 ; 12Se
3:250
102
FINITE-ELEMENT METHODS
b14 b15 b16 b22 b23 b24 b25 b26 b33 b34 b35 b36 b44 b45
@N4 RR dx dy 1 2 b41 ; @y 3Se e
@N1 @N5 dx dy 0 b51 ; @y e @y
@N1 @N6 RR dx dy 1 3 b61 ; @y @y 3Se e 2
@N2 R2 dx dy 2 ; 4Se e @y
@N2 @N3 R2 R3 b32 ; dx dy @y 12Se e @y
@N2 @N4 RR dx dy 1 2 b42 ; @y @y 3Se e
@N2 @N5 RR dx dy 2 3 b52 ; @y @y 3Se e
@N2 @N6 dx dy 0 b62 ; @y e @y 2
@N3 R2 dx dy 3 ; 4Se e @y
@N3 @N4 dx dy 0 b43 ; @y e @y
@N3 @N5 RR dx dy 2 3 b53 ; @y @y 3Se e
@N3 @N6 RR dx dy 3 1 b63 ; @y @y 3Se e 2
@N4 2 dx dy
R2 R1 R2 R22 ; 3Se 1 e @y
@N4 @N5 dx dy @y e @y
@N1 @y
1
R R R22 2R1 R3 R1 R2 b54 ; 3Se 2 3
3:251
3:252
3:253
3:254
3:255
3:256
3:257
3:258
3:259
3:260
3:261
3:262
3:263
3:264
3.5
b46
b55 b56
b66 C.
e Ne Ne
e
@N6 @y
MATRIX ELEMENTS
103
@N6 dx dy @y
1
R R 2R2 R3 R21 R1 R3 b64 ; 3Se 1 2 2
@N5 2 dx dy
R23 R2 R3 R22 ; 3S @y e e
@N5 @N6 dx dy @y e @y 1
R R R23 2R1 R2 R2 R3 b65 ; 3Se 1 3 2
@N6 2 dx dy
R21 R1 R3 R23 : @y 3S e e
T
dx dy The component 0 c11 B c21 B
B c31 Ce Ne Ne T dx dy B B c41 e B @ c51 c61
3:265
3:266
3:267
3:268
representation of the matrix is 1 c12 c13 c14 c15 c16 c22 c23 c24 c25 c26 C C c32 c33 c34 c35 c36 C C; c42 c43 c44 c45 c46 C C c52 c53 c54 c55 c56 A c62 c63 c64 c65 c66
3:269
and the integrand is 0 2 N1 BN N B 2 1 B B N3 N1 T Ne Ne B BN N B 4 1 B @ N5 N1 N6 N1
N1 N2 N22 N3 N2 N4 N2 N5 N2 N6 N2
N1 N3 N2 N3 N32 N4 N3 N5 N3 N6 N3
N1 N4 N2 N4 N3 N4 N42 N5 N4 N6 N4
N1 N5 N2 N5 N3 N5 N4 N5 N52 N6 N5
1 N1 N6 N2 N6 C C C N3 N6 C C:
3:270 N4 N6 C C C N5 N6 A N62
Using the integration formula shown in Eq. (3.184), we obtain the following matrix elements:
S c11 N12 dx dy L21
2L1 12 dx dy e ;
3:271 30 e e
N1 N2 dx dy
2L1 1L1
2L2 1L2 dx dy c12 e
Se c21 ; 180
e
3:272
104
FINITE-ELEMENT METHODS
c13 c14
e
Se c31 ;
180
c15
c23
e
e
e
2L1
1L1
2L3
1L3 dx dy
3:273
N1 N4 dx dy
N1 N5 dx dy
Se c51 ;
45
c16 c22
N1 N3 dx dy
e
e
2L1
1L1 4L1 L2 dx dy 0 c41 ;
3:274
2L1
1L1 4L2 L3 dx dy
3:275
N1 N6 dx dy
2L1 1L1 4L3 L1 dx dy 0 c61 ;
3:276 e
S N22 dx dy
2L2 12 L22 dx dy e ;
3:277 30 e e
N2 N3 dx dy
2L2 1L2
2L3 1L3 dx dy e
e
e
Se c32 ; 180
N2 N4 dx dy
2L2 e e
N2 N5 dx dy
2L2 e e
N2 N6 dx dy
2L2
3:278
c24 c25 c26
e
Se c62 ;
45
1L2 4L2 L3 dx dy 0 c52 ;
3:280 1L2 4L3 L1 dx dy
3:281
S
2L3 1L23 dx dy e ; 30
e
e
N3 N4 dx dy
2L3 1L3 4L1 L2 dx dy N32 dx dy
c33 c34
e
1L2 4L1 L2 dx dy 0 c42 ;
3:279
e
Se c43 ;
45 c35 c36
e
e
N3 N5 dx dy N3 N6 dx dy
3:282
e
3:283
e
e
2L3
1L3 4L2 L3 dx dy 0 c53 ;
3:284
2L3
1L3 4L3 L1 dx dy 0 c63 ;
3:285
3.6
c46 c55 c56 c66
N42 dx dy
105
8S
4L1 L2 2 dx dy e ; 45 e e
4S N4 N5 dx dy 4L1 L2 4L2 L3 dx dy e c54 ; 45 e e
4S N4 N6 dx dy 4L1 L2 4L3 L1 dx dy e c64 ; 45 e e
8S N52 dx dy
4L2 L3 2 dx dy e ; 45 e e
4S N5 N6 dx dy 4L2 L3 4L3 L1 dx dy e c65 ; 45 e e
8S N62 dx dy
4L3 L1 2 dx dy e : 45 e e
c44 c45
PROGRAMMING
3:286
3:287
3:288
3:289
3:290
3:291
We have thus obtained the matrixes Ae , Be , and Ce for the secondorder triangular element e. We can form the global matrixes K and M by substituting Ae , Be , and Ce into Eqs. (3.166) and (3.167) and summing them. Because Ae , Be , and Ce are symmetrical 6 6 matrixes, the global matrixes are also symmetrical.
3.6 PROGRAMMING As described above, when using an FEM, we ®rst obtain the matrix elements for element e. We then obtain the global matrixes for the eigenvalue matrix equation by summing the contributions of all the elements. This section discusses how computer programs based on the ®rst- and second-order triangular elements can be written by using
K
b2 M ffg f0g;
3:292
which is the eigenvalue matrix equation. Here, K, M, and ffg are de®ned as P f Z2e
Ae Be k02 x2e Ce g; e P M Z2e Ce ; e P ffg ffe g K
e
3:293
3:294
3:295
106
FINITE-ELEMENT METHODS
and Ae , Be , and Ce as
Ae
Be Ce
@Ne @Ne T dx dy; @x e @x @Ne @Ne T dx dy; @y e @y e
Ne Ne T dx dy:
3:296
3:297
3:298
3.6.1 First-Order Triangular Elements Figure 3.10 shows an example of an optical waveguide whose buried structure has been divided into 18 ®rst-order triangular elements e1 ; . . . ; e18 . The core with width W comprises two elements e9 and e10 , and Fig. 3.11 shows the local coordinates for element e9 . In this ®gure, the local coordinates of the node numbers 6, 7, and 10Ðwhose coordinates are
x6 ; y6 ,
x7 ; xy7 , and
x10 ; y10 Ðare respectively 1, 2, and 3. Thus, the node number can be determined from the element number and the local coordinate. The actual programming ¯ow is as follows: 1. Divide the whole analysis region into a number of meshes by using ®rst-order triangular elements.
FIGURE 3.10. Mesh formed by ®rst-order triangular elements.
3.6
PROGRAMMING
107
FIGURE 3.11. Local coordinates for the nodes of a ®rst-order triangular element.
2. As shown in Fig. 3.11, any node number can be identi®ed by specifying the element number and the local coordinate. For example, from the node number of element e9 and the local coordinate 1, we get the node number 7 and the coordinate
x7 ; y7 . 3. Using Eqs. (3.169)±(3.171), calculate the 3 3 matrixes Ae , Be , and Ce for each element e. 4. Add the calculated results for matrixes Ae , Be , and Ce to the global matrixes K and M , whose row and column numbers correspond to the combinations of the element numbers and the local coordinates. For example, since the local coordinates 1 and 2 respectively correspond to nodes 7 and 10, the matrix elements with the ®rst row and the second column of Ae , Be , and Ce are added to the matrix elements of the 7th row and the 10th column of both matrixes K and M . Since the matrixes used in the scalar FEM are sparse and symmetrical, the amount of memory required for the matrixes can be reduced. 5. As shown in Fig. 3.10, node 6 belongs to six triangular elements (e2 , e3 , e4 , e7 , e8 , and e9 ). Since each node belongs to more than one triangular element, obtain the global matrixes K and M by summing the calculated matrix elements of Ae , Be , and Ce for all triangular elements. 6. When imposing the boundary conditions, which will be discussed in Section 3.7, incorporate them into matrixes K and M . 7. Obtain the propagation constant b or the effective index neff by solving the eigenvalue matrix equation (3.165).
108
FINITE-ELEMENT METHODS
FIGURE 3.12. General meshes formed by ®rst-order triangular elements.
Figure 3.12 shows an example of general meshes formed by ®rst-order triangular elements. The numbers of nodes in the vertical and horizontal directions are respectively My and Mx, which means that the total number of nodes is Mx My . To discuss sparsity, we consider element e1 , in which there are three nodes; the numbers are 1, 2, and My 1. Any node whose number is larger than My 1 is not related to node 1. Figure 3.13 shows matrixes K and M corresponding to the meshes shown in Fig. 3.12. The physical meanings of the matrix elements can be inferred from the wave functions f1 ; f2 ; . . . ; fM and in Fig. 3.13. Matrix element aij
i 6 j is related to the interaction between wave functions fi and fj, and matrix element aii is related to the self-interaction of wave function fi . Thus, as shown in Fig. 3.13, matrixes K and M have nonzero elements until the My 1 column. The number My 1 is called the bandwidth of the sparse matrix.
3.6.2 Second-Order Triangular Elements Figure 3.14 shows an example of general meshes formed by the secondorder triangular elements, and Fig. 3.15 illustrates the correspondence
3.6
PROGRAMMING
109
FIGURE 3.13. Forms of global matrixes K and M , M Mx My .
between the node numbers and the local coordinates for element e1 . A computer program based on second-order triangular elements is basically the same as one based on the ®rst-order triangular elements, but it should be noted that each second-order triangular element has six local coordinates and that the corresponding matrixes K and M have a bandwidth of 2My 1. When the number of the elements is the same, the total number of nodes and the bandwidth are larger for second-order triangular elements than they are for ®rst-order triangular elements. Since the second-order elements approximate the unknown wave functions by quadratic curved surface functions, they are more accurate than the ®rst-order elements, which approximate the unknown wave functions by plane functions. Thus, the second-order elements are numerically more ef®cient.
110
FINITE-ELEMENT METHODS
FIGURE 3.14. General meshes formed by second-order triangular elements.
FIGURE 3.15. Local coordinates for the nodes of a second-order triangular element.
3.7 BOUNDARY CONDITIONS In this section, we discuss the boundary conditions that should be applied to the nodes on the edges of the analysis region. The eigenvalue matrix equation was shown in Eq. (3.154) as
K
b2 M ffg
G
Z2 f
@f dG f0g: @n
3:299
3.7
BOUNDARY CONDITIONS
111
Here, we discuss the two conditions used most widely.
3.7.1 Neumann Condition The Neumann condition requires that the derivative of the wave function be set to zero, which means that the variation of the wave function at the boundaries would be negligibly small. Thus, we get @fi 0: @n
3:300
Substituting Eq. (3.300) into (3.299), we get the familiar eigenvalue matrix equation
K
b2 M ffg f0g:
3:301
Here, we mention another important application of boundary conditions. Figure 3.10 shows the whole analysis region. Analyzing the whole region, we simultaneously obtain even modes including a dominant mode and odd modes whose ®elds are zero at the mirror-symmetrical plane of the structure. On the other hand, we can obtain the solutions for only the even modes or the odd modes by analyzing the half-plane structure (Fig. 3.16) with the Neumann condition or the Dirichlet condition applied at the mirror-symmetrical plane at the center. This is convenient when we analyze the de®nite modes of optical waveguides.
3.7.2 Dirichlet Condition The Dirichlet condition requires that the wave functions at the boundaries be set to zero: fi 0:
3:302
Thus, we also get the eigenvalue matrix equation (3.301). The Dirichlet condition requires a further process. Since Eq. (3.302) has to hold for the
112
FINITE-ELEMENT METHODS
FIGURE 3.16. Boundary conditions on a mirror-symmetrical plane.
ith component fi of eigenvector ffg, some matrix elements other than the diagonal terms for matrixes K and M have to be set to zero: 00 BB BB BB BB BB 0 0 BB BB BB @@
0 0 Kii 0 0
1 C C C C 00C C C C A
0
0
B B B 0 B 2B b B 0 0 Mii B 0 B @ 0
11
1 f 1 CC CCB .. C CCB . C CCB C CB fi C f0g: 00C CCB C CCB .. C CC@ . A AA fM 0
3:303 Since the ith-row elements of the matrixes in Eq. (3.303) satisfy the equation
Kii
b2 Mii fi 0;
3:304
Eq. (3.302) holds. Another way to implement the Dirichlet condition is to simply omit the nodes at the boundaries because under the Dirichlet condition they do not in¯uence the other nodes. This requires less computer memory than is required when Eq. (3.303) is used.
3.7
BOUNDARY CONDITIONS
113
PROBLEMS 1. In the derivation of the eigenvalue matrix equations (3.40) and (3.56), it was assumed that the wave function fe and its normal derivative Z2e @fe =@n are each continuous at the boundaries of two neighboring elements. This was the basis on which the line integral calculus terms in Eq. (3.51) were canceled out at the boundaries. Discuss the validity of the assumption. ANSWER x mode: a. Epq
fe !
At horizontal interfaces Ex is continuous:
At vertical interfaces Ex is discontinuous: 8 @E > > At horizontal interfaces x
/ Hz is continuous: < @y @f Z2e e ! @Ex er2 > @n > : At vertical interfaces / Ez is discontinuous: @x er1 y mode: b. Epq At horizontal interfaces Hx is continuous: fe ! At vertical interfaces Hx is discontinuous: 8 1 @Hx > > At horizontal interfaces
/ Ez is continuous: < er @y 2 @fe Ze ! > 1 @Hx 1 @n > : At vertical interfaces / Hz is discontinuous: er @x er
Thus, at the horizontal interfaces the assumption is valid for both the x mode and the Epq mode. At the vertical interfaces, on the other hand, x mode are the wave function fe and the derivative Z2e @fe =@n for the Epq y 2 discontinuous and the derivative Ze @fe =@n for the Epq mode is discontinuous. From the above discussion, the conclusion is that when the ratio of the width of the core to the thickness of the core is not large, which implies that the in¯uences of the vertical boundaries cannot be neglected, the accuracy of the SC-FEM is degraded. This is especially true for largeindex-difference optical waveguides, such as those made from semiconductor materials. x Epq
114
FINITE-ELEMENT METHODS
FIGURE P3.1. Simple example of ®rst-order triangular elements.
2. Figure P3.1 shows a simple example of ®rst-order triangular elements, where the total node number is 6. Show the form of the matrix equation for this example. ANSWER 00
K11
BB K BB 21 BB BB K31 BB BB 0 BB BB @@ 0 0
K13
0
0
K22
K23
K24
0
K32
K33
K34
K35
K42
K43
K44
K45
0
K53
K54
K55
0 C C C 0 C C K46 C C C K56 A
0 K64 Symmetry
K65
K66
0 0
1
K12
M11
BM B 21 B B M31 b2 B B 0 B B @ 0 0
0
110
M12
M13
0
0
M22
M23
M24
0
M32
M33
M34
M35
M42
M43
M44
M45
0
M53
M54
M55
1 f1 CB C 0 C CCB f 2 C CCB C 0 CCB f 3 C CCB C f0g:
P3:1 CB C M46 C CCB f 4 C CCB C M56 AA@ f5 A
0
0
M64
M65
M66
Symmetry
0
f6
REFERENCES
115
REFERENCES [1] O. C. Zienkiewitz, The Finite Element Method, 3rd ed., McGraw-Hill, New York, 1973. [2] M. Koshiba, H. Saitoh, M. Eguchi, and K. Hirayama, ``Simple scalar ®nite element approach to optical waveguides,'' IEE Proc. J., vol. 139, pp. 166± 171, 1992. [3] M. Koshiba, Optical Waveguide Theory by the Finite Element Method, KTK Scienti®c Publishers and Kluwer Academic Publishers, Dordrecht, Holland, 1992. [4] K. Kawano, S. Sekine, H. Takeuchi, M. Wada, M. Kohtoku, N. Yoshimoto, T. Ito, M. Yanagibashi, S. Kondo, and Y. Noguchi, ``4 4 InGaAlAs=InAlAs MQW directional coupler waveguide switch modules integrated with spotsize converters and their 10 Gbit=s operation,'' Electron. Lett., vol. 31, pp. 96±97, 1995. [5] T. Itoh and R. Mittra, ``Spectral domain approach for calculating the dispersion characteristics of microstrip lines,'' IEEE Trans. Microwave Theory Tech., vol. MTT-21, pp. 496±499, 1973. [6] K. Kawano, T. Kitoh, H. Jumonji, T. Nozawa, M. Yanagibashi, and T. Suzuki, ``Spectral domain approach of coplanar waveguide traveling-wave electrodes and their applications to Ti : LiNbO3 Mach±Zehnder optical modulators,'' IEEE Trans. Microwave Theory Tech., vol. 39, pp. 1595±1601, 1991.
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 4
FINITE-DIFFERENCE METHODS
Since the semivectorial ®nite-difference methods (SV-FDMs) developed by Stern [1, 2] are numerically ef®cient methods taking polarization into consideration and providing accurate results, they are widely used in the computer-aided design (CAD) software currently available for 2D crosssectional analyses of optical waveguides. Finite-difference schemes are also used in the ®nite-difference beam propagation methods (FD-BPMs), which are of course also widely used in CAD software and which are discussed in detail in Chapter 5. The present chapter will help readers understand SV-FDMs and their formulation. It will also help readers become familiar with how to program them. Furthermore, it will teach users of CAD software not only how to identify the main causes of errors but also how to decrease the size of errors. When the FEMs were discussed in Chapter 3, the wave equations themselves were not solved, but instead the functional was introduced and the variational principle was used, or the Galerkin method, which is a weighted residual method, was used. The FDMs dealt with in this chapter, in contrast, are more direct approaches to solving the wave equations. They solve eigenvalue matrix equations for electric ®elds or magnetic ®elds, equations derived from ®nite-difference approximations for the wave equations. This chapter brie¯y describes the ®nite-difference approximations and then derives the vectorial wave equations. It then obtains the semivectorial wave equations by ignoring the terms for the interaction between two polarized ®eld components in the vectorial wave equations. 117
118
FINITE-DIFFERENCE METHODS
This chapter then discusses the formulation of the SV-FDMs, the errors caused by the ®nite-difference approximations, and SV-FDM programming. Although Stern's formulation uses equidistant discretization, this chapter uses the more versatile nonequidistant discretization.
4.1 FINITE-DIFFERENCE APPROXIMATIONS In the FDMs discussed in this chapter, eigenvalue matrix equations are derived by using ®nite-difference schemes to approximate the wave equations. Let us ®rst brie¯y examine the ®nite-difference approximations for the derivatives and then examine the accuracy of these approximations. Assume that a 1D function f
x is continuous and smooth. As shown in Fig. 4.1, the function values f1 , f2 , and f3 at x h1 , h2 , 0 are expressed as f1 f
h1 ;
4:1
f2 f
h2 ;
4:2
f3 f
0:
4:3
Next, we can write f1 and f2 as Taylor series expansions around x 0: f1 f
h1 f
0 1 3 h f 3! 1
3
1 3 h f 3! 2
3
1
0
1 2 h f 2! 1
2
0
0 O
h41 ;
f2 f
h2 f
0
1 h f 1! 1 1 h f 1! 2
1
4:4
0
1 2 h f 2! 2
2
0
0 O
h42 ;
FIGURE 4.1. Difference approximations for derivatives.
4:5
4.1
FINITE-DIFFERENCE APPROXIMATIONS
119
where f
n is an nth derivative de®ned as f
n
x
d n f
x : dxn
4:6
Subtracting Eq. (4.4) from Eq. (4.5), we get an expression for the ®rst derivative: f2
f1
h2 h1 f
1
1
0
h22 2
h21 f
2
0 O
h3 :
Thus, f
1
0
f2 f1 1
h h1 h2 2 2
h1 f
2
0 O
h2 :
4:7
As shown in Eq. (4.7), the error caused by approximating the ®rst derivative at x 0 with the differential expression f
1
0
f2 f1 h1 h2
4:8
is O
h2 when h1 h2 (equidistant discretization) and is O
h when h1 6 h2 (nonequidistant discretization). The error caused by approximating the second derivative with the expression f
2
0
2 h2 f 1 h1 h2
h1 h2 f3 h1 f2
h1 h2
4:9
is also O
h2 when h1 h2 and is O
h when h1 6 h2 (see Problem 1). Thus, when CAD software is used, equidistant discretization is preferred whenever there is enough computer memory. When discretization is nonequidistant, we have to be careful that the ratio of h1 to h2 does not change drastically, since a drastic change would greatly increase the size of the errors.
120
FINITE-DIFFERENCE METHODS
4.2 WAVE EQUATIONS 4.2.1 Vectorial Wave Equations In this section, the ®nite-difference expressions are obtained for the semivectorial wave equations. The approximations used in the process are made clear by starting from the fully vectorial forms of the equations. The vectorial wave equation [Eq. (1.34)] for the electric ®eld E is =er ? E k02 er E 0: H E= er 2
4:10
Let us consider a structure uniform in the z direction. In this case, the derivative of relative permittivity with respect to z is zero: @er 0: @z
4:11
Thus, the second term in Eq. (4.10) can be written as =
=er 1 @er 1 @er ?E = Ex Ey : er @x er @y er
4:12
After substituting Eq. (4.12) into Eq. (4.10), we separate Eq. (4.10) into the x and y components. [The derivative with respect to z is indicated in Eqs. (4.13)±(4.30) in the form of @=@z because that is the formulation used in the beam propagation methods dealt with in Chapter 5.] Thus, we obtain the vectorial wave equation using the electric ®eld components Ex and Ey . Its x component is @2 Ex @ 1 @er @2 Ex @2 Ex @ 1 @er 2 k e E E E 0; 0 r x @x er @x x @x er @y y @x2 @y2 @z2
4:13 and its y component is @2 Ey @2 Ey @ 1 @er @2 Ey @ 1 @er 2 2 E 2 k0 er Ey E 0: @y er @y y @y er @x x @x2 @y @z
4:14
4.2
WAVE EQUATIONS
121
Because @ 1 @ @2 Ex @ 1 @er
e E 2 E @x er @x r x @x er @x x @x
4:15
@2 Ey @ 1 @er @ 1 @
e E 2 E ; @y er @y r y @y er @y y @y
4:16
and
Eqs. (4.13) and (4.14) can be rewritten as @ 1 @ @2 Ex @2 Ex @ 1 @er 2
e E 2 2 k0 er Ex E 0
4:17 @x er @x r x @x er @y y @y @z and @ 2 Ey @ 1 @ @2 Ey @ 1 @er 2
e E 2 k0 er Ey E 0:
4:18 @y er @y r y @y er @x x @x2 @z Let us next derive the corresponding components of the equation for the magnetic ®eld H. The vectorial wave equation [Eq. (1.40)] for the magnetic ®eld H is H2 H
=er 3
=3H k02 er H 0: er
4:19
Here, the second term in Eq. (4.19) is investigated in detail. Recall that we are considering a structure uniform in the z direction and that Eq. (4.11) therefore holds. When i, j, and k are respectively assumed to be unit vectors in the x, y, and z directions, we can obtain i j k @e @er r 0 =er 3
=3H @y @x
=3Hx
=3Hy
=3Hz
@er @er
=3Hz i
=3Hz j @y @x @er @er
=3Hy
=3Hx k @x @y
4:20
122
FINITE-DIFFERENCE METHODS
by assuming that @er =@z 0 and using the expressions
=3Hx
@Hz @y
@Hy ; @z
4:21
=3Hy
@Hx @z
@Hz ; @x
4:22
=3Hz
@Hy @x
@Hx : @y
4:23
The substitution of Eqs. (4.21), (4.22), and (4.23) into Eq. (4.20) results in @er @Hy @Hx @er @Hy @Hx =er 3
=3H i j @y @x @y @x @x @y @er @Hx @Hz @er @Hz @Hy k: @x @z @x @y @y @z
4:24
Substituting Eq. (4.24) into Eq. (4.19) and separating the result into the x and y components, we obtain the vectorial wave equation using the magnetic ®eld components Hx and Hy . Its x component is @2 Hx @2 Hx 2 @x2 @y
1 @er @Hx @2 Hx 1 @er @Hy 2 k02 er Hx 0;
4:25 er @y @y er @y @x @z
and its y component is @2 Hy @x2
1 @er @Hy @2 Hy @2 Hy 1 @er @Hx 2 2 k02 er Hy 0:
4:26 er @x @x er @x @y @y @z
And because @ 1 @Hx @2 H 2x er @y er @y @y
1 @er @Hx er @y @y
4:27
@2 Hy @ 1 @Hy er 2 @x er @x @x
1 @er @Hy ; er @x @x
4:28
and
4.2
WAVE EQUATIONS
123
Eqs. (4.25) and (4.26) can be rewritten as @2 Hx @ 1 @Hx @2 Hx 1 @er @Hy 0 e k02 er Hx r @y er @y er @y @x @x2 @z2
4:29
@ 2 Hy @ 2 Hy @ 1 @Hy 1 @er @Hx er 2 2 k02 er Hy 0: @x er @x er @x @y @y @z
4:30
and
Furthermore, since what we are concerned with here is a 2D crosssectional analysis, the derivatives of the electric and magnetic ®elds with respect to z are constant: @ @z
jb;
4:31
where b is a propagation constant. Thus, using Eqs. (4.13) and (4.14), we obtain for the x component @2 Ex @ 1 @er @2 Ex
k02 er E @x er @x x @x2 @y2
@ 1 @er b Ex E 0 @x er @y y
4:32 2
and for the y component @2 Ey @2 Ey @ 1 @er E
k02 er 2 @y er @y y @x2 @y
@ 1 @er E 0: b Ey @y er @x x
4:33 2
They can be rewritten as @ 1 @ @2 E
er Ex 2x
k02 er @x er @x @y
@ 1 @er E 0
4:34 b Ex @x er @y y 2
and @2 Ey @ 1 @
e E
k02 er @y er @y r y @x2
@ 1 @er b Ey E 0:
4:35 @y er @x x 2
124
FINITE-DIFFERENCE METHODS
The last terms in Eqs. (4.32)±(4.35) correspond to the interactions between the x-directed electric ®eld component Ex and the y-directed electric ®eld component Ey . Using Eqs. (4.25) and (4.26), on the other hand, we obtain for the x component @2 Hx @2 Hx 2 @x2 @y
1 @er @Hx
k02 er er @y @y
b2 Hx
1 @er @Hy 0 er @y @x
4:36
b2 Hy
1 @er @Hx 0; er @x @y
4:37
and for the y component @2 Hy @x2
1 @er @Hy @2 Hy 2
k02 er er @x @x @y
which can be rewritten as @2 Hx @ 1 @Hx
k02 er er @y er @y @x2
b2 Hx
1 @er @Hy 0 er @y @x
4:38
@ 2 Hy @ 1 @Hy 2
k02 er @x er @x @y
b2 Hy
1 @er @Hx 0: er @x @y
4:39
and er
Similar to what we saw in the corresponding equations for the electric ®eld E, the last terms in Eqs. (4.36)±(4.39) correspond to the interactions between the x-directed magnetic ®eld component Hx and the y-directed magnetic ®eld component Hy . 4.2.2 Semivectorial Wave Equations In equations for the ®elds that propagate in optical waveguides, the terms corresponding to the interaction between the x-directed electric ®eld component Ex and the y-directed electric ®eld component Ey , @ 1 @er E and @x er @y y
@ 1 @er E in Eqs:
4:32
4:35; @y er @x x
4.2
WAVE EQUATIONS
125
and the terms corresponding to the interaction between the x-directed magnetic ®eld component Hx and the y-directed magnetic ®eld component Hy , 1 @er @Hy er @y @x
and
1 @er @Hx er @x @y
in Eqs:
4:36
4:39;
are usually small. Ignoring these terms for the interaction, we can decouple the vectorial wave equations for the x- and y-directed ®eld components and reduce them to semivectorial wave equations, which can be solved in a way that is numerically ef®cient. Semivectorial analyses that neglect the terms for the interaction are therefore widely used when designing optical waveguide devices for which the coupling between the x- and y-directed polarizations does not have to be taken into consideration. As shown in Fig. 4.2, these analyses can be divided into the quasi-TE mode analysis, in which the principal ®eld component is Ex or Hy , and the quasi-TM mode analysis, in which the principal ®eld component is Ey or Hx .
FIGURE 4.2. Principal ®eld components for (a) quasi-TE and (b) quasi-TM modes.
126
FINITE-DIFFERENCE METHODS
A. Quasi-TE Mode The principal ®eld component in the electric ®eld representation for the quasi-TE mode, where the y-directed electric ®eld component Ey is assumed to be zero, is the x-directed electric ®eld component Ex . So according to Eq. (4.32), the semivectorial wave equation for the quasi-TE mode is @2 Ex @ 1 @er @ 2 Ex E
k02 er x @x er @x @x2 @y2
b2 Ex 0;
4:40
which, according to Eq. (4.34), can be rewritten as @ 1 @ @2 E
er Ex 2x
k02 er @x er @x @y
b2 Ex 0:
4:41
The principal ®eld component in the magnetic ®eld representation for the quasi-TE mode, on the other hand, where the x-directed magnetic ®eld component Hx is assumed to be zero, is the y-directed magnetic ®eld component Hy . So according to Eq. (4.37), the semivectorial wave equation is @2 Hy @x2
1 @er @Hy @2 Hy 2
k02 er er @x @x @y
b2 Hy 0;
4:42
which, according to Eq. (4.39), can be rewritten as @2 Hy @ 1 @Hy 2
k02 er er @x er @x @y
b2 Hy 0:
4:43
B. Quasi-TM Mode The principal ®eld component in the electric ®eld representation for the quasi-TM mode, where the x-directed electric ®eld component Ex is assumed to be zero, is the y-directed electric ®eld component Ey . According to Eq. (4.33), the semivectorial wave equation for the quasi-TM mode is therefore @2 Ey @2 Ey @ 1 @er 2 E
k02 er @y er @y y @x2 @y
b2 Ey 0;
4:44
4.3
FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
which, according to Eq. (4.35), can be rewritten as @2 Ey @ 1 @
e E
k02 er b2 Ey 0: @y er @y r y @x2
127
4:45
The principal ®eld component for the quasi-TM mode in the magnetic ®eld representation, where the y-directed magnetic ®eld component Hy is assumed to be zero, is the x-directed magnetic ®eld component Hx . As before, the semivectorial wave equation based on Eq. (4.36) is @2 Hx @2 Hx 2 @x2 @y
1 @er @Hx
k02 er er @y @y
b2 Hx 0;
which, according to Eq. (4.38), can be rewritten as @2 Hx @ 1 @Hx er
k02 er b2 Hx 0: @y er @y @x2
4:46
4:47
4.2.3 Scalar Wave Equation In the vectorial and semivectorial wave equations discussed above, the derivatives of relative permittivity er with respect to the x and y coordinates are taken into consideration. If we assume these derivatives to be zero, or that @er 0 @x
and
@er 0; @y
4:48
the wave equations can be reduced to the scalar wave equation @2 f @2 f
k02 er @x2 @y2
b2 f 0;
4:49
where f is a wave function and designates a scalar ®eld.
4.3 FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS We can derive the ®nite-difference expressions for the semivectorial wave equations by using the ®nite differences discussed in Section 4.1 to approximate the derivatives of the semivectorial wave equations for the
128
FINITE-DIFFERENCE METHODS
electric ®eld or the magnetic ®eld representations for the quasi-TE and quasi-TM modes. The currently available CAD software for 2D crosssectional analyses and 3D beam propagation analyses makes use of the ®nite-difference expressions discussed in this section. Figure 4.3 shows a discretization used in a 2D cross-sectional analysis of optical waveguides. Here, the pair
p; q is assumed to correspond to the
x; y coordinates of a node. It should be noted that the interface of two materials is set midway between two nodes in order to minimize the error caused by the difference approximation. Although the discretization used by Stern [1, 2] was an equidistant one, the more versatile scheme described here is nonequidistant, with discretization widths e and w in the x direction and n and s in the y direction. The scalar case is also brie¯y discussed here. Although the variable transformations x xk0 and y yk0 are useful for reducing the round-off error in an actual calculation, they are not used here for simplicity. Reasonable results for the facet re¯ectivities of 3D semiconductor optical waveguides [3] have been obtained by incorporating the semivectorial ®nite-difference expressions discussed in this section into the bidirectional method of line BPM (MoL-BPM) [4].
4.3.1 Quasi-TE Mode A. Ex Representation The Ex representation wave equation derived for the quasi-TE mode [Eq. (4.40)] is @2 Ex @ 1 @er @ 2 Ex
k02 er E x @x er @x @x2 @y2
b2 Ex 0:
4:50
FIGURE 4.3. Nonequidistant discretization for the ®nite-difference method.
4.3
FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
129
What we want to do now is derive the ®nite-difference expression for this equation. As shown in Fig. 4.3, the ®eld components and coordinates at nodes are expressed as Ep1;q E
xp1 ; yq ;
4:51
Ep;q E
xp ; yq ;
4:52
E
xp 1 ; yq ;
4:53
Ep;q1 E
xp ; yq1 ;
4:54
Ep;q
4:55
Ep
1;q
1
E
xp ; yq 1 ;
n yq
yq 1 ;
4:56
s yq1
yq ;
4:57
e xp1
xp ;
4:58
xp 1 :
4:59
w xp Using for Ep1;q, Ep around
p; q, we get Ep1;q Ep
1;q
Ep;q1 Ep;q
1
1;q ,
Ep;q1 , and Ep;q
1
Taylor series expansions
1 @E 1 @2 E Ep;q e e2 O
e3 ; 1! @x p;q 2! @x2 p;q 1 @E 1 @2 E Ep;q w w2 O
w3 ; 1! @x p;q 2! @x2 p;q 1 @E 1 @2 E Ep;q s s2 O
s3 ; 1! @y p;q 2! @y2 p;q 1 @E 1 @2 E Ep;q n n2 O
n3 : 1! @y p;q 2! @y2 p;q
4:60
4:61
4:62
4:63
First, we derive the ®nite-difference expression for the ®rst term in Eq. (4.50). Multiplying Eq. (4.60) by w and Eq. (4.61) by e and then adding them, we ®nd that wEp1;q eEp
1;q
@2 E e2 w w2 e @x2 p;q 2! e3 w w3 e @4 E e4 w w4 e 4 : 3! @x 4!
w eEp;q @3 E 3 @x p;q
p;q
130
FINITE-DIFFERENCE METHODS
Therefore @2 E 2! fwEp1;q eEp 1;q
e wEp;q g 2 @x p;q ew
e w @3 E 2! ew
e w
e w @x3 p;q ew
e w 3! @4 E 2! ew
e w
e2 ew w2 @x4 p;q ew
e w 4! 2 fwEp1;q eEp 1;q
e wEp;q g ew
e w @3 E e w @4 E e2 ew w2 @x3 p;q 3 @x4 p;q 12 and
@2 E 2 2 Ep1;q E 2 @x p;q e
e w w
e w p
2 E : ew p;q
1;q
4:64
4:65
As shown in Section 4.1, the error caused by the ®nite-difference approximation in Eq. (4.65) is O
e w when e 6 w (nonequidistant discretization) and is O
Dx2 when e w Dx (equidistant discretization). We use a similar procedure to obtain the ®nite-difference expression for the derivative with respect to y [the third term in Eq. (4.50)]. Multiplying Eq. (4.62) by n and Eq. (4.63) by s and adding them, we get @2 E s2 n n2 s nEp;q1 sEp;q 1
n sEp;q 2 @y p;q 2! 3 3 3 4 @ E s n n s @ E s 4 n n4 s 3 4 : @y 3! @y 4! p;q
p;q
Therefore @2 E 2! fnEp;q1 sEp;q 1
n sEp;q g @y2 p;q ns
n s @3 E 2! ns
s n
s n 3 @y p;q ns
n s 3! 4 @ E 2! ns
n s
n2 ns s2 @y4 p;q ns
n s 4!
4:66
4.3
FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
131
and @2 E 2 2 jp;q Ep;q1 E 2 @y s
n s n
n s p;q
1
2 E : sn p;q
4:67
According to Eq. (4.66), the error caused by the ®nite-difference approximation in Eq. (4.67) is O
s n when n 6 s (nonequidistant discretization) and is O
Dy2 when s n Dy (equidistant discretization). Now, let us consider the second term in Eq. (4.50), which includes a derivative of relative permittivity er. For simplicity here, expression (4.6) is used to represent derivatives such as @=@x and @2 =@x2 . The difference center of the equations under discussion is
p; q. Here, we introduce
p 12 ; q as a hypothetical difference center between nodes
p; q and
p 1; q and introduce
p 12 ; q as a hypothetical difference center between nodes
p 1; q and
p; q. Using these two hypothetical difference centers, we can get the following two Taylor series expansions:
1 @er E er @x x
p1=2;q
1 @er E er @x x
1 1 1 @er e Ex p;q 1! er @x p;q 2
2 1 1 @er e 2 Ex 2! er @x p;q 2
1 @er E er @x x
p 1=2;q
3 1 1 @er e 3 ; Ex 3! er @x p;q 2
1 @er E er @x x
p;q
4:68
1 1 1 @er w Ex 1! er @x p;q 2
2 1 1 @er w 2 Ex 2! er @x p;q 2
3 1 1 @er w 3 Ex : 3! er @x p;q 2
4:69
132
FINITE-DIFFERENCE METHODS
Subtracting Eq. (4.69) from Eq. (4.68), we get
1 @er E er @x x
p1=2;q
1 @er E er @x x
1 @er E er @x x
p 1=2;q
1
2 e w 1 @er 1
e w
e Ex 2 er @x 4 p;q p;q 2!
1 @er E er @x x
3
1
e w
e2 ew 8 p;q 3!
w2
w
:
Thus, the ®rst derivative is ( @ 1 @er 2 1 @er 1 @er E E E @x er @x x p;q e w er @x x p1=2;q er @x x p
1 @er E er @x x
1 @er E er @x x
2
e
p;q
1=2;q
w 4
p;q
3
)
e2
ew 24
w2
:
4:70
Finally, we get ( @ 1 @er 2 1 @er 1 @er E E E @x er @x x p;q e w er @x x p1=2;q er @x x p
) 1=2;q
:
4:71 According to Eq. (4.70), the error caused by the ®nite-difference approximation in Eq. (4.71) is O
e w when e 6 w (nonequidistant discretization) and is O
Dx2 when e w Dx (equidistant discretization). The next step is to derive the ®nite-difference expression for Eq. (4.71), and this is done by calculating the two terms within the brackets on the right-hand side.
4.3
FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
133
Calculating Taylor series expansions of Ep1;q and Ep;q around the hypothetical difference center
p 12 ; q, we get Ep1;q
Ep;q
e 1 1 @E Ep1=2;q 1! @x p1=2;q 2 2! e 3 1 @3 E O
e4 ; 3! @x3 p1=2;q 2 e 1 1 @E Ep1=2;q 1! @x p1=2;q 2 2! e 3 1 @3 E O
e4 : 3! @x3 p1=2;q 2
e 2 @2 E @x2 p1=2;q 2
4:72 e 2 @2 E @x2 p1=2;q 2
4:73
Adding Eq. (4.72) to Eq. (4.73), we get Ep1;q Ep;q
e2 @4 E @2 E 1 e 4 2Ep1=2;q 2 4 : @x p1=2;q 2 @x p1=2;q 12 2
Therefore Ep1=2;q 12
Ep1;q Ep;q O
e2 :
4:74
Using the expression er
p; q for er
xp ; yq and calculating Taylor series expansions of er
p 1; q and er
p; q around the hypothetical difference center
p 12 ; q, we get er
p 1; q er p
1 2;q
e 1 @2 e e 2 1 @er r 1! @x p1=2;q 2 2! @x2 p1=2;q 2
e 3 1 @3 er O
e4 ;
4:75 3! @x3 p1=2;q 2 e 1 @2 e e 2 1 @er r 1 er
p; q er p 2 ; q 1! @x p1=2;q 2 2! @x2 p1=2;q 2 e 3 1 @3 er O
e4 :
4:76 3! @x3 p1=2;q 2
134
FINITE-DIFFERENCE METHODS
And adding Eq. (4.75) to Eq. (4.76), we get e 2 1 @2 er er
p 1; q er
p; q 2er p ; q 2 2 @x p1=2;q 2 4 @e 1 e 4 4r : @x 12 2
p1=2;q
Therefore er
p 12 ; q 12 fer
p 1; q er
p; qg O
e2 :
4:77
On the other hand, subtracting Eq. (4.76) from Eq. (4.75), we get er
p 1; q
@er @3 er 2 e 3 er
p; q e : @x p1=2;q @x3 p1=2;q 3! 2
Thus @er 1 fe
p 1; q @x p1=2;q e r
er
p; qg O
e2 :
4:78
Combining Eqs. (4.77) and (4.78), we get 1 1 @er 1 2 fe
p 1; q e
p; qg O
e r er @x p1=2;q 2 r 1 2 fer
p 1; q er
p; qg O
e e 2 2 O
e er
p 1; q er
p; q 1 2 fer
p 1; q er
p; qg O
e e
2 er
p 1; q er
p; q O
e: e er
p 1; q er
p; q
4:79
4.3
FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
135
Finally, we can derive the following equation by using Eqs. (4.74) and (4.79): 1 @er 2 er
p 1; q er
p; q E O
e er @x p1=2;q e er
p 1; q er
p; q Ep1;q Ep;q 2 O
e 2
1 er
p 1; q er
p; q Ep;q O
e:
E e er
p 1; q er
p; q p1;q
4:80
Now that we have the ®rst term within the brackets on the right-hand side of Eq. (4.71), we can derive the second term by using the same procedure. Calculating Taylor series expansions of Ep;q and Ep 1;q around the hypothetical difference center
p 12 ; q, we get Ep;q
Ep
1;q
w 1 1 @E Ep 1=2;q 1! @x p 1=2;q 2 2! w3 1 @3 E O
w4 ; 3! @x3 p 1=2;q 2 w 1 1 @E Ep 1=2;q 1! @x p 1=2;q 2 2! w3 1 @3 E O
w4 : 3! @x3 p 1=2;q 2
@2 E @x2 p
1=2;q
w2 2
4:81
@2 E @x2 p
1=2;q
w2 2
4:82
Adding Eq. (4.81) to Eq. (4.82), we get Ep
1;q
Ep;q
w2 @2 E 2Ep 1=2;q 2 @x p 1=2;q 2 4 @E 1 w4 4 : @x 12 2 p 1=2;q
Therefore Ep
1=2;q
12
Ep;q Ep
1;q
O
w2 :
4:83
136
FINITE-DIFFERENCE METHODS
Similarly, calculating Taylor series expansions of er
p 1; q and er
p; q around the hypothetical difference center
p 12 ; q, we get w 1 1 @er er
p 1; q er p 1! @x p 1=2;q 2 2! w2 1 @3 e w3 r O
w4 ; 2 3! @x3 p 1=2;q 2 w 1 1 1 @er ;q er
p; q er p 2 1! @x p 1=2;q 2 2! w2 1 @3 e w3 r O
w4 : 2 3! @x3 p 1=2;q 2
@2 er @x2 p
1 ;q 2
1=2;q
4:84 @2 er @x2 p
1=2;q
4:85
Adding Eq. (4.84) to Eq. (4.85) gives
er
p
1; q er
p; q 2er
p @4 er 4 @x p ;
er
p
1 2 ; q
12 fer
p
1 2 ; q
1=2;q
@2 er @x2 p
1=2;q
w2 2
1 w4 ; 12 2
1; q er
p; qg O
w2 ;
4:86
whereas subtracting Eq. (4.85) from Eq. (4.86) gives
er
p; q ;
er
p @er @x p
@er 1; q @x p
1=2;q
@3 er w 3 @x p 1=2;q
1 fe
p; q w r
er
p
1=2;q
2 w3 ; 3! 2
1; qg O
w2 :
4:87
4.3
FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
137
Combining Eqs. (4.86) and (4.87), we get 1 1 @er 1 2 fer
p 1; q er
p; qg O
w er @x p 1=2;q 2 1 2 fe
p; q er
p 1; qg O
w w r 2 2 O
w er
p; q er
p 1; q 1 2 fer
p; q er
p 1; qg O
w e
2 er
p; q er
p e er
p; q er
p
1; q O
w: 1; q
4:88
Finally, we can derive the following equation by using Eqs. (4.83) and (4.88): 1 @er 2 er
p; q er
p 1; q E O
w er @x p 1=2;q w er
p; q er
p 1; q Ep;q Ep 1;q 2 O
w 2
1 er
p; q er
p w er
p; q er
p
1; q
E Ep 1; q p;q
1;q
O
w:
4:89
Now that we have obtained the two terms within the brackets on the right-hand side of Eq. (4.71) by using Taylor series expansions, let us further our understanding by deriving equations for these terms without using Taylor series expansions. With respect to the ®rst term, the difference center is
p 12 ; q. To set the difference center at
p 12 ; q, we need to take the average of
p 1; q and
p; q. Thus, we get the relations Ep1=2;q 12
Ep1;q Ep;q ; er
p 12 ; q 12 fer
p 1; q er
p; qg; @er 1 fer
p 1; q er
p; qg: @x p1=2;q e
4:90
4:91
4:92
138
FINITE-DIFFERENCE METHODS
We can immediately derive the equation 1 @er 1 er
p 1; q er
p; q Ep;q : E
E er @x p1=2;q e er
p 1; q er
p; q p1;q
4:93
With respect to the second term, on the other hand, the hypothetical difference center is
p 12 ; q. To set the difference center at
p 12 ; q, we need to take the average of
p; q and
p 1; q. Thus, we get the relations Ep er
p
@er @x p
1=2;q
12
Ep;q Ep
1 2 ; q
12 fer
p; q er
p
1=2;q
1 fe
p; q w r
1;q ;
er
p
4:94 1; qg;
4:95
1; qg:
4:96
Again we can immediately derive the equation 1 @er E er @x p
1=2;q
1 er
p; q er
p w er
p; q er
p
1; q
E Ep 1; q p;q
1;q :
4:97
We can see that Eqs. (4.93) and (4.97) are equivalent to Eqs. (4.80) and (4.89). Let us return here to the main topic. Substituting Eqs. (4.80) and (4.89) into Eq. (4.71), we get @ 1 @er 2 1 er
p 1; q er
p; q Ep;q E
E @x er @x e w e er
p 1; q er
p; q p1;q 1 er
p; q er
p w er
p; q er
p
1; q
E Ep 1; q p;q
1;q :
4:98
4.3
FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
139
Substituting Eqs. (4.65), (4.67), and (4.98) into Eq. (4.50), we get 2 2 2 2 Ep1;q Ep 1;q Ep;q e
e w w
e w ew ew 1 er
p 1; q er
p; q Ep;q
E e er
p 1; q er
p; q p1;q 1 er
p; q er
p 1; q
E Ep 1;q w er
p; q er
p 1; q p;q 2 2 2 E E E fk02 er
p; q s
s n p;q1 n
s n p;q 1 ns p;q 2 er
p; q er
p 1; q 1 E w
e w er
p; q er
p 1; q p 1;q 2 er
p; q er
p 1; q 1 E e
e w er
p; q er
p 1; q p1;q 2 2 er
p; q er
p 1; q ew w
e w er
p; q er
p 1; q 2 er
p; q er
p 1; q E e
e w er
p; q er
p 1; q p;q
2 E n
s n p;q
1
2 E s
s n p;q1
2 2er
p 1; q E w
e w er
p; q er
p 1; q p
1 E fk02 er
p; q ns p;q 1;q
2 2er
p 1; q E e
e w er
p; q er
p 1; q p1;q 2 2 er
p; q er
p 1; q ew w
e w er
p; q er
p 1; q 2 er
p; q er
p 1; q E e
e w er
p; q er
p 1; q p;q
2 E n
s n p;q
fk02 er
p; q
1
2 E s
s n p;q1
b2 gEp;q 0:
1 E ns p;q
b2 gEp;q
b2 gEp;q
140
FINITE-DIFFERENCE METHODS
Thus, we get the following ®nite difference expression for Eq. (4.50): aw Ep
1;q
ae Ep1;q an Ep;q
1
as Ep;q1
ax ay Ep;q fk02 er
p; q
b2 gEp;q 0;
4:99
where aw
2 2er
p 1; q ; w
e w er
p; q er
p 1; q
4:100
ae
2 2er
p 1; q ; e
e w er
p; q er
p 1; q
4:101
an
2 ; n
n s
4:102
as
2 ; s
n s
4:103
ax
2 ew
2 er
p; q er
p w
e w er
p; q er
p
1; q 1; q
2 er
p; q er
p 1; q e
e w er
p; q er
p 1; q ay
4 ae aw ; ew
4:104
2 ns
4:105
an
as :
B. Hy Representation The Hy representation wave equation derived for the quasi-TE mode [Eq. (4.43)] is @2 Hy @ 1 @Hy 2
k02 er er @x er @x @y
b2 Hy 0:
4:106
Now, let us derive the ®nite-difference expression for this equation. To do this, we start with the ®rst term of Eq. (4.106). Again using
p 12 ; q as the hypothetical ®nite-difference center between nodes
p; q and
4.3
141
FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
p 1; q and using
p 12 ; q as the hypothetical ®nite-difference center between nodes
p 1; q and
p; q, we get
1 1 @Hy 1 @Hy 1 1 @Hy e er @x p1=2;q er @x p;q 1! er @x p;q 2
2 1 1 @Hy e 2 2! er @x p;q 2
3 1 1 @Hy e 3 ;
4:107 3! er @x p;q 2
1 1 @Hy 1 @Hy 1 1 @Hy w er @x p 1=2;q er @x p;q 1! er @x p;q 2
2 1 1 @Hy w 2 2! er @x p;q 2
3 1 1 @Hy w 3 :
4:108 3! er @x p;q 2 Subtracting Eq. (4.107) from Eq. (4.106), we get 1 @Hy 1 @Hy er @x p1=2;q er @x p 1=2;q
1 1 @Hy e w er @x p;q 2
2 1 @Hy 1
e w
e w er @x p;q 2! 4
3 1 @Hy 1
e w
e2 ew w2 : er @x p;q 3! 8 Thus, ( ) @ 1 @Hy 2 1 @Hy 1 @Hy @x er @x p;q e w er @x p1=2;q er @x p 1=2;q
2
3 2 1 @Hy e w 1 @Hy e ew er @x p;q 4 er @x p;q 24
w2
:
4:109
142
FINITE-DIFFERENCE METHODS
Finally, we obtain ( @ 1 @Hy 2 1 @Hy 1 @Hy @x er @x p;q e w er @x p1=2;q er @x p
) 1=2;q
:
4:110
According to Eq. (4.109), the error caused by the ®nite-difference approximation in Eq. (4.110) is O
e w when e 6 w (nonequidistant discretization) and is O
Dx2 when e w Dx (equidistant discretization). The next step is to derive the ®nite-difference expression for Eq. (4.110) by calculating the two terms within the brackets on the righthand side. Now, we calculate the ®rst term. Calculating Taylor series expansions of Hp1;q and Hp;q around the hypothetical difference center
p 12 ; q, we get Hp1;q
Hp;q
e 1 1 @H Hp1=2;q 1! @x p1=2;q 2 2! e 3 1 @3 H O
e4 ; 3! @x3 p1=2;q 2 e 1 1 @H Hp1=2;q 1! @x p1=2;q 2 2! e 3 1 @3 H O
e4 : 3! @x3 p1=2;q 2
e 2 @2 H @x2 p1=2;q 2
4:111 e 2 @2 H @x2 p1=2;q 2
4:112
Subtracting Eq. (4.112) from Eq. (4.111), we get Hp1=2;q
Hp;q
@H @3 H e3 : e @x p1=2;q @x3 p1=2;q 24
Therefore @H 1
Hp1;q @x p1=2;q e
Hp;q O
e2 :
4:113
4.3
143
FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
Recall that the relative permittivity at
p 12 ; q [Eq. (4.77)] is er
p 12 ; q 12 fer
p 1; q er
p; qg O
e2 :
4:114
Thus, according to Eqs. (4.113) and (4.114), 1 @Hy 12 fer
p 1; q er
p; qg O
e2 er @x p1=2;q 1 2
Hp1;q Hp;q O
e e 2 O
e2 er
p 1; q er
p; q 1 2
Hp1;q Hp;q O
e e
1 2
H e er
p 1; q er
p; q p1;q
1
Hp;q O
e:
4:115
Next, we calculate the second term within the brackets in Eq. (4.110). Calculating Taylor series expansions of Hp;q and Hp 1;q around the hypothetical difference center
p 12 ; q, we get Hp;q Hp
1=2;q
1 @H 1! @x p
1 @3 H 3! @x3 p Hp
1;q
Hp
1=2;q
1=2;q
w3 2
1 @H 1! @x p
1 @3 H 3! @x3 p
1=2;q
1=2;q
w 1 @2 H 2 2! @x2 p
2
w2 2
O
w4 ;
w 1 @2 H 2 2 2! @x 1=2;q p
w3
1=2;q
O
w4 :
4:116
1=2;q
w2 2
4:117
144
FINITE-DIFFERENCE METHODS
Subtracting Eq. (4.117) from Eq. (4.116), we get Hp;q
Hp
1;q
@H @x p
1=2;q
w
@3 H @x3 p
1=2;q
w3 : 24
Therefore @H @x p
1=2;q
1
H w p;q
Hp
1;q
O
w2 :
Recalling that the relative permittivity at
p er
p
1 2 ; q
1 2 ; q
[Eq. (4.86)] is
1; qg O
w2
12 fer
p; q er
p
4:118
4:119
and using Eqs. (4.118) and (4.119), we get 1 @Hy er @x p
1=2;q
12 fer
p; q er
p
1
H w p;q
1; qg O
w2
1
Hp 1;q O
w
2 er
p; q er
p 1 Hp
H w p;q
2
1; q
O
w2
1;q O
w
1 2 w er
p; q er
p
2
1; q
Hp;q
Hp
1;q
O
w:
4:120
We now have the ®nite-difference expressions for the ®rst and second terms of Eq. (4.110). Although it is not shown here, these ®nite-difference expressions can be derived more easily with the procedures used in deriving Eqs. (4.93) and (4.97) (see Problem 3).
4.3
145
FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
Substitution of Eqs. (4.115) and (4.120) into Eq. (4.110) results in @ 1 @Hy 2 1 2
H @x er @x e w e er
p 1; q er
p; q p1;q 1 2 w er
p; q er
p
1; q
Hp;q
Hp
Hp;q
1;q :
Thus, we get the ®nite-difference expression @ 1 @Hy 2 1 2er
p; q er
H @x er @x e w e er
p 1; q er
p; q p1;q 1 2er
p; q
H w er
p; q er
p 1; q p;q
Hp
Hp;q
1;q
2 2er
p; q H e
e w er
p; q er
p 1; q p1;q 2 2er
p; q e
e w er
p; q er
p 1; q 2 2er
p; q H w
e w er
p; q er
p 1; q p;q
2 2er
p; q H w
e w er
p; q er
p 1; q p
1;q :
4:121
On the other hand, the ®nite-difference expression for the derivative with respect to y [Eq. (4.67)] is @2 H 2 2 Hp;q1 H 2 @y p;q s
n s n
n s p;q
1
2 H : sn p;q
4:122
Here, the error caused by the ®nite-difference approximation in Eq. (4.122) is O
s n when n 6 s (nonequidistant discretization) and is O
Dy2 when s n Dy (equidistant discretization).
146
FINITE-DIFFERENCE METHODS
Substituting Eqs. (4.121) and (4.122) into Eq. (4.106), we get the following ®nite-difference expressions for Eq. (4.106): 2 1 2er
p; q
H e w e er
p 1; q er
p; q p1;q
1 2er
p; q
H w er
p; q er
p 1; q p;q
2 2 Hp;q1 H s
n s n
n s p;q
Hp;q
Hp
1;q
2 H fk02 er
p; q ns p;q
1
b2 gHp;q
2 2er
p; q H e
e w er
p; q er
p 1; q p1;q
2 ns
2 2er
p; q e
e w er
p; q er
p 1; q
2 2er
p; q H w
e w er
p; q er
p 1; q p;q
2 2er
p; q H w
e w er
p; q er
p 1; q p
2 2 Hp;q1 H s
n s n
n s p;q
1
1;q
fk02 er
p; q
b2 gHp;q
0
Thus, we get the following ®nal ®nite-difference expression for Eq. (4.106):
aw Hp
1;q
ae Hp1;q an Hp;q
1
as Hp;q1
ax ay Hp;q fk02 er
p; q
b2 gHp;q 0;
4:123
4.3
FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
147
where 2 2er
p; q ; w
e w er
p; q er
p 1; q 2 2er
p; q ; e
e w er
p; q er
p 1; q 2 ; n
n s 2 ; s
n s 2 2er
p; q w
e w er
p; q er
p 1; q 2 2er
p; q e
e w er
p; q er
p 1; q ae aw ; 2 an as : ns
aw
4:124
ae
4:125
an as ax
ay
4:126
4:127
4:128
4:129
It should be noted that since the forms of the derivatives with respect to x differ slightly between the wave equations of the electric ®eld and magnetic ®eld representations, the resultant ®nite-difference expressions for aw, ae , and ax also differ between the two representations. 4.3.2 Quasi-TM Mode A. Ey Representation The Ey representation wave equation for the quasi-TM mode [Eq. (4.44)] is @2 Ey @2 Ey @ 1 @er 2 E
k02 er @y er @y y @x2 @y
b2 Ey 0:
4:130
Comparing this equation with the Ex representation wave equation for the quasi-TE mode [Eq. (4.50)], we ®nd that we can obtain the ®nitedifference expression for the Ey representation for the quasi-TM mode [Eq. (4.130)] by making the following replacements in Eqs. (4.100)± (4.105): x $ y;
4:131
Ex $ Ey :
4:132
148
FINITE-DIFFERENCE METHODS
Thus, we get the following ®nite-difference expression for Eq. (4.130): aw Ep
1;q
ae Ep1;q an Ep;q
1
as Ep;q1
ax ay Ep;q fk02 er
p; q
b2 gEp;q 0;
4:133
where aw
2 ; w
e w
4:134
ae
2 ; e
e w
4:135
an
2 2er
p; q 1 n
n s er
p; q er
p; q
;
4:136
as
2 2er
p; q 1 ; s
n s er
p; q er
p; q 1
4:137
ax
2 ew
ay
2 ns
ae
1
aw ;
2 er
p; q er
p; q n
n s er
p; q er
p; q
4:138 1 1
2 er
p; q er
p; q 1 s
n s er
p; q er
p; q 1
4 an as : ns
4:139
B. Hx Representation The Hx representation wave equation for the quasi-TM mode [Eq. (4.43)] is @ 2 Hx @ 1 @Hx er
4:140
k02 er b2 Hx 0: 2 @y er @y @x Comparing this equation with the Hy representation for the quasi-TE mode [Eq. (4.106)], we ®nd that the ®nite-difference expression for the Hx representation for the quasi-TM mode [Eq. (4.140)] can be obtained by making the following replacements in Eqs. (4.124)±(4.129): x $ y;
4:141
Hy $ Hx :
4:142
4.3
FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
149
Thus, we get the following ®nite-difference expression for the Hx representation: aw Hp
1;q
ae Hp1;q an Hp;q
ax ay Hp;q
fk02 er
1
as Hp;q1
p; q
b2 gHp;q 0;
4:143
where aw
2 ; w
e w
4:144
ae
2 ; e
w e
4:145
an
2 2er
p; q n
n s er
p; q er
p; q
;
4:146
as
2 2er
p; q ; s
n s er
p; q er
p; q 1
4:147
ax
2 ew
ay
2 2er
p; q n
n s er
p; q er
p; q
ae
1
aw ;
4:148 1
2 2er
p; q s
n s er
p; q er
p; q 1
an
as :
4:149
4.3.3 Scalar Mode The scalar wave equation [Eq. (4.49)] is @2 f @2 f
k02 er @x2 @y2
b2 f 0:
4:150
Since the derivatives of relative permittivity with respect to x or y in the semivectorial wave equations are set to zero for the scalar analysis, we can
150
FINITE-DIFFERENCE METHODS
easily get the ®nite-difference expression for the scalar ®nite-difference method (SC-FDM): aw fp
1;q
ae fp1;q an fp;q
1
as fp;q1
ax ay fp;q fk02 er
p; q
b2 gfp;q 0;
4:151
where 2 ; w
e w 2 ; e
w e 2 ; n
n s 2 ; s
n s 2 ae ew 2 an ns
aw
4:152
ae
4:153
an as ax ay
4:154
4:155 aw ;
4:156
as :
4:157
4.4 PROGRAMMING Now, we look at how an eigenvalue matrix equation can be solved using the FDM. The procedure is almost the same as that for the solution using the FEM, described in detail in Chapter 3, so only important differences are dealt with here. The ®nite-difference expression for the semivectorial wave equation was obtained by approximating the derivatives with the difference expressions: aw fp
1;q
ae fp1;q an fp;q
f
ax ay
k02 er
1
as fp;q1
p; qgfp;q
b2 fp;q 0;
4:158
where
p; q corresponds to coordinates
x; y and where fp;q corresponds, for the quasi-TE mode, to the ®eld component Ex or Hy and corresponds, for the quasi-TM mode, to the ®eld component Ey or Hx .
4.4
PROGRAMMING
151
FIGURE 4.4. Meshes for the ®nite-difference method.
Figure 4.4 shows a mesh model in a ®nite-difference scheme in which the whole analysis area is divided into a number of meshes and the nodes are numbered from top to bottom and from left to right. Calculating Eq. (4.158) for each node in an Mx My matrix, we obtain the following eigenvalue matrix equation: Affg b2 ffg:
4:159
Here, b2 is an eigenvalue and ffg is an eigenvector expressed as ffg
f1
f2
f3
fM T ;
4:160
where M is Mx My . It should be noted that, if the variable transformations x xk0 and y yk0 mentioned in Section 4.4 are used, the eigenvalue in Eq. (4.159) is n2eff , where neff
b=k0 is the effective index. Figure 4.5 shows the global matrix A corresponding to the meshes shown in Fig. 4.4.
152
FINITE-DIFFERENCE METHODS
FIGURE 4.5. Form of global matrix A. Here, M Mx My .
We can obtain the propagation constant and ®eld distribution by solving the eigenvalue matrix equation (4.159). In the SV-FDMs, the global matrix A is a nonsymmetric sparse matrix. In the SC-FDM, on the other hand, the global matrix is symmetric, so only half of it has to be calculated. Taking these facts into account and considering the interaction between nodes shown in Figs. 4.4 and 4.5, we can easily understand that the bandwidth of the global matrix is 2My 1 in the SV-FDM and is My 1 in the SC-FDM. The latter bandwidth is the same as that of the global matrix in the ®rst-order SC-FEM. In the actual programming, the node number r is used instead of
p; q. When nodes are numbered from top to bottom and from left to right as shown in Fig. 4.4, the node number r for
p; q can be designated as follows: r
p
1My q;
4:161
where 1 p Mx and 1 q My . The rth row in the eigenvalue matrix equation can be expressed as ar;r
My fr My
ar;r 1 fr
ar;rMy frMy
1
ar;r fr ar;r1 fr1
b2 fr 0;
4:162
4.5
BOUNDARY CONDITIONS
153
where ai;j is an element of the global matrix A. The coef®cients of Eq. (4.158) and Eq. (4.162) correspond as follows: aw $ ar;r
ax ay
k02 er
My ;
4:163
ae $ ar;rMy ;
4:164
an $ ar;r 1 ;
4:165
as $ ar;r1 ;
4:166
p; q $ ar;r :
4:167
The eigenvalue matrix equation is formed in basically the same way as that used to form the eigenvalue matrix equation for the FEM discussed in Section 3.6. There is, however, one important difference. In the FEM, the whole analysis area is divided into a number of elements, and the variational principle or the Galerkin method is applied to each element. Therefore, as shown in Eqs. (3.293) and (3.294), when the FEM is used, the formation of the global matrix requires a summation over all the elements. In the FDM, on the other hand, the derivatives in the wave equations are approximated with ®nite differences. The FDM thus does not require a summation in order to form the global matrix.
4.5 BOUNDARY CONDITIONS In the actual programming, we have to impose boundary conditions on the nodes on the edge of the analysis window. In other words, it is necessary that the effect of nodes outside the analysis window be taken into account at the edge of the window. Here, the Dirichlet, Neumann, and analytical boundary conditions will be discussed. DIRICHLET CONDITION
set to zero. Thus,
A wave function outside the analysis window is f 0:
4:168
NEUMANN CONDITION The normal derivative of a wave function at the edge of the analysis window is set to zero. Thus,
@f 0: @n
4:169
154
FINITE-DIFFERENCE METHODS
In other words, it is assumed that the value of a hypothetical wave function outside the analysis window is equal to that of an actual wave function at the edge of the analysis window. When we assume that the wave number in a vacuum, the effective index, the discretization width at the edge of the analysis window, and the relative permittivity are respectively k0 , neff , D and er
p; q, the analytical wave function outside the analysis window to be connected with a wave function at the edge of the analysis window is assumed to decay exponentially with the decay constant p 2 k0 jneff er
p; qj: ANALYTICAL BOUNDARY CONDITION
exp
q k0 jn2eff er
p; qj D :
4:170
Let us consider here how the boundary conditions can be written into a computer program. Although Eq. (4.162) has to be used for programming, for simplicity we will instead consider Eq. (4.158), written here as
WRITING BOUNDARY CONDITIONS INTO PROGRAMS
aw fp
1;q
ae fp1;q an fp;q
1
as fp;q1
f
ax ay k02 er
p; qgfp;q
b2 fp;q 0:
4:171
a. Left-Hand Boundary (p 1 and except at corners) Consider the left-hand boundary shown as 1 in Fig. 4.6. Here, the discretization width is Dx. When we assume that
p; q is a node on the boundary, the hypothetical node outside the analysis window is
p 1; q and we get fp
1;q
gL fp;q :
4:172
For the Dirichlet, Neumann, and analytical conditions, the coef®cient gL is expressed as 8 > <0 gL 1 p > : exp
k0 jn2eff er
p; qj Dx
Dirichlet;
Neumann;
analytical:
4:173
4.5
BOUNDARY CONDITIONS
155
FIGURE 4.6. Nodes on boundaries and hypothetical nodes outside boundaries.
Substituting Eqs. (4.172) and (4.173) into Eq. (4.171), we get ae fp1;q an fp;q
1
as fp;q1
faw gL
ax ay k02 er
p; qgfp;q
b2 fp;q 0:
4:174
b. Right-Hand Boundary (p Mx and except at corners) Consider the right-hand boundary shown as 2 in Fig. 4.6. Here, the discretization width is again Dx. When we assume that
p; q is a node on the boundary, the hypothetical node outside the analysis window is
p 1; q and we get fp1;q gR fp;q :
4:175
For the Dirichlet, Neumann, and analytical conditions, the coef®cient gR is expressed as 8 > <0 gR 1 p > : exp
k0 jn2eff er
p; qj Dx
Dirichlet;
Neumann;
analytical:
4:176
156
FINITE-DIFFERENCE METHODS
Substituting Eqs. (4.175) and Eq. (4.176) into Eq. (4.171), we get aw fp
1;q
an fp;q
1
as fp;q1 b2 fp;q 0:
fae gR
ax ay k02 er
p; qgfp;q
4:177
c. Top Boundary (q 1 and except at corners) Consider the top boundary shown as 3 in Fig. 4.6. Here, the discretization width is Dy. When we assume that
p; q is a node on the boundary, the hypothetical node outside the analysis window is
p; q 1 and we get fp;q
1
gU fp;q :
4:178
For the Dirichlet, Neumann, and analytical conditions, the coef®cient gU is expressed as 8
Dirichlet; > <0
Neumann; gU 1 p > : 2 exp
k0 jneff er
p; qj Dy
analytical:
4:179
Substituting Eqs. (4.178) and (4.179) into Eq. (4.171), we get aw fp
1;q
ae fp1;q as fp;q1
fan gU
ax ay k02 er
p; qgfp;q
b2 fp;q 0:
4:180
d. Bottom Boundary (q My and except at corners) Consider the bottom boundary shown as 4 in Fig. 4.6. Here, the discretization width is again Dy. When we assume that
p; q is a node on the boundary, the hypothetical node outside the analysis window is
p; q 1 and we get fp;q1 gD fp;q : For the Dirichlet, Neumann, and analytical gD is expressed as 8 > <0 gD 1 p > : exp
k0 jn2eff er
p; qj Dy
4:181 conditions, the coef®cient
Dirichlet;
Neumann;
analytical:
4:182
4.5
157
BOUNDARY CONDITIONS
Substituting Eqs. (4.181) and (4.182) into Eq. (4.171), we get a w fp
1;q
ae fp1;q an fp;q
1
fas gD
ax ay k02 er
p; qgfp;q
b2 fp;q 0:
4:183
e. Left-Top Corner ( p q 1) Consider the left-top corner shown as 5 in Fig. 4.6. Here, the discretization widths are Dx and Dy. When we assume that
p; q is the node at the corner, the hypothetical nodes outside the analysis window are
p 1; q to the left and
p; q 1 to the top. We thus get fp
1;q
fp;q
1
gL fp;q ;
4:184
gU fp;q :
4:185
For the Dirichlet, Neumann, and analytical, conditions, the coef®cients gL and gU are expressed as 8
Dirichlet; > <0
Neumann; gL 1
4:186 p > : 2 exp
k0 jneff er
p; qj Dx
analytical; 8
Dirichlet; > <0
Neumann;
4:187 gU 1 p > : 2 exp
k0 jneff er
p; qj Dy
analytical: Substituting Eqs. (4.184) to (4.187) into Eq. (4.171), we get ae fp1;q as fp;q1 faw gL an gU
ax ay k02 er
p; qgfp;q
b2 fp;q 0:
4:188
f. Left-Bottom Corner ( p 1; q My ) Consider the left- bottom corner shown as 6 in Fig. 4.6. Here, the discretization widths are again Dx and Dy. When we assume that
p; q is the node at the corner, the hypothetical nodes outside the analysis window are
p 1; q to the left and
p; q 1 to the bottom. We thus get gL fp;q ;
4:189
fp;q1 gD fp;q :
4:190
fp
1;q
158
FINITE-DIFFERENCE METHODS
For the Dirichlet, Neumann, and analytical conditions, the coef®cients gL and gD are expressed as 8 > <0 gL 1 p > : exp
k0 jn2eff er
p; qj Dx 8 > <0 gD 1 p > : exp
k0 jn2eff er
p; qj Dy
Dirichlet;
Neumann;
4:191
analytical;
Dirichlet;
Neumann;
4:192
analytical:
Substituting Eqs. (4.189) to (4.192) into Eq. (4.171), we get ae fp1;q an fp;q
1
faw gL as gD
ax ay k02 er
p; qgfp;q
b2 fp;q 0:
4:193
g. Right-Top Corner ( p Mx ; q 1) Consider the right-top corner shown as 7 in Fig. 4.6. Here, the discretization widths are again Dx and Dy. When we assume that
p; q is the node at the corner, the hypothetical nodes outside the analysis window are
p 1; q to the right and
p; q 1 to the top. We thus get fp1;q gR fp;q ;
4:194
gU fp;q :
4:195
fp;q
1
For the Dirichlet, Neumann, and analytical conditions, the coef®cients gR and gU are expressed as 8 > <0 gR 1 p > : exp
k0 jn2eff er
p; qj Dx 8 > <0 gU 1 p > : exp
k0 jn2eff er
p; qj Dy
Dirichlet;
Neumann;
4:196
analytical;
Dirichlet;
Neumann;
analytical:
4:197
4.5
159
BOUNDARY CONDITIONS
Substituting Eqs. (4.194)±(4.197) into Eq. (4.171), we get aw fp
1;q
as fp;q1
fae gR an gU
ax ay k02 er
p; qgfp;q
b2 fp;q 0:
4:198
h. Right-Bottom Corner ( p Mx ; q My ) Consider the right-bottom corner shown as 8 in Fig. 4.6. Here, the discretization widths are again Dx and Dy. When we assume that
p; q is the node at the corner, the hypothetical nodes outside the analysis window are
p 1; q to the right and
p; q 1 to the bottom. We thus get fp1;q gR fp;q ;
4:199
fp;q1 gD fp;q :
4:200
For the Dirichlet, Neumann, and analytical conditions, the coef®cients gR and gD are expressed as 8 > <0 gR 1 p > : exp
k0 jn2eff er
p; qj Dx 8 > <0 gD 1 p > : exp
k0 jn2eff er
p; qj Dy
Dirichlet;
Neumann;
4:201
analytical;
Dirichlet;
Neumann;
4:202
analytical:
Substituting Eqs. (4.199)±(4.202) into Eq. (4.171), we get aw fp
1;q
an fp;q
1
fae gR as gD
ax ay k02 er
p; qgfp;q
b2 fp;q 0:
4:203
In a way similar to that described in Section 3.7 for the FEM, we can calculate the even or the odd mode by assuming Neumann or Dirichlet conditions at the symmetry plane. This increases the numerical ef®ciency in terms of central processing unit (CPU) time and computer memory.
160
FINITE-DIFFERENCE METHODS
4.6 NUMERICAL EXAMPLE The ®nite-difference expressions in this chapter are used in CAD software currently available on the market. Readers can also develop software by themselves. This section brie¯y discusses a calculation model and results calculated using SV-FDM software. Figure 4.7 shows a calculation model that has a 0.4-mm2 core. The refractive indexes for a wavelength of 1.55 mm are 3.5 for the core and 3.1693 for the cladding. The nonequidistant discretization scheme was used in this example. The number of nodes Mx in the horizontal direction and the number of nodes My in the vertical direction were both 96, and the minimum and maximum discretization widths in both directions were respectively 0.025 and 0.05 mm. The calculated effective index neff for both the quasi-TE and quasi-TM modes was 3.2172, and Fig. 4.8 is a three-dimensional plot of the electric ®eld component Ex calculated for the quasi-TE mode. It should be noted that since the normal component of the electric ¯ux density Dx er Ex is continuous at the interface between two media, as shown in Eq. (1.55), the normal component of the electric ®eld Ex is not continuous at the interface. In Fig. 4.8, we can clearly see the discontinuities of Ex . Since, as pointed out in Sections 4.1 and 4.3, the ®nite-difference expressions have the errors of the order of h for nonequidistant discretization and of the order of h2 for equidistant discretization, equidistant discretization is preferable. But because extremely ®ne meshes are necessary for dealing with interfaces at which the refractive index changes abruptly, as in this calculation model, nonequidistant discretization must
FIGURE 4.7. Calculation model.
PROBLEMS
161
FIGURE 4.8. The x-directed electric ®eld component Ex .
be used in order to reduce the amounts of computer memory and CPU time required. The scalar FEM did not give the degenerated results for the square core due to inaccurate boundary conditions, as discussed in Chapter 3. The semivectorial FDMs, on the other hand, can overcome this dif®culty for a small-aspect core structure. PROBLEMS 1. Show that when the second derivative is approximated by Eq. (4.9), f
2
0
2 h2 f 1 h1 h2
h1 h2 f3 h1 f2 ;
h1 h2
P4:1
the error is O
h2 when h1 h2 (equidistant discretization) and is O
h when h1 6 h2 (nonequidistant discretization). ANSWER Multiplying Eq. (4.4) by h2 and Eq. (4.5) by h1 and adding them, we get f
2
0
2 h2 f 1 h1 h2
h1 h2 f3 h1 f2
h1 h2
1
h 3 2
h1 f
3
0 O
h2 ;
P4:2
from which the answer can easily be obtained.
162
FINITE-DIFFERENCE METHODS
2. Derive the ®nite-difference expression (4.121) without using the Taylor series expansion.
ANSWER Since the hypothetical difference center is
p 12 ; q for the ®rst term within the brackets on the right-hand side of Eq. (4.110), we can derive the equations er
p 12 ; q 12 fer
p 1; q er
p; qg
P4:3
and @Hy 1 fHy
p 1; q @x p1=2;q e
Hy
p; qg;
P4:4
from which we can immediately derive 1 @Hy 1 2
H er @x p1=2;q e er
p 1; q er
p; q p1;q
Hp;q :
P4:5
And since the hypothetical difference center is
p 12 ; q for the second term within the brackets, we can derive the equations er
p
1 2 ; q
12 fer
p; q
er
p
1; qg
P4:6
@Hy @x p
1 fH
p; q w y
Hy
p
1; qg;
P4:7
and
1=2;q
from which we can immediately derive 1 @Hy er @x p
1=2;q
1 2 w er
p; q er
p
1; q
Hp;q
Hp
1;q :
P4:8
Substituting Eqs. (P4.5) and (P4.8) into Eq. (4.110) and using some mathematical manipulations, we can easily derive Eq. (4.121).
PROBLEMS
163
FIGURE P4.1. Simple example of the ®nite-difference method.
3. A simple calculation model for the semivectorial FDM is shown in Fig. P4.1. Show the form of the global matrix A of Eq. (4.159).
ANSWER The global matrix A is shown in Fig. P4.2. It should be noted that the interactions between nodes 3 and 4, which correspond to a3;4 and a4;3, and the interactions between nodes 6 and 7, which correspond to a6;7 and a7;6, have to be set to zero. Since the number of nodes in the y direction is 3, the bandwidth is 7. The reason for this is explained in Section 4.4. 4. Calculate effective indexes neff for the quasi-TE and quasi-TM modes of the strip-loaded optical waveguide shown in Fig. P4.3. Assume that the refractive indexes for a wavelength of 1.55 mm are 3.3884 for the InGaAsP of the 1.3-mm band-gap wavelength (1:3Q) core and 3.1693 for the InP cladding.
FIGURE P4.2. Form of global matrix A.
164
FINITE-DIFFERENCE METHODS
FIGURE P4.3. Calculation example of a strip-loaded optical waveguide.
ANSWER The results you get will slightly depend on the CAD software used to calculate them, but the effective index for the quasi-TE mode is 3.2576 and that for the quasi-TM mode is 3.2471. REFERENCES [1] M. Stern, ``Semivectorial polarized ®nite difference method optical waveguides with arbitrary index pro®les,'' IEE Proc. J., vol. 135, pp. 56±63, 1988. [2] M. Stern, ``Semivectorial polarized H ®eld solutions for dielectric waveguides with arbitrary index pro®les,'' IEE Proc. J., vol. 135, pp. 333±338, 1988. [3] K. Kawano, T. Kitoh, M. Kohtoku, T. Takeshita, and Y. Hasumi, ``3-D semivectorial analysis to calculate facet re¯ectivities of semiconductor optical waveguides based on the bi-directional method of line BPM (MoL- BPM),'' IEEE Photon. Technol. Lett., vol. 10, pp. 108±110, 1998. [4] B. Gerdes, B. Lunitz, D. Benish, and R. Pregla, ``Analysis of slab waveguide discontinuities including radiation and absorption effects,'' Electron. Lett., vol. 28, pp. 1013±1015, 1992.
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 5
BEAM PROPAGATION METHODS The analysis methods discussed in the preceding chapters assumed the structures of the optical waveguides to be uniform in the propagation direction. A lot of the waveguides used in actual optical waveguide devices, however, have nonuniform structures such as bends, tapers, and crosses in the propagation direction. In this chapter, we discuss the beam propagation methods (BPMs) that have been developed for the analysis of such nonuniform structures. Various kinds of BPMs, such as the fast Fourier transform (FFT-BPM) [1±4], the ®nite difference (FD-BPM) [4±11], and the ®nite element (FEBPM) [12], have been developed. For the derivatives with respect to the coordinates in the lateral directions, they respectively make use of the fast Fourier transform (FFT), the ®nite-difference (FD) approximation, and the ®nite-element (FE) approximation. The FFT-BPM and the FD-BPM will be discussed here. Beam propagation CAD software is widely available on the market.
5.1 FAST FOURIER TRANSFORM BEAM PROPAGATION METHOD The FFT-BPM [1] had been widely applied to design optical waveguides until the FD-BPM [4] was developed. The FFT-BPM has the following disadvantages due to the nature of the FFT: (1) it requires a long 165
166
BEAM PROPAGATION METHODS
computation time, (2) the discretization widths in the lateral directions must be uniform, (3) the simple transparent boundary condition cannot be used at the analysis boundaries, (4) very small discretization widths cannot be used in the lateral directions, (5) the polarization cannot be treated, (6) it is inadequate for large-index-difference optical waveguides, (7) the number of sampling points must be a power of 2, and (8) the propagation step has to be small. But it is investigated here because the FFT-BPM is historically important and the line of thinking it exempli®es is very interesting and useful. 5.1.1 Wave Equation The scalar Helmholtz equation is expressed as H2 c
x; y; z k02 n2
x; y; zc
x; y; z 0;
5:1
where H2 is the Laplacian H2
@2 @2 @2 2 2 2 @x @y @z
5:2
and k0 is the wave number in a vacuum. Here, the slowly varying envelope approximation (SVEA) is used to approximate the wave function c
x; y; z of the light propagating in the z direction. In this approximation, c
x; y; z is separated into the slowly varying envelope function f
x; y; z and the very fast oscillatory phase term exp
jbz as follows: c
x; y; z f
x; y; z exp
jbz:
5:3
b neff k0 ;
5:4
Here,
where neff is the reference index, for which the refractive index of the substrate or cladding is usually used. Substituting the second derivative of the wave function c
x; y; z with respect to z, @2 c @2 f 2 exp
jbz @z2 @z
2jb
@f exp
jbz @z
b2 f exp
jbz;
5:5
5.1
FAST FOURIER TRANSFORM BEAM PROPAGATION METHOD
167
into Eq. (5.1) and dividing both sides by the exponential term exp
jbz, we get @2 f @z2
2jb
@f H2? f
k02 n2 @z
b2 f 0;
5:6
where H2? is a Laplacian in the lateral directions (i.e., the x and y directions) and is expressed as H2?
@2 @2 : @x2 @y2
5:7
Or, using the relation k02 n2
b2 k02
n2
n2eff ;
5:8
we get 2jb
@f @z
@2 f H2? f k02
n2 @z2
n2eff f:
5:9
Since the second derivative of the wave function f with respect to z is not neglected, Eq. (5.9) is a wide-angle formulation. On the other hand, when the second derivative is neglected, that is, when we assume @2 f 0; @z2
5:10
the wave equation (5.9) is reduced to 2jb
@f H2? f k02
n2 @z
n2eff f:
5:11
The assumption that the second derivative of the wave function f with respect to z can be neglected is called the Fresnel approximation or the para-axial approximation. The equality of the Fresnel approximation to the para-axial approximation will be discussed in Section 5.1.5.
168
BEAM PROPAGATION METHODS
5.1.2 Fresnel Approximation Let us try to solve the wave equation (5.11), which is based on the Fresnel approximation. First, we separate the variables of the wave function f of the Fresnel wave equation into the propagation direction and the lateral directions: f
x; y; z A
x; y exp
gz:
5:12
Substituting Eq. (5.12) into Eq. (5.11) and dividing both sides by f, we get 2jbg H2? k02
n2
n2eff ;
and therefore g
j fH2 k02
n2 2b ?
n2eff g:
5:13
Substituting g in Eq. (5.13) into Eq (5.12), we get f
x; y; z A
x; y exp
j fH2 k02
n2 2b ?
n2eff gz :
5:14
Thus, the wave function f
x; y; z Dz, which advances further than f
x; y; z by Dz in the propagation direction, can be written as f
x; y; z Dz exp
j fH2 k02
n2 2b ?
n2eff g
Dz f
x; y; z:
5:15
We separate the exponential term into the following two terms: f
x; y; z Dz exp
Dz 2 j H? exp
jwf
x; y; z; 2b
5:16
5.1
FAST FOURIER TRANSFORM BEAM PROPAGATION METHOD
169
where w
1 2 2 k
n 2b 0
n2eff Dz
k02 f
n Dn2 2k0 neff eff
n2eff g Dz
k0 Dn Dz:
5:17
Here, we used the relation n neff Dn and, assuming that Dn is suf®ciently small, neglected
Dn2 . Since H2? in Eq. (5.16) is a derivative operator, the following relation holds for the general function f : H2? Dn f
Dn H2? f
H2? Dn f Dn
H2? f
H2? Dn f 6 0:
Dn H2? f
5:18
This relation implies that the ®rst and second operators of Eq. (5.16) cannot be interchanged (i.e., they are not commutable). However, we symmetrize the operators in Eq. (5.16) as f
x; y; z Dz exp
Dz 2 Dz 2 j H? exp
jw exp j H? f
x; y; z: 4b 4b
5:19
Although the reasons are not discussed here because of space limitations, Eq. (5.19) results in errors of the order of
Dz3 . This is one of the reasons that the propagation step Dz in the FFT-BPM has to be small. It should also be pointed out that Eq. (5.16), which generates the unsymmetrical operators, results in larger errors: errors of the order of
Dz2 [2]. Now, let us discuss the physical meaning of each term in Eq. (5.19). When the refractive index is assumed to be uniform in the analysis region, w is equal to zero. Thus, the wave equation Eq. (5.19) is reduced to Dz 2 Dz 2 f
x; y; z Dz exp j H? 1 exp j H? f
x; y; z 4b 4b Dz exp j H2? f
x; y; z:
5:20 2b
170
BEAM PROPAGATION METHODS
This implies that the operator exp
Dz j H2? 2b
5:21
corresponds to the propagation over Dz in free space. Therefore, the ®rst and the third terms of Eq. (5.19) correspond to the free-space propagation of the light over Dz=2. Thus, Eq. (5.19) implies that the wave function at z Dz can be obtained by ®rst advancing the wave function at z by Dz=2 in free space, then giving a phase shift
w due to a phase-shift lens, and ®nally advancing the wave function by another Dz=2 in free space. Next, we will obtain the explicit expression of the free-space propagation operator (5.21) by actually applying it to the wave function. The discrete Fourier transform (i.e., the spectral domain wave function) is h mx nyi MP1 NP1 f
x; y; z exp j2p ; f~ mn
z X Y i0 h0
5:22
where x Dx i;
y Dy h;
0iM
1;
M M m 2 2
1;
0hN
X Dx M ;
Y Dy N ;
1;
N N n 2 2
5:23 1:
Here, X and Y are the widths in the x and y directions. The inverse discrete Fourier transform, on the other hand, is f
x; y; z
MP =2 1
h mx nyi ; f~ mn
z exp j2p X Y N =2
NP =2 1
m M =2 n
5:24
where the coef®cient 1=MN is omitted for simplicity. According to the operator (5.21) and the above discussion, when the wave propagates over Dz=2 in free space, we get the following wave function at z Dz=2:
Dz Dz 2 exp j H? f
x; y; z: f x; y; z 2 4b
5:25
5.1
171
FAST FOURIER TRANSFORM BEAM PROPAGATION METHOD
We can get the wave function at z Dz=2 by replacing z by z Dz=2 in Eq. (5.24):
MP =2 1 Dz f x; y; z 2 m M =2
h mx nyi Dz ~f z exp j2p : mn 2 X Y N =2
NP =2 1 n
5:26 On the other hand, substituting the right-hand side of Eq. (5.24) for f
x; y; z on the left-hand side of Eq. (5.25), we get
=2 1 Dz Dz 2 MP exp j H? f x; y; z 2 4b m M =2 h mx nyi exp j2p : X Y
NP =2 1 n N =2
f~ mn
z
5:27
As the right-hand side of this equation can be rewritten as h mx nyi Dz 2 j H? exp j2p 4b X Y M =2 n N =2 ( ) 2 2 2 MP =2 1 NP =2 1
2p m n f~ mn
z exp j Dz X Y 4b m M =2 n N =2 h mx nyi ; exp j2p X Y
MP =2 1 m
NP =2 1
f~ mn
z exp
we get another expression for the wave function at z Dz=2:
MP =2 1 Dz f x; y; z 2 m M =2 (
NP =2 1 n N =2
f~ mn
z
)
2p2 m2 n 2 Dz exp j X Y 4b h mx nyi exp j2p : X Y
5:28
172
BEAM PROPAGATION METHODS
Since the wave functions f
x; y; z Dz=2 given in Eqs. (5.26) and (5.28) have to be equal to each other, we get the important relation f~ mn
( ) 2 2 2 Dz
2p m n z f~ mn
z exp j Dz : 2 X Y 4b
5:29
Equation (5.29) shows the relation between the spectral domain wave function f~ mn
z Dz=2 at z Dz=2 and the spectral domain wave function f~ mn
z at z. The exponential term on the right-hand side of Eq. (5.29), (
)
2p2 m2 n 2 Dz ; exp j X Y 4b corresponds to the propagation over Dz=2 in free space. We also ®nd that Eq. (5.28) is the inverse discrete Fourier transform of the function (
) 2 2 2
2p m n f~ mn
z exp j Dz : X Y 4b From these discussions, we draw the conclusion that the application of the operator
Dz 2 exp j H? ; 4b
5:30
which corresponds to the propagation over Dz=2 in free space, to the space-domain wave function f
x; y; z at z is equivalent to application of the mathematical procedure T
1
1 2 2 exp j
kx ky Dz T 4b
5:31
to the space-domain wave function f
x; y; z. Here, the symbols T and T 1 respectively represent the discrete Fourier transform and the inverse discrete Fourier transform. The variables kx and ky are expressed as kx
2pm X
and ky
2pn : Y
5:32
5.1
FAST FOURIER TRANSFORM BEAM PROPAGATION METHOD
173
FIGURE 5.1. Calculation of one period in the FFT-BPM.
Thus, the FFT-BPM calculation procedures for a period Dz can be summarized as follows, where steps 1±5 correspond to the labels in Fig. 5.1: 1. At the propagation position z, calculate the spectral domain wave function f~ mn
z in the Fourier transform domain by taking the Fourier transform of the space-domain wave function f
x; y; z. 2. To get the transformed wave function f~ mn
z Dz=2 at z Dz=2, multiply (
)
2p2 mx2 ny2 exp j Dz X Y 4b
5:33
by the spectral domain wave function f~ mn
z obtained in step 1. This multiplication corresponds to the propagation over the distance Dz=2 in free space. 3. Taking the inverse Fourier transform of the spectral domain wave function f~ mn
z Dz=2 obtained in step 2, obtain the space-domain wave function f
x; y; z Dz=2 just in front of the phase-shift lens. Then, multiplying the phase-shift term exp
jw due to the phaseshift lens by the space-domain wave function f
x; y; z Dz=2, obtain the space-domain wave function just after the phase-shift lens: Dz : exp
jwf x; y; z 2
5:34
174
BEAM PROPAGATION METHODS
4. Taking the Fourier transform of the space-domain wave function just after the phase-shift lens and multiplying it by (
)
2p2 mx2 ny2 Dz ; exp j X Y 4b
5:35
corresponding to the propagation over Dz=2 in free space, obtain the spectral domain wave function f~ mn
z Dz at z Dz. 5. When the space-domain wave function f
x; y; z Dz at z Dz is necessary, take the inverse Fourier transform of the spectral domain wave function f~ mn
z Dz obtained in step 4. Repeating steps 1±5, we can get the space-domain wave function at the target propagation position. It should be noted that if the space-domain wave function at each z Dz is not necessary, one should return directly to step 2 from step 4 and repeat steps 2±4. 5.1.3 Wide-Angle Formulation Up to this stage (i.e., in the Fresnel approximation), the second derivative of the slowly varying envelope function f with respect to z has been neglected. Now, we return to the wide-angle wave equation (5.9), which contains the second derivative. Readers who feel they have no need for such a discussion of the wide-angle formulation for the FFT-BPM can skip this section. Substituting the wave function given by Eq. (5.12) into Eq. (5.9) and dividing both sides by f, we get 2jbg
g2 H2? k02
n2
n2eff :
Therefore g2
2jbg H2? k02
n2
n2eff 0
and q b2 fH2? k02
n2 n2eff g q jb j b2 fH2? k02
n2 n2eff g:
g jb
5:36
5.1
FAST FOURIER TRANSFORM BEAM PROPAGATION METHOD
175
Since the wave is supposed to propagate in the z direction, the minus sign in Eq. (5.36) should be used so that q
5:37 gj b b2 fH2? k02
n2 n2eff g : Substituting this g into Eq. (5.12), we get the wave function at z: q f
x; y; z A
x; y exp j b b2 fH2? k02
n2 n2eff g z :
5:38 Finally, we get the wave function at z Dz: q f
x; y; z Dz exp j b b2 fH2? k02
n2 n2eff g Dz f
x; y; z:
5:39 Now, we expand the second term inside the exponential function of Eq. (5.39):
b2 fH2? k02
n2
n2eff g1=2 1=2 k 2
n2 n2eff
b2 H2? 1=2 1 0 2 b H2? 1 k02
n2 n2eff
b2 H2? 1=2 1 2 b2 H2? ;
k02
n2
n2eff b2 H2?
b2 H2? 1=2 12 k02
n2
n2eff
b2 H2?
1=2
:
5:40
The Laplacian H2? in the lateral directions corresponds to the square of the wave numbers in the lateral directions. The wave numbers in the lateral directions are much smaller than the wave number in the propagation direction (i.e., the propagation constant). Thus, the assumption H2? b2 is concluded to be valid. In addition, taking
n2 n2eff
n neff
n neff 2neff
n neff and b k0 neff into consideration, we can simplify the right-hand side of Eq. (5.40) to
b2 H2? 1=2 k02
neff
n b
neff
b2 H2? 1=2 k0
n
neff :
5:41
176
BEAM PROPAGATION METHODS
Substituting Eq. (5.41) into Eq. (5.39), we get f
x; y; z Dz exp j b exp
jfb
q b2 fH2? k02
n2 n2eff g Dx f
x; y; z
b2 H2? 1=2
exp jk0
n
k0
n
neff g Dzf
x; y; z 2
neff Dz exp
jf
b H2? 1=2
bg Dzf
x; y; z:
5:42
Since the propagation constant b has the order of the inverse of the wavelength, it is a large number. Inside the exponential function,
b2 H2? 1=2 , which is also the large number, is subtracted by b. This numerical process can therefore cause large round-off errors. To reduce the errors, we modify the term inside the exponential function:
b2 H2? 1=2
b
f
b2 H2? 1=2 bg f
b2 H2? 1=2 bg
b2 H2? 1=2 b H2? :
b2 H2? 1=2 b
Symmetrizing the operators as we do in the wave equation (5.19) based on the Fresnel approximation, we ®nally get f
x; y; z Dz exp jk0
n
neff Dz exp j Dz
H2? f
x; y; z
b2 H2? 1=2 b
Dz H2? exp j exp
jw 2
b2 H2? 1=2 b Dz H2? f
x; y; z: exp j 2
b2 H2? 1=2 b
5:43
This is a wide-angle formulation, so the operator for the propagation over Dz=2 in the Fresnel approximation corresponds to that in the wideangle formulation as follows: 0 1 0 1 B exp@
H2? C B Dz C $ exp j A @ A: 2
b2 H2? 1=2 b Fresnel approximation Wide-angle formulation j
Dz 2 H 4b ?
5:44
5.1
FAST FOURIER TRANSFORM BEAM PROPAGATION METHOD
177
The calculation procedures for the beam propagation based on the wide-angle formulation are exactly the same as those for the beam propagation method based on the Fresnel approximation.
5.1.4 Analytical Boundaries To reduce the re¯ections at analysis windows, we need some arti®cial boundary conditions. Since the simple transparent boundary condition (TBC) that is normally used in the FD-BPM (and which will be discussed in a later section) cannot be used in the FFT-BPM, some other arti®cial boundary conditions using complex refractive index materials or window functions have to be used to make the propagating ®elds decay properly near the edges of the analysis window. These arti®cial boundaries in the FFT-BPM usually require some experiences in optimizing the parameters to minimize the re¯ections.
5.1.5 Further Investigation The free-space propagation operators for the Fresnel approximation and for the wide-angle formulation that were given in expression (5.44) are further examined in this section. According to Eqs. (5.12) and (5.24), the slowly varying envelope function f
x; z in a 2D case is expressed as f
x; z
P m
fm
x; z;
5:45
where fm
x; z f~ m exp
jkx x exp
gz
5:46
and kx
2pm : X
5:47
Here, X is the total width of the analysis region and m is an integer. First, let us verify that the Fresnel approximation is equivalent to the para-axial approximation by using a 1D case, where we reduce the
178
BEAM PROPAGATION METHODS
derivative operator H2? in Eq. (5.44) to the square of a wave number in a lateral direction, kx2 . The free-space propagation operators for the Fresnel approximation and the wide-angle formulation are respectively Dz 2 exp j kx ; 4b Dz exp j 2
b2
! kx2 : kx2 1=2 b
5:48
5:49
Thus, the phase term jF for the Fresnel approximation and the phase term jW for the wide-angle formulation are jF
Dz 2 k ; 4b x
jW
Dz 2
b2
5:50 kx2 : kx2 1=2 b
5:51
Equation (5.4) can be used to express b as a function of the reference index neff . Assuming that the reference index neff is equal to the effective index, we can, as shown in Fig. 5.2, reduce b to the z-directed component of the wave number k of a whole wave (i.e., the propagation constant). Since k and b are generally much larger than the x-directed wave number kx, the approximation k b is valid. Thus, kx can be approximated as kx k sin y b sin y:
5:52
FIGURE 5.2. Relation between k, kx , and b. Here, y is the propagation angle.
5.1
FAST FOURIER TRANSFORM BEAM PROPAGATION METHOD
179
Substituting Eq. (5.52) into Eqs. (5.50) and Eq. (5.51), we can reduce the phase term jF for the Fresnel approximation and the phase term jW for the wide-angle formulation to jF
Dz 2 2 Dz y y b sin y 4b2 sin2 cos2 ; 4b 4b 2 2
jW
Dz 2
b2
kx2 Dz 1=2 2
b2 kx2 b
5:53
b2 sin2 y b2 sin2 y1=2 b
Dz b2 4 sin2
y=2 cos2
y=2 Dz b2 4 sin2
y=2 cos2
y=2 2 b
cos y 1 2 2b cos2
y=2
Dz 2 2 y 4b sin : 4b 2
5:54
Comparing Eq. (5.53) with Eq. (5.54), we ®nd that the Fresnel approximation is a good approximation for the wide-angle formulation only when the propagation angles are so small that cos2
y=2 can be considered to be nearly equal to 1. Next, we discuss the discretization width in the lateral directions. Since the free-space propagation operator (5.48) for the Fresnel approximation is purely imaginary, it satis®es the unitary condition. The free-space propagation operator (5.49) for the wide-angle formulation, on the other hand, has to satisfy b2 kx2
5:55
if the unitary condition is to be satis®ed. That is, when 2 2pm
k0 neff X 2
5:56
is not satis®ed, the denominator of the argument of the operator (5.49) becomes complex. Thus, the free-space propagation operator (5.49) itself also becomes complex and does not satisfy the unitary condition. This causes the power of the propagating optical wave to dissipate. Assuming the maximum number of m to be M =2, we can rewrite condition (5.56) as 2 2 2p M p p 2 ;
k0 neff X 2 X =M Dx 2
5:57
180
BEAM PROPAGATION METHODS
where Dx X =M is the discretization width. Since the relation k0 2p=l0 holds for the wave number in a vacuum (l0 is the wavelength in the vacuum), Eq. (5.57) implies that the discretization width Dx in the lateral direction has to satisfy the condition l0 Dx 2neff
5:58
if the power of the beam is to be conserved. The condition (5.58) implies that a very small discretization cannot be used in the wide-angle FFTBPM. For 3D cases, the conditions (5.57) and (5.58) are modi®ed to
2p M
k0 neff X 2 2
2
p 2 X =M
2
2
p Dx
2
5:59
and l p 0 Dx; 2neff
5:60
where we assume that Dx Dy.
5.2 FINITE-DIFFERENCE BEAM PROPAGATION METHOD The FD-BPM is very powerful and has been widely used for optical waveguide design. Of the various FD-BPMs that have been developed, the one with the implicit scheme developed by Chung and Dagli [4] is stateof-the-art from the viewpoints of accuracy, numerical ef®ciency, and stability. Its unconditional stability is particularly advantageous not only because it allows us to use the method in actual design without danger of diversion but also because it allows us to set the propagation step relatively large. In addition, the TBC [13], which is simple and requires no special experience to use, has been developed for the FD-BPM by Hadley [5]. A wide-angle scheme using Pade approximant operators [5, 6] has also been developed by him. These contributions greatly advanced the FD-BPM and have enabled it to be used even in the design of optical waveguides made of high-contrast-index materials, such as semiconductor optical waveguides.
5.2
FINITE-DIFFERENCE BEAM PROPAGATION METHOD
181
5.2.1 Wave Equation To clarify the formulation of the FD-BPM, we must ®rst derive the wave equation. In this section, as in Chapter 2, we assume that the structure of the optical waveguide is uniform in the y direction. The necessary wave equations can be derived from the semivectorial wave equations given in Chapter 4, but let us further our understanding by deriving them from Maxwell's equations. The component representations of Maxwell's equations =3E
jom0 H;
= 3 H joe0 er E
5:61
5:62
were given in Eqs. (2.5)±(2.10), where the relative permeability mr was assumed to be equal to 1. Since the structure in the y direction is assumed to be uniform, the derivatives with respect to y can be set to zero. Thus, Eqs. (2.5)±(2.10) are reduced to
@Ex @z
@Hx @z
@Ey @z
jom0 Hx ;
5:63
@Ez @x
jom0 Hy ;
5:64
@Ey @x
jom0 Hz ;
5:65
@Hy joe0 er Ex ; @z
5:66
@Hz joe0 er Ey ; @x
5:67
@Hy joe0 er Ez : @x
5:68
A. TE Mode Figure 5.3 shows the principal ®eld components for the TE mode. Since both Ey and Hx are the principal ®elds, we need to have the wave equations for both. In the TE mode, as discussed in Section 2.1.1, the x- and z-directed electric ®eld components and the y-directed magnetic ®eld component are zero: Ex Ez Hy 0:
5:69
182
BEAM PROPAGATION METHODS
FIGURE 5.3. Principal ®eld components for TE mode are Ey and Hx .
Substituting Eq. (5.69) into Eqs. (5.63)±(5.68), we obtain for the TE mode the equations
@Hx @z
@Ey jom0 Hx ; @z @Ey jom0 Hz ; @x @Hz joe0 er Ey : @x
5:70
5:71
5:72
1. Ey Representation First, we derive the wave equation for the ydirected electric ®eld component Ey . Substituting the x- and z-directed magnetic ®eld components
Hx Hz
1 @Ey ; jom0 @z 1 @Ey jom0 @x
5:73
5:74
5.2
FINITE-DIFFERENCE BEAM PROPAGATION METHOD
183
obtained from Eqs. (5.70) and (5.71) into Eq. (5.72) yields the following wave equation for the principal electric ®eld component Ey : @2 Ey @2 Ey 2 k02 er Ey 0; @z2 @x
5:75
where k02 o2 e0 m0 . 2. Hx Representation Next, we derive the wave equation for the xdirected magnetic ®eld component Hx . Differentiating Eq. (5.72) with respect to z, we get @2 Hx @z2
@Ey @2 Hz joe0 er : @z @x @z
5:76
In deriving Eq. (5.76), we have assumed the variation of the relative permittivity er along the propagation axis to be negligibly small. That is, @er 0: @z
5:77
This results in the approximation @Ey @Ey @ @e
er Ey r Ey er er : @z @z @z @z
5:78
Thus, it should be noted that the approximation (5.77) is implicitly used in the BPM. From a calculation of @Eq. (5.70)=@x @Eq. (5.70)=@z or the magnetic divergence equation = ? H 0; we get @Hx @Hz 0: @x @z That is, @Hz @z
@Hx : @x
5:79
184
BEAM PROPAGATION METHODS
Substituting this equation and Eq. (5.70) into Eq. (5.76), we can eliminate Hz in Eq. (5.76). This results in the wave equation for the principal magnetic ®eld component Hx : @ 2 Hx @ 2 Hx 2 k02 er Hx 0: @z2 @x
5:80
B. TM Mode Figure 5.4 shows the principal ®eld components for the TM mode. Since both Ex and Hy are principal ®elds, we need to have the wave equations for both. In the TE mode, as discussed in Section 2.1.2, the x- and z-directed magnetic ®eld components and y-directed electric ®eld component are zero: Hx Hz Ey 0:
5:81
Substituting Eq. (5.81) into Eqs. (5.63)±(5.68), we obtain for the TM mode the equations @Ex @z
@Ez @x
jom0 Hy ;
5:82
@Hy joe0 er Ex ; @z
5:83
@Hy joe0 er Ez : @x
5:84
FIGURE 5.4. Principal ®eld components for TM mode are Ex and Hy .
5.2
FINITE-DIFFERENCE BEAM PROPAGATION METHOD
185
1. Ex Representation First, we derive the wave equation for the x-directed electric ®eld component Ex . Differentiating Eq. (5.82) with respect to z, we get @2 Ex @z2
@ 2 Ez @z @x
jom0
@Hy : @z
5:85
Substituting Eq. (5.83) into Eq. (5.85), we can eliminate Hy and obtain @ 2 Ex @z2
@2 Ez @z @x
jom0
joe0 er Ex
k02 er Ex :
5:86
From a calculation of @Eq. (5.83)=@x @Eq. (5.84)=@z or the divergence equation for the electric ¯ux density = ?
er E 0;
5:87
we get @ @
er Ex
er Ez 0: @x @z That is, @Ez @z
1 @
e E : er @x r x
Substituting this equation into Eq. (5.86), we get the wave equation for the principal electric ®eld component Ex : @2 Ex @ 1 @
e E k02 er Ex 0:
5:88 @x er @x r x @z2 2. Hy Representation Next, we derive the wave equation for the ydirected magnetic ®eld component Hy . Substituting the x- and z-directed electric ®eld components Ex Ez
1 @Hy ; joe0 er @z 1 @Hy ; joe0 er @x
5:89
5:90
which are obtained from Eqs. (5.83) and (5.84), into Eq. (5.82) yields @ 1 @Hy @ 1 @Hy
5:91 jom0 Hy @z joe0 er @z @x joe0 er @x
186
BEAM PROPAGATION METHODS
and therefore @ 1 @Hy @ 1 @Hy @z er @z @x er @x
o2 e 0 m 0 Hy
k02 Hy :
Making use of the approximation @ 1 @Hy @z er @z
1 @er @Hy 1 @2 Hy 1 @2 Hy ; e2r @z @z er @z2 er @z2
which follows from Eq. (5.77), we get the wave equation for the principal magnetic ®eld component Hy : @ 2 Hy @ 1 @Hy e k02 er Hy 0: r @x er @x @z2
5:92
5.2.2 FD-BPM Formulation Next, let us discuss the FD-BPM formulation based on the implicit scheme developed by Chung and Dagli [4]. Since the discussions here are limited to 2D problems, the amount of memory required is not large. Thus, the equidistant discretization is used to ensure the second-order accuracy. Further improvement of accuracy has been achieved by Yamauchi et al. [7]. A. TE Mode The wave equation for the y-directed electric ®eld Ey
x; y; z is @2 Ey @2 Ey 2 k02 er Ey 0: @z2 @x
5:93
As in the FFT-BPM, using the slowly varying envelope approximation, we divide the principal ®eld Ey
x; y; z propagating in the z direction into the slowly varying envelope function f
x; y; z and the very fast oscillatory phase term exp
jbz as follows: Ey
x; y; z f
x; y; z exp
jbz;
5:94
5.2
FINITE-DIFFERENCE BEAM PROPAGATION METHOD
187
where b neff k0 :
5:95
Here, k0 is the wave number in the vacuum and neff is the reference index, for which the effective index is usually used. Substituting @2 Ey @2 f 2 exp
jbz @z @z2
2jb
@f exp
jbz @z
b2 f exp
jbz;
5:96
which is obtained from Eq. (5.94), into Eq. (5.93) and dividing both sides of the resultant equation by the exponential term exp
jbz, we get @2 f @z2
2jb
@f @2 f
k02 n2 @z @x2
b2 f 0
5:97
n2eff f;
5:98
or 2jb
@f @z
@2 f @2 f 2 k02
er @z2 @x
where we used the relation er n2 . As discussed in the section covering the FFT-BPM, Eq. (5.98) is the wide-angle formulation. When we assume that @2 f 0; @z2
5:99
Eq. (5.98) is reduced to the Fresnel wave equation 2jb
@f @2 f 2 k02
er @z @x
n2eff f:
5:100
First, we discuss the FD expression for the Fresnel approximation. That for the wide-angle formulation will be covered in Section 5.3. When we use the discretization of the x and z coordinates x p Dx;
5:101
z l Dz;
5:102
188
BEAM PROPAGATION METHODS
where p and q are integers, the following notations are used for the wave function f
x; z and the relative permittivity er
x; z: f
x; z ! flp ;
5:103
er
x; z ! elr
p:
5:104
The next step is the discretization of the Fresnel wave equation (5.100). First, we discretize it in the x direction. The discretization number l, which corresponds to the z coordinate, will be discussed later. The ®rst and the second terms on the right-hand side of Eq. (5.100) are expressed as 1
0 B @2 f 1 B B @x2 Dx B @
fp1 fp Dx |{z}
1 Difference center is p 2
fp1
2fp fp 2
C C C C A
fp
fp 1 Dx |{z} Difference center is p
1 2
1
5:105
Dx |{z} Difference center is p
and k02
er
n2eff f k02 er
p n2eff fp : |{z}
5:106
Difference center is p
Substituting Eqs. (5.105) and (5.106) into Eq. (5.100), we get 2jb
@fp fp1 2fp fp @z
Dx2 aw fp
1
1
k02 er
p
n2eff fp
ax fp ae fp1 k02 er
p
n2eff fp :
Thus, the discretization of the wave equation (5.100) is 2jb
@fp a w fp @z
1
fax k02 er
p
n2eff gfp ae fp1 ;
5:107
5.2
189
FINITE-DIFFERENCE BEAM PROPAGATION METHOD
where we used the de®nitions aw
1 ;
Dx2
5:108
ae
1 ;
Dx2
5:109
ax
2 :
Dx2
5:110
The next step is the discretization of Eq. (5.107) with respect to z. Discretizing the left-hand side of Eq. (5.107) with respect to z, we get
2jb
fl1 flp p : Dz
5:111
It should be noted that, as shown in expression (5.111), the difference center of the left-hand side of Eq. (5.107) is the point l 12 midway between l and l 1. The difference center of the right-hand side of Eq. (5.107) discretized with respect to z should be l 12. Thus, we modify Eq. (5.107) to
2jb
fl1 flp 1 l l p 2 aw fp Dz
1
falx k02 elr
p
n2eff gflp ale flp1
l1 l1 12 al1 k02 el1 w fp 1 fax r
p l1 al1 e fp1 :
n2eff gfl1 p
5:112
Rewriting this equation so that the terms on the left- and right-hand sides respectively contain l 1 and l, we get al1 2jb 1 2 l1 al1 x e 2 k0 er
p neff fl1 fl1 p Dz 2 2 2 p1 l alw l ax 2jb 1 2 l al 2 fp 1 k0 er
p neff flp e flp1 : Dz 2 2 2 2
al1 w fl1 2 p 1
190
BEAM PROPAGATION METHODS
Multiplying both sides of this equation by 2, we get the FD expression for the TE mode: l1 al1 w fp 1
alw flp
al1 x
4jb Dz
k02 el1 r
p
4jb alx k02 elr
p 1 Dz
n2eff
fl1 p
l1 al1 e fp1
n2eff flp ale flp1 :
5:113
Although in Eq. (5.113) we use the wave number in a vacuum, k0 , for ease of understanding, it is recommended that in actual programming the coordinates (i.e., x, y, and z) be multiplied by k0 and the propagation constant b be divided by k0 in order to reduce the round-off errors. The resultant formulation corresponds to dividing both sides of Eq. (5.113) by k02. B. TM Mode The wave equation (5.92) for the principal magnetic ®eld component Hy is @ 2 Hy @ 1 @Hy k02 er Hy 0: er @x er @x @z2
5:114
As in Eq. (5.94), the principal ®eld Hy
x; y; z propagating in the z direction is divided into the slowly varying envelope function f
x; y; z and the very fast oscillatory phase term exp
jbz as follows: Hy
x; y; z f
x; y; z exp
jbz:
5:115
Substituting the second derivative of Eq. (5.115), which corresponds to Eq. (5.96), into Eq. (5.114) and dividing the resultant equation by the exponential term exp
jbz, we get for the TM mode the wide-angle wave equation @f 2jb @z
@2 f @ 1 @f er k02
er @z2 @x er @x
n2eff f:
5:116
When we assume that @2 f 0; @z2
5:117
5.2
FINITE-DIFFERENCE BEAM PROPAGATION METHOD
191
we get the Fresnel wave equation @f @ 1 @f er k02
er 2jb @z @x er @x
n2eff f:
5:118
To discretize the wave equation (5.118), we express the x and z coordinates, the wave function f
x; z, and the relative permittivity er
x; z as follows: x p Dx;
5:119
z l Dz;
5:120
f
x; z flp ;
5:121
er
x; z elr
p:
5:122
We ®rst discretize the Fresnel wave equation (5.118) in the x direction. The ®rst term on the right-hand side of Eq. (5.118) is discretized as follows: @ 1 @f er @x er @x 0 er
p
1
fp1 fp 1 B 1 B B Dx @er
p 1=2 Dx |{z} Difference center is p1=2
fp1 C C C; er
p 1=2 Dx A |{z} 1
fp
Difference center is p 1=2
5:123
where
1 e
p 1 er
p er p r ; 2 2 1 e
p er
p 1 er p r : 2 2
5:124
5:125
192
BEAM PROPAGATION METHODS
Thus, we get @ 1 @f er @x er @x 0 er
p
1
fp1 fp 1 B 2 B B Dx @er
p 1 er
p Dx |{z} Difference center is p1=2
fp1 fp 2er
p er
p 1 er
p
Dx2
fp fp 1 C 2 C C A er
p er
p 1 Dx |{z} Difference center is p 1=2
2er
p er
p er
p
fp 1
fp
Dx
2
1 2er
p 1 2er
p f f 2 e
p 1 e
p p1 2 e
p e
p 1 p
Dx r
Dx r r r 2er
p 2er
p 1 f er
p 1 er
p er
p er
p 1
Dx2 p
aw fp
1
1
1
ax fp ae fp1 ;
5:126
where aw
1 2er
p 2 e
p e
p
Dx r r
;
5:127
ae
1 2er
p ; 2 e
p e
p 1
Dx r r
5:128
1 2er
p 2 e
p e
p
Dx r r
ax
ae
1
1
1 2er
p 2 e
p e
p 1
Dx r r
aw :
5:129
We also get k02
er
n2eff f k02
er
p n2eff fp : |{z}
5:130
Difference center is p
Substituting Eqs. (5.126) and (5.130) into Eq. (5.118), we get 2jb
@f aw fp @z
1
ax fp ae fp1 k02 er
p
n2eff fp
and therefore 2jb
@f aw fp @z
1
fax k02 er
p
n2eff gfp ae fp1 :
5:131
5.2
FINITE-DIFFERENCE BEAM PROPAGATION METHOD
193
The next step is the discretization of Eq. (5.131) with respect to z. Discretizing the left-hand side of Eq. (5.131) with respect to z, we get 2jb
fl1 flp p : Dz
5:132
As with expression (5.111), since the difference center of the left-hand side of Eq. (5.131) is the point l 12 midway between l and l 1 as shown in Eq. (5.131), the difference center of the right-hand side of Eq. (5.131) should be l 12. Using the same procedure, we used for the TE mode, we get the FD expression 4jb l1 l1 l1 l1 2 l1 2 ax al1 k0 er
p neff fl1 aw fp 1 e fp1 p Dz 4jb l l l l 2 l 2 l l k0 er
p neff fp ae fp1 :
5:133 aw fp 1 ax Dz 5.2.3 Nonequidistant Discretization Scheme In the above discussions, we ®rst discretized the Fresnel wave equation in the x direction as 2jb
@f aw fp @z
1
fax k02 er
p
n2eff gfp ae fp1 :
5:134
Then, discretizing Eq. (5.134) in the z direction, we got the FD expression for the Fresnel wave equation for a 1D nonequidistant discretization, which is shown in Fig. 5.5: 4jb l1 l1 l1 l1 2 l1 2 k0 er
p neff fl1 ax al1 aw fp 1 e fp1 p Dz 4jb l l l 2 l 2 aw fp 1 ax k0 er
p neff flp ale flp1 ; Dz
5:135 where the difference centers of both sides of Eq. (5.134) are the same in the z direction.
FIGURE 5.5. One-dimensional nonequidistant discretization.
194
BEAM PROPAGATION METHODS
a. TE modeÐEy representation and Hx representation: aw
2 ; w
e w
5:136
ae
2 ; e
e w
5:137
ax
2 ew 4 ae aw ew ae aw :
5:138
b. TM mode: Ex representation: aw
2 2er
p 1 ; w
e w er
p er
p 1
5:139
ae
2 2er
p 1 ; e
e w er
p er
p 1
5:140
2 ew
ax
2 er
p er
p w
e w er
p er
p
1 1
2 er
p er
p 1 e
e w er
p er
p 1 4 ae aw : ew
5:141
Hy representation: aw
2 2er
p w
e w er
p er
p
;
5:142
ae
2 2er
p ; e
e w er
p er
p 1
5:143
ax
2 2er
p w
e w er
p er
p
1
1
2 2er
p e
e w er
p er
p 1
ae aw :
5:144
5.2
FINITE-DIFFERENCE BEAM PROPAGATION METHOD
195
5.2.4 Stability Condition Up to this stage, in the discretization of Eqs. (5.100) and (5.118) in the z direction, the difference centers were assumed to be the same (i.e., the point z Dz=2 midway between z and z Dz) for both the left- and the right-hand sides. Here, we further discuss the in¯uence of the difference center of the right-hand side in the z direction on the stability along the beam propagation by using the procedure discussed in Ref. [11]. The Fresnel equation is 2jb
@f
x; z @2 f
x; z k02
er @z @x2
n2eff f
x; z:
5:145
For simplicity, uniform media are assumed here. Since the reference index p neff can be set to the refractive index er of the medium, we can assume n2eff er . Thus, the wave equation (5.145) can be simpli®ed to 2jb
@f
x; z @2 f
x; z : @z @x2
5:146
Introducing the difference parameter a, which determines the difference center in the z direction, we modify the wave equation (5.146) to get 2jb
f
x; z Dz Dz
f
z
a
@2 f
x; z Dz
1 @x2
a
@2 f
x; z :
5:147 @x2
Here, it should be noted that a 0:5 has been assumed in the preceding discussions and that the scheme with a 0:5 is called the Crank± Nicolson scheme. When for a plane wave the slowly varying envelope function is expressed as f
x; z f0 exp
jkx x exp
jbz;
5:148
the second derivative of the wave function with respect to x is @2 f
x; z @x2
kx2 f
x; z:
5:149
196
BEAM PROPAGATION METHODS
Substituting Eqs. (5.148) and (5.149) into (5.147) and dividing the resultant equation by f
x; z, we get 2jb
1 exp
jb Dz Dz
kx2 fa exp
jb Dz
1
1
ag:
Therefore
2jb 2jb 2 kx a exp
jb Dz Dz Dz
1
akx2
and exp
jb Dz
2jb=Dz
1 akx2 : 2jb=Dz akx2
5:150
Since the exponential term exp
jb Dz given by Eq. (5.150) is a propagation term, we can clarify the in¯uence of the parameter a on the stability of the beam propagation by investigating how the absolute value of the exponential function changes when the propagation distance Dz changes. The absolute value of the propagation term is expressed as l j exp
jb Dzj ( )1=2
2b=Dz2
1 a2 kx4
2b=Dz2 a2 kx4 ( )1=2 C 2
1 a2 kx4 ; C 2 a2 kx4
5:151
where C
2b : Dz
5:152
The absolute value of the propagation term is related to the value of a as follows: CASE a 0:5
C 2 kx4 =4 l C 2 kx4 =4
1=2 1:
5:153
5.2
197
FINITE-DIFFERENCE BEAM PROPAGATION METHOD
Since the absolute value of the propagation term is always equal to 1, the propagating ®eld does not diverge as the beam propagates. There is, however, the possibility of oscillation. CASE 0:5 < a 1
a2 < a2 ; ( )1=2 C 2
1 a2 kx4 ; l < 1: C 2 a2 kx4
1
5:154
Since the absolute value of the propagation term is always less than 1, the propagating ®eld is unconditionally stable. However, the propagating ®eld decays as the beam propagates. CASE 0 a < 0:5
(
1
a2 > a2 ;
C 2
1 a2 kx4 ; l C 2 a2 kx4
)1=2 > 1:
5:155
Since the absolute value of the propagation term is always greater than 1, the propagating ®eld grows larger and larger as the beam propagates. It will ®nally diverge. A difference parameter a less than 0.5 thus should not be used, and a 0:5 is usually used in the calculation. The Fresnel wave equation with the difference parameter a is expressed as 2jb
fl1 flp p a
alw flp Dz
1
1
falx k02 elr
p
n2eff flp ale flp1
l1 l1 a
al1 k02 el1 w fp 1 fax r
p
l1 al1 e fp1 :
n2eff gfl1 p
5:156
Equation (5.156) is solved in exactly the same way that Eqs. (5.112) and (5.133), which are based on the Crank±Nicolson sheme, are solved. 5.2.5 Transparent Boundary Condition An in®nitely wide area would have no analysis boundary and there would thus be no re¯ections at boundaries. Because such an area cannot be assumed in actual design, however, a limited analysis window has to be
198
BEAM PROPAGATION METHODS
used. In actual structures, radiated waves are re¯ected at the boundaries and return to the core area, where they interact with the propagating ®elds. This interaction disturbs the propagating ®elds and greatly degrades the calculation accuracy. In this section, we discuss the TBC. The TBC was developed by Hadley [5] as a way to ef®ciently suppress the re¯ections at boundaries, and it is easy to implement into computer programs. Although the TBC will be applied to 2D problems here, it can easily be extended and applied to 3D problems. As shown in Fig. 5.6, the analysis window contains nodes at p 1 to p M . The hypothetical nodes at p 0 and p M 1 are assumed to be outside the analysis window. Following Hadley's line of thinking [5], we incorporate the in¯uences of the nodes at p 0 and p M 1 into the nodes at p 1 and p M . In what follows, we consider how the boundary conditions can be written into a computer program. A. Left-Hand Boundary Consider the left-hand boundary in Fig. 5.6. We incorporate the in¯uence of the hypothetical node at p 0 (outside the analysis window) into the node at p 1 (inside the analysis window). The wave function for the left-traveling wave with the x-directed wave number kx is expressed as f
x; z A
z exp
jkx x:
5:157
We denote the x coordinates and the ®elds of the nodes at p 0, 1, 2 as x0 , x1 , x2 and as f0 , f1 , f2 , and we assume that f0 A
z exp
jkx x0 ;
5:158
f1 A
z exp
jkx x1 ;
5:159
f2 A
z exp
jkx x2 :
5:160
The ®elds f1 and f2 are inside the analysis window, and f0 is a hypothetical ®eld whose in¯uence should be incorporated into the ®eld inside the analysis window.
FIGURE 5.6. Nodes p 0 and p M 1 are outside the analysis area.
5.2
199
FINITE-DIFFERENCE BEAM PROPAGATION METHOD
Dividing Eq. (5.160) by Eq. (5.159) and Eq. (5.159) by Eq. (5.158), we get
where Dx x2
x1 x1
exp
jkx Dx
f2 ; f1
5:161
exp
jkx Dx
f1 ; f0
5:162
x0 . Substituting the ratio of f2 to f1 , Z1
f2 ; f1
5:163
which is equal to exp
jkx Dx, into Eq. (5.162), we get f0
f1 : Z1
5:164
Equation (5.164) can also be derived by substituting the x-directed wave number kx
1 ln
Z1 ; j Dx
5:165
which is obtained from Eq. (5.161), into f0 f1 exp
jkx Dx;
5:166
which is obtained from Eq. (5.162). It should be noted that since the wave travels leftward, the real part of the x-directed wave number kx, Re
kx , should be negative. When it is positive, which implies re¯ection at the left-hand boundary, the sign should be changed from plus to minus. B. Right-Hand Boundary Consider the right-hand boundary in Fig. 5.6. We incorporate the in¯uence of the hypothetical node at p M 1 (outside the analysis window) into the node at p M (inside the analysis window).
200
BEAM PROPAGATION METHODS
The wave function of the right-traveling wave with the x-directed wave number kx is expressed as f
x; z A
z exp
jkx x:
5:167
We denote the x coordinates and the ®elds of the nodes p M 1, M , M 1 as xM 1 , xM , xM 1 and fM 1, fM , fM 1 , and we assume that fM
1
A
z exp
jkx xM
1 ;
5:168
fM A
z exp
jkx xM ;
5:169
fM 1 A
z exp
jkx xM 1 ;
5:170
where fM 1 and fM are the ®elds inside the analysis window and fM 1 is the hypothetical ®eld whose in¯uence should be incorporated into the ®eld inside the analysis window. Dividing Eq. (5.169) by Eq. (5.168) and Eq. (5.170) by Eq. (5.169), we get fM 1 fM f exp
jkx Dx M ; fM 1 exp
jkx Dx
where Dx xM fM 1 ,
xM
1
xM 1
5:171
5:172
xM . Substituting the ratio of fM to
ZM
fM ; fM 1
5:173
which is equal to exp
jkx Dx, into Eq. (5.172), we get fM 1 fM ZM :
5:174
Equation (5.174) can also be derived by substituting the x-directed wave number kx
1 ln
ZM ; j Dx
5:175
5.2
FINITE-DIFFERENCE BEAM PROPAGATION METHOD
201
which is obtained from Eq. (5.171), into fM 1 fM exp
jkx Dx:
5:176
Similar to what we saw in the case of the left-hand boundary, since the wave travels rightward, the real part of the x-directed wave number kx should be negative. When it is positive, which implies re¯ection occurs at the right-hand boundary, the sign should be changed from plus to minus. 5.2.6 Programming We now look at how the TBC is written into the program. For the FD scheme in the propagation direction, the most widely used Crank± Nicolson scheme (i.e., a 0:5) in Eq. (5.147) is used. The problem is to obtain the unknown ®eld fl1 at z Dz by using the known ®eld fl, where superscripts l and l 1 respectively correspond to z and z Dz. The following equation has to be solved:
4jb l1 2 l1 2 k0 er
p neff fl1 al1 e fp1 p Dz 4jb l l l 2 l 2 aw fp 1 ax k0 er
p neff flp ale flp1 :
5:177 Dz
l1 al1 w fp 1
al1 x
l1 l1 The unknown ®elds to be obtained at z Dz are fl1 p 1 , fp , and fp1 and l l l the known ®elds at z are fp 1 , fp , and fp1 for p 1 and M. Assuming the unknown coef®cients A
p, B
p, and C
p and the known value D
p to be
A
p B
p C
p
al1 w ; al1 x al1 e ;
D
p alw flp
1
4jb Dz
alx
k02 el1 r
p
2 neff ;
4jb k02 elr
p Dz
2 neff flp ale flp1 ;
5:178
5:179
5:180
5:181
we simplify Eq. (5.177) to l1 l1 A
pfl1 p 1 B
pfp C
pfp1 D
p:
5:182
202
BEAM PROPAGATION METHODS
In the following, we will discuss Eq. (5.182) with respect to p, which represents the lateral position of the node. A. Left-Hand Boundary
p 1 The ®eld f1 of the node on the lefthand boundary is in¯uenced by the ®eld f0 of the hypothetical node outside the analysis window. As shown in Eq. (5.164), the ®eld f0 of the hypothetical node is expressed as f0 f1 gL ;
5:183
where gL
1 1 l l: Z1 f2 =f1
5:184
Here, since the parameter gL is determined by the known ®elds at z (i.e., l), gL is known. Thus, Eq. (5.182) is reduced to l1 B0
1fl1 D
1; 1 C
1f2
5:185
where 0
B
1
al1 w gL
C
1
al1 e ;
al1 x
D
1 alw gL fl1 alx
4jb Dz
k02 el1 r
1
4jb k02 elr
1 Dz
n2eff
;
2 neff fl1 ale fl2 :
5:186
5:187
5:188
B. Right-Hand Boundary
p M The ®eld fM of the node on the right-hand boundary is in¯uenced by the ®eld fM 1 of the hypothetical node outside the analysis window. As shown in Eq. (5.174), the ®eld fM 1 of the hypothetical node is expressed as fM 1 fM gR ;
5:189
where gR ZM
1 flM =flM 1
:
5:190
5.2
203
FINITE-DIFFERENCE BEAM PROPAGATION METHOD
Here, since the parameter gR is determined by the known ®elds at z (i.e., l), gL is known. Thus, Eq. (5.182) is reduced to l1 0 A
Mfl1 M 1 B
M fM D
M ;
5:191
where A
M
al1 w ;
B0
M
al1 e gR
D
M
alw flM 1
al1 x
ale gR
4jb Dz
alx
k02 el1 r
M
4jb k02 elr
M Dz
2 neff ; n2eff
flM :
5:192
5:193
5:194
Summarizing the above discussions, we get the algebraic equations 0
B0
1
B B A
2 B B B B B B B B B B B B @
C
1 B
2
C
2
A
3
B
3
0
D
1
1
C B B D
2 C C B C B B D
3 C C B C: B C B .. C B . C B C B B D
M 1 C A @
C
3
A
M
10 1 fl1 1 CB CB l1 C CB f2 C C CB CB l1 C CB f3 C C CB C CB CB . C CB .. C C CB C CB CB fl1 C C C
M 1 C A@ M 1 A fl1 M B0
M
5:195
D
M Since Eq. (5.195) is a tridiagonal matrix equation, we can obtain the l1 unknown wave ®elds fl1 1 ; . . . ; fM by using the Thomas method [14].
204
BEAM PROPAGATION METHODS
5.3 WIDE-ANGLE ANALYSIS USING PADE APPROXIMANT OPERATORS Up to this stage, the discussions have been based on the Fresnel equation (i.e., the para-axial wave equation). Here, we discuss the wide-angle beam propagation method based on Pade approximant operators [5] and discuss the multistep method [6], both of which were developed by Hadley.
5.3.1 Pade Approximant Operators When the second derivative with respect to z is not neglected, the wave equation is 2jb
@f @z
@2 f Pf; @z2
5:196
where for the TE mode P
@2 k02
er @x2
n2eff
5:197
and for the TM mode @ 1 @ P er k02
er @x er @x
n2eff :
5:198
Solving Eq. (5.196) formally, we get @ j @ 1 f @z 2b @z
jP f 2b
and therefore @f jP=2b f: @z 1
j=2b
@=@z
5:199
5.3
WIDE-ANGLE ANALYSIS USING PADEÂ APPROXIMANT OPERATORS
205
When the derivative with respect to z is neglected, Eq. (5.199) is reduced to the Fresnel equation. Here, we regard the derivative with respect to z in Eq. (5.199) as the recurrence formula @ jP=2b : @z n 1
j=2b
@=@z n 1
5:200
Next, we will specify the explicit expressions for the various orders of the recurrence formula. First, we de®ne the starting equation @ @z
1
0:
5:201
The explicit expressions for the corresponding wide-angle (WA) orders are shown below. 1. WA-0th order (Fresnel approximation): @ @z 0
jP=2b j @ 1 2b @z
j
P : 2b
5:202
1
2. WA-1st order: @ @z 1
jP=2b jP=2b j jP j @ 1 1 2b 2b 2b @z 0
j
P=2b :
5:203 1 P=4b2
3. WA-2nd order: @ @z 2
jP=2b jP=2b j jP=2b j @ 1 1 2b 1 P=4b2 2b @z 1 j
P=2b P2 =8b3 : 1 P=2b2
5:204
206
BEAM PROPAGATION METHODS
4. WA-3rd order: @ @z 3
jP=2b j @ j 1 1 2b @z 2 2b
jP=2b j
P=2b P2 =8b3 1 P=2b2
jP=2b
1 P=2b2 1 P=2b2 P=4b2 P2 =16b4
j
P=2b P2 =4b3 : 1 3P=4b2 P2 =16b4
5:205
5. WA-4th order: @ @z 4
jP=2b jP=2b j @ j j
P=2b P2 =4b3 1 1 2b @z 3 2b 1 3P=4b2 P2 =16b4
jP=2b
1 3P=4b2 P2 =16b4 1 3P=4b2 P2 =16b4 P=4b2 P2 =8b4 j
P=2b 3P2 =8b3 P3 =32b5 : 1 P=b2 3P2 =16b4
5:206
6. WA-5th order: @ @z 5
jP=2b j @ j 1 1 2b @z 4 2b
jP=2b j
P=2b 3P2 =8b3 P3 =32b5 1 P=b2 3P2 =16b4
jP=2b
1 P=b2 3P2 =16b4 1 P=b2 3P2 =16b4 P=4b2 3P2 =16b4 P3 =64b6 j
P=2b P2 =2b3 3P3 =32b5 : 1 5P=4b2 3P2 =8b4 P3 =64b6
5:207
5.3
207
WIDE-ANGLE ANALYSIS USING PADEÂ APPROXIMANT OPERATORS
7. WA-6th order: @ @z 6
jP=2b jP=2b j @ j j
P=2b P2 =2b3 3P3 =32b5 1 1 2b @z 5 2b 1 5P=4b2 3P2 =8b4 P3 =64b6
jP=2b
1 5P=4b2 3P2 =8b4 P3 =64b6 1 5P=4b2 3P2 =8b4 P3 =64b6 P=4b2 P2 =4b4 3P3 =64b6
j
P=2b 5P2 =8b3 3P3 =16b5 P4 =128b7 1 3P=2b2 5P2 =8b4 P3 =16b6
P=2b 5P2 =8b3 3P3 =16b5 P4 =128b7 j : 1 3P=2b2 5P2 =8b4 P3 =16b6
5:208
8. WA-7th order: @ @z 7
jP=2b j @ 1 2b @z 6
1
j 2b
jP=2b j
P=2b 5P =8b3 3P3 =16b5 P4 =128b7 1 3P=2b2 5P2 =8b4 P3 =16b6 2
jP=2b
1 3P=2b2 5P2 =8b4 P3 =16b6 1 3P=2b2 5P2 =8b4 P3 =16b6 P=4b2 5P2 =16b4 3P2 =32b6 P4 =256b8 j
P=2b 3P2 =4b3 5P3 =16b5 P4 =32b7 : 1 7P=4b2 15P2 =16b4 5P3 =32b6 P4 =256b8
5:209
Thus, the recurrence formula (5.200) can be reduced to an expression that includes only the operator P: @f @z
j
N f; D
5:210
208
BEAM PROPAGATION METHODS
where N and D are both polynomials of the operator P. The wide-angle orders correspond to the orders for the Pade orders as follows: WA
0 $
1; 0;
WA
1 $
1; 1;
WA
2 $
2; 1;
WA
3 $
2; 2;
WA
4 $
3; 2;
WA WA
5 $
3; 3; 6 $
4; 3;
WA
7 $
4; 4:
Differentiating Eq. (5.210) based on the Crank±Nicolson scheme, we get @f 1 !
fl1 fl ; @z Dz N N 1 l1
f fl ; j f! j D D 2
Left-hand side: Right-hand side:
5:211
where the right-hand side was averaged by l and l 1 so that the difference center of the left-hand side coincides with that of the righthand side. From Eq. (5.211), we get 1 l1
f Dz
fl
j
N 1 l1
f fl : D 2
Therefore D
fl1 and
fl
1 jN Dz
fl1 fl 2
Dz l1 Dj N f D 2
Dz j N fl ; 2
5:212
which can be rewritten as fl1
D j
Dz=2N l f: D j
Dz=2N
5:213
5.3
WIDE-ANGLE ANALYSIS USING PADEÂ APPROXIMANT OPERATORS
209
Since, as shown in Eqs. (5.202)±(5.209), the coef®cients of the polynomials D and N in Eq. (5.213) are real, D and N themselves are real. Thus, Eq. (5.213) can be written as D j
Dz=2N l f D j
Dz=2N * Pn x Pi Pni0 i i fl : i i0 x*P
fl1
5:214
In the following, we show the coef®cients xi for the wide-angle orders WA-0 to WA-7. 1. WA-0th order [PadeÂ(1,0): Fresnel approximation]: From Eq. (5.202), we get D 1;
N
P 2b
and therefore D
j
Dz N 1 2
j
Dz P : 2 2b
5:215
Thus, we get x0 1;
x1
j
Dz : 4b
5:216
2. WA-1st order [PadeÂ(1,1)]: From Eq. (5.203), we get D1
P ; 4b2
N
P 2b
and therefore D
j
Dz P N 1 2 2 4b
jD
P 1 1 2
1 4b 4b
jb DzP:
5:217
210
BEAM PROPAGATION METHODS
Thus, we get x0 1;
x1
1
1 4b2
jb Dz:
5:218
3. WA-2nd order [PadeÂ(2,1)]: From Eq. (5.204), we get D1
P ; 2b2
P P2 3 2b 8b
N
and therefore Dz P P2 j 2 2b 8b3
Dz P j N 1 2 2 2b
D
1
1
2 4b2
jb DzP
j
Dz 2 P : 16b3
5:219
Thus, we get x0 1;
x1
1
2 4b2
jb Dz;
x2
j
Dz : 16b3
5:220
4. WA-3rd order [PadeÂ(2,2)]: From Eq. (5.205), we get D1
3P P2 ; 4b2 16b4
and therefore D
Dz 3P P2 j N 1 2 2 16b4 4b 1
1
3 4b2
N
P P2 3 2b 4b
Dz P P2 j 2 2b 4b3
jb DzP
1
1 16b4
j2b DzP2 :
5:221
Thus, we get x0 1;
x1
1
3 4b2
jb Dz;
x2
1
1 16b4
j2b Dz:
5:222
211
WIDE-ANGLE ANALYSIS USING PADEÂ APPROXIMANT OPERATORS
5.3
5. WA-4th order [PadeÂ(3,2)]: From Eq. (5.206), we get D1
P 3P2 ; b2 16b4
N
P 3P2 P3 3 2b 8b 32b5
and therefore D
Dz P 3P2 j N 1 2 2 16b4 b 1 j
1
4 4b2
Dz P 3P2 P3 j 2 2b 8b3 32b5
jb DzP
1
3 16b4
j3b DzP2
Dz 3 P : 64b5
5:223
Thus, we get x0 1; x3
j
x1
1
4 4b2
jb Dz;
x2
1
3 16b4
j3b Dz;
Dz : 64b5
5:224
6. WA-5th order [PadeÂ(3,3)]: From Eq. (5.207), we get D1
5P 3P2 P3 ; 4b2 8b4 64b6
N
P P2 3P3 ; 3 2b 2b 32b5
and therefore D
Dz 5P 3P2 P3 j N 1 2 4 2 8b 64b6 4b 1
1
5 4b2
1
1 64b6
jb DzP j3b DzP3 :
Dz P P2 3P2 j 2 2b 2b3 32b5 1
3 8b4
j2b DzP2
5:225
212
BEAM PROPAGATION METHODS
Thus, we get
x0 1; x3
x1
1
1 64b6
1
5 4b2
jb Dz;
x2
1
3 8b4
j2b Dz;
j3b Dz:
5:226
7. WA-6th order [PadeÂ(4,3)]: From Eq. (5.208), we get
D1 N
3P 5P2 P3 ; 2b2 8b4 16b6
P 5P2 3P3 P4 3 ; 2b 8b 16b5 128b7
and therefore
D
j
Dz 3P 5P2 P3 N 1 2 4 2 8b 16b6 2b Dz P 5P2 3P3 P4 j 2 2b 8b3 16b5 128b7 1
1
6 4b2
1
2 32b6
jb DzP j3b DzP3
1
10 16b4
j5b DzP2
j Dz 4 P : 256b7
5:227
Thus, we get
x0 1; x3
1
2 32b6
1
6 4b2
jb Dz;
x2
j3b Dz;
x4
j Dz : 256b7
x1
1
10 16b4
j5b Dz;
5:228
213
WIDE-ANGLE ANALYSIS USING PADEÂ APPROXIMANT OPERATORS
5.3
8. WA-7th order [PadeÂ(4,4)]: From Eq. (5.209), we get D1 N
7P 15P2 5P3 P4 ; 4b2 16b4 32b6 256b8
P 3P2 5P3 P4 3 ; 5 2b 4b 16b 32b7
and therefore D
j
Dz 7P 15P2 5P3 P4 N 1 2 2 16b4 32b6 256b8 4b Dz P 3P2 5P3 P4 j 2 2b 4b3 16b5 32b7 1
1
7 4b2
1
5 32b6
jb DzP
1
15 16b4
j5b DzP3
1
1 256b8
j6b DzP2 j4b DzP4 :
5:229
Thus, we get x0 1; x3
1
5 32b6
1
7 4b2
jb Dz;
j5b Dz;
x4
x1
x2 1
1 256b8
1
15 16b4 j4b Dz:
j6b Dz;
5:230
Since through the above discussions the explicit expressions for D j
Dz=2N are obtained, those for D j
Dz=2N can also be obtained. Both sides of Eq. (5.212) are clari®ed and the unknown ®eld fl1 can be calculated. 5.3.2 Multistep Method The reason the matrix of the Fresnel approximation is tridiagonal is that the order of the operator P, which contains the second derivative with respect to the x coordinate and can be approximated by the FD scheme with three terms as shown in Eq. (4.9), is 1.
214
BEAM PROPAGATION METHODS
Since the wide-angle formulations written in Eqs. (5.204)±(5.209) or Eqs. (5.219)±(5.230) include the powers of the operator P higher than 2, the column width of nonzero matrix elements is greater than 3. Thus, the numerically ef®cient Thomas method cannot be used to solve the ®nal algebraic matrix equation. In this section, we discuss the multistep method, which was developed by Hadley [6] in order to solve this problem. Consider Eq. (5.214). The numerator of the factor on the right-hand side of Eq. (5.214) is obtained as shown in Eqs. (5.215)±(5.230). The denominator can also be obtained, since it is simply the complex conjugate of the numerator. Since x0 is equal to 1, the numerator of the term on the right-hand side of Eq. (5.214) can be factorized as n P i0
xi Pi
1 an P
1 a2 P
1 a1 P;
5:231
where the coef®cients a1 ; a2 ; . . . ; an can be obtained by solving the algebraic equation D
j
n P Dz xi Pi 0: N 2 i0
5:232
The denominator of the term on the right-hand side of Eq. (5.214) can be obtained by using the complex conjugates of the coef®cients a1 , a2 ; . . . ; an : n P i0
i x*P
1 an P*
1 a2 P*
1 a1 P* i
1 an*P
1 a2*P
1 a1*P;
5:233
Thus, the unknown ®eld fl1 at z Dz is related to the known ®eld fl at z as follows: fl1
1 an P
1 a2 P
1 a1 P l f:
1 an*P
1 a2*P
1 a1*P
5:234
Next, we discuss how to solve Eq. (5.234). First, we rewrite it as
1 an*P
1 a2*P l1 1 a1 P l f f:
1 an P
1 a2 P 1 a1*P
5:235
5.3
WIDE-ANGLE ANALYSIS USING PADEÂ APPROXIMANT OPERATORS
215
Then, de®ning the ®eld fl1=n as fl1=n
1 an*P
1 a2*P l1 f ;
1 an P
1 a2 P
5:236
1 a1 P l f: 1 a1*P
5:237
we rewrite Eq. (5.235) as fl1=n
Since fl is known, we can obtain fl1=n by solving Eq. (5.237). Using , we rewrite Eq. (5.236) as f l1=n
1 an*P
1 a3*P l1 1 a2 P l1=n f f :
1 an P
1 a3 P 1 a2*P
5:238
Then, de®ning the ®eld fl2=n as fl2=n
1 an*P
1 a3*P l1 f ;
1 an P
1 a3 P
5:239
1 a2 P l1=n : f 1 a2*P
5:240
we rewrite Eq. (5.238) as fl2=n
Since fl1=n is known, we can obtain fl2=n by solving Eq. (5.240). Repeating this procedure, we ®nally get the unknown ®eld fl1 at z Dz by solving fl1
1 an P l
n f 1 an*P
1=n
:
5:241
That is, the known ®eld fl1 can be obtained from the known ®eld fl by successively solving fli=n
1 ai P l
i f 1 a*P i
1=n
5:242
when i 1; 2; . . . ; n. The advantage of the multistep method is that the matrix equation to be solved in each step is the same size as the Fresnel equation and for 2D problems is tridiagonal. Thus, the calculation procedure is very easy. The method can be easily extended to 3D problems, and it has also been used in the wide-angle FE-BPM [12].
216
BEAM PROPAGATION METHODS
5.4 THREE-DIMENSIONAL SEMIVECTORIAL ANALYSIS The preceding discussions were limited to the 2D BPM, which assumes the 1D cross-sectional structure in the lateral direction. When, however, the propagating ®eld is widely spread in the 2D cross section, the 3D beam propagation method is required. Since the formulation for the 2D BPM in Section 5.2.2 can be straightforwardly extended to the 3D BPM, a much more numerically ef®cient 3D BPM formulation based on the alternate-direction implicit (ADI) method [8±10] will be shown here. In the ADI-BPM, the calculation for one step, z ! z Dz, is divided into two steps, z ! z Dz=2 and z Dz=2 ! z Dz, and the two steps are solved successively in the x and y directions. Since solving a 3D problem can be reduced to solving a 2D problem twice by using the ADI method, instead of a large matrix equation we have only to solve tridiagonal matrix equations twice. Thus, high numerical ef®ciency is attained especially for large-size waveguides, such as spotsize-converterintegrated structures [15±17]. In this section, the semivectorial formulation is used to analyze large-index-difference waveguides and to treat the polarization. Nonuniform discretization is also assumed for versatility of analysis. Neglecting the terms for the interaction between polarizations in the vectorial wave equations (4.10) and (4.19) in Section 4.2, we get the semivectorial wave equation @2 c Pc 0: @z2
5:243
Here, Pc and c for the quasi-TE mode are obtained from Eqs. (4.17) and (4.30) as @ 1 @ @2 c
er c 2 k02 er c
electric ®eld representation; Pc @x er @x @y c Ex ; Pc er
5:244
@ 1 @c @2 c 2 k02 er c
magnetic ®eld representation; @x er @x @y
c Hy :
5:245
5.4
217
THREE-DIMENSIONAL SEMIVECTORIAL ANALYSIS
And Pc and c for the quasi-TM mode are obtained from Eqs. (4.18) and (4.29) as @2 c @ 1 @ Pc 2
e c k02 er c
electric ®eld representation; @x @y er @y r c Ey ;
5:246 @2 c @ 1 @c @2 c Pc 2 er 2 k02 er c
magnetic ®eld representation; @x @y er @y @z
c Hx :
5:247
Using the slowly varying envelope approximationÐin this case, that the wave function c
x; y; z propagating in the z direction can be separated into a slowly varying envelope function f
x; y; z and a very fast oscillating phase term exp
jbzÐwe get c
x; y; z f
x; y; z exp
jbz;
5:248
where k0 , neff , and b
neff k0 are respectively the wave number in a vacuum, the reference index, and the propagation constant. Assuming the Fresnel approximation @2 f 0; @z2
5:249
we reduce Eq. (5.243) to the Fresnel wave equation 2jb
@f Pf: @z
5:250
It should be noted that Pc in Eq. (5.250) differs by b2 from that in Eqs. (5.243)±(5.247) and for the quasi-TE mode is expressed as @ 1 @ @2 f
er f 2
k02 er Pf @x er @x @y
b2 f
5:251
in the electric ®eld representation and as @ 1 @f @2 f Pf er 2
k02 er @x er @x @y
b2 f
5:252
218
BEAM PROPAGATION METHODS
in the magnetic ®eld representation. For the quasi-TM mode it is expressed as @2 f @ 1 @
e f
k02 er b2 f Pf 2
5:253 @x @y er @y r in the electric ®eld representation and as @2 f @ 1 @f
k02 er Pf 2 er @x @y er @y
b2 f
5:254
in the magnetic ®eld representation. The nonequidistant discretization mesh shown in Fig. 4.3 for the 2D cross-sectional FDM is also used in the lateral directions for the 3D FDBPM. Here, the subscripts for the lateral positions x and y are respectively p and q and the superscript for position z in the propagation direction is l. Thus, using Eqs. (4.51)±(4.59), we can write the ®elds, the discretization widths, and the relative permittivity as follows: flp;q f
xp ; yq ; zl ; flp;q1 f
xp ; yq1 ; zl ; e xp1
xp ;
flp1;q f
xp1 ; yq ; zl ; n yq w xp
yq 1 ;
xp 1 ;
s yq1
yq ;
elr
p; q er
xp ; yq ; zl :
5:255
First, we discuss the discretization with respect to x and y. Discretizing Eqs. (5.251)±(5.253), by deduction from Eqs. (4.99), (4.123), (4.133), and (4.143), we get P alw flp
1;q
ale flp1;q aln flp;q
alx aly flp;q k02
elr
p; q
1
als flp;q1 n2eff flp;q ;
where b k0 neff . Thus, discretizing only the right-hand side of Eq. (5.250) with respect to x and y, we can reduce Eq. (5.250) to 2jb
@f aw fp 1;q ae fp1;q an fp;q 1 as fp;q1 @z
ax ay fp;q k02 er
p; q n2eff fp;q :
5:256
5.4
THREE-DIMENSIONAL SEMIVECTORIAL ANALYSIS
219
This equation can be rewritten as 2jb
@f
derivative with respect to x
aw fp 1;q ae fp1;q ax fp;q @z
an fp;q 1 as fp;q1 ay fp;q
derivative with respect to y k02 er
p; q
n2eff fp;q :
5:257
Now, we move to the discretization of the derivative with respect to z on the left-hand side of Eq. (5.257). A sensitive problem in discretization is the difference centers of the right-hand side and the left-hand side of the equation in the z direction. In the ADI-BPM, the calculation for the step z ! z Dz is divided into two steps, z ! z Dz=2 and z Dz=2 ! z Dz. In the following, we describe the explicit calculation procedure. 5.4.1 First Step: z ! z Dz=2
l ! l 12 The derivative with respect to x on the right-hand side of Eq. (5.257) is written by the implicit FD expression using the unknown ®elds at l 12 as l1=2 l1=2 fl1=2 fp1;q al1=2 fl1=2 al1=2 w x p;q : p 1;q ae
5:258
The derivative with respect to y, on the other hand, is written by the explicit FD expression with the known ®elds at l as aln flp;q
1
als flp;q1 aly flp;q :
5:259
The remaining part on the right-hand side is expressed by using an average of l and l 12 as
p; q k02
el1=2 r
n2eff
flp;q fl1=2 p;q ; 2
5:260
where el1=2
elr el1 r r =2. With respect to the left-hand side of Eq. (5.257), we get l1=2
2jb
l
fp;q fp;q @f ! 2jb : @z Dz=2
5:261
220
BEAM PROPAGATION METHODS
Thus, from expressions (5.258)±(5.261), we get fl1=2 flp;q p;q l1=2 l1=2 2jb fl1=2 fp1;q al1=2 fl1=2
al1=2 w x p;q p 1;q ae Dz=2
aln flp;q
1
als flp;q1 aly flp;q
k02
erl1=2
p; q
n2eff
flp;q fl1=2 p;q : 2
5:262
So that the terms on the left- and right-hand sides respectively contain l 12 and l, we rewrite Eq. (5.262) as fl1=2 al1=2 w p 1;q
al1=2 x
4jb Dz
k02 l1=2
e
p; q 2 r
al1=2 fl1=2 e p1;q 4jb k02 l1=2 l l an fp;q 1 aly
p; q
er Dz 2
n2eff
n2eff
l1=2 fp;q
flp;q als flp;q1 :
5:263
5.4.2 Second Step: z Dz=2 ! z Dz
l 12 ! l 1 The derivative with respect to y on the right-hand side of Eq. (5.257) is written by the implicit FD expression using the unknown ®elds at l 1 as l1 l1 l1 l1 l1 al1 n fp;q 1 as fp;q1 ay fp;q :
5:264
The derivative with respect to x, on the other hand, is written by the explicit FD expression using the known ®elds at l 12 as l1=2 l1=2 al1=2 fl1=2 fp1;q al1=2 fl1=2 w x p;q : p 1;q ae
5:265
The remaining part in the right-hand side is expressed by using an average of l 12 and l 1 as
p; q k02 el1=2 r
n2eff
fl1=2 fl1 p;q P;q : 2
5:266
5.4
221
THREE-DIMENSIONAL SEMIVECTORIAL ANALYSIS
With respect to the left-hand side of Eq. (5.257), we get fl1 f11=2 @f p;q p;q ! 2jb 2jb @z Dz=2
5:267
Thus, from Eqs. (5.264)±(5.267), we get fl1 fl1=2 p;q p;q l1 l1 l1 l1 l1 2jb
al1 n fp;q 1 as fp;q1 ay fp;q Dz=2 l1=2 l1=2 fl1=2 fp1;q al1=2 fl1=2
al1=2 w x p;q p 1;q ae
k02
el1=2
p; q r
n2eff
fl1=2 fl1 p;q p;q : 2
5:268
So that the terms on the left- and right-hand sides respectively contain l 1 and l 12, we rewrite Eq. (5.268) as l1 al1 n fp;q 1
l1 al1 s fp;q1
awl1=2 fpl1=2 1;q
al1=2 f11=2 e p1;q :
al1 y
al1=2 x
4jb Dz
k02 l1=2
p; q
e 2 r
4jb k02 l1=2
p; q
er Dz 2
n2eff
n2eff
fl1 p;q
fl1=2 p;q
5:269
As discussed above, since the actual calculation in the two steps of the ADI-BPM is the 2D BPM, it is very numerically ef®cient. The mode mismatch (obtained from the overlap integral) between the propagating ®eld calculated by the semivectorial ADI-BPM and the initial eigen®eld calculated by the scalar FEM is shown in Fig. 5.7 as a function of the propagation distance. Since the propagating ®eld and the initial ®eld are obtained by the semivectorial ADI-BPM and the scalar FEM, the mode mismatch shown in Fig. 5.7 corresponds to the difference between the eigen®eld shapes of the FD scheme and the FE scheme. As shown in this ®gure, the difference between the eigen®elds obtained by the FDM and the FEM is very small even though the concepts of the two methods are different.
222
BEAM PROPAGATION METHODS
FIGURE 5.7. Eigen®eld mismatch between the FEM and FD-BPM.
5.5 THREE-DIMENSIONAL FULLY VECTORIAL ANALYSIS Since a full discussion of the 3D fully vectorial beam propagation method is beyond the scope of this book, we roughly cover the formulation, emphasizing the relations between the wave equations. For details, refer to Ref. [11]. 5.5.1 Wave Equations As shown in Eqs. (4.17) and (4.18), the 3D vectorial wave equations for the electric ®eld representation are for the x component @ 1 @ @2 E @2 E @ 1 @er Ey 0
er Ex 2x 2x k02 er Ex @y @z @x er @x @x er @y
5:270 and for the y component @ 2 Ey @ 1 @ @ 2 Ey @ 1 @er 2
e E 2 k0 er Ey E 0: @y er @y r y @y er @x x @x2 @z
5:271 As shown in Eqs. (4.29) and (4.30), the 3D vectorial wave equations for the magnetic ®eld representation are for the x component @ 2 Hx @ 1 @Hx @ 2 Hx 1 @er @Hy e k02 er Hx 0
5:272 r 2 2 @y er @y er @y @x @x @z
5.5 THREE-DIMENSIONAL FULLY VECTORIAL ANALYSIS
223
and for the y component @2 Hy @2 Hy @ 1 @Hy 1 @er @Hx 2 2 k02 er Hy er 0: @x er @x er @x @y @y @z
5:273
First, we discuss the electric ®eld representation. Using the SVEAÐin this case the wave functions Ex
x; y; z and Ey
x; y; z propagating in the z direction can be separated into the slowly varying envelope functions E~ x
x; y; z and E~ y
x; y; z and the vary fast oscillating phase term exp
jbzÐwe get Ex
x; y; z E~ x
x; y; z exp
jbz;
5:274
Ey
x; y; z E~ y
x; y; z exp
jbz:
5:275
Substituting the second derivatives with respect to z, @2 Ex @2 E~ x 2 exp
jbz @z2 @z @2 Ey @2 E~ y 2 exp
jbz @z2 @z
@E~ x exp
jbz @z @E~ y 2jb exp
jbz @z 2jb
b2 E~ x exp
jbz;
5:276
b2 E~ y exp
jbz
5:277
into Eqs. (5.274) and (5.275) and dividing the results by the phase term exp
jbz, we get the wide-angle equations for the electric ®eld representation: 2jb 2jb
@ @E~ x Pxx E~ x Pxy E~ y ; @z @z @ @E~ y Pyy E~ y Pyx E~ x : @z @z
5:278
5:279
The matrix expression for Eqs. (5.278) and (5.279) is 2jb
@ @ E~ x Pxx ~ Pyx @z @z Ey
Pxy Pyy
E~ x ; E~ y
5:280
224
BEAM PROPAGATION METHODS
where @ 1 @ @2 E~ ~ ~
er Ex 2x
k02 er b2 E~ x ; Pxx Ex @x er @x @y @2 E~ y @ 1 @er ~ @ 1 @ ~ ~ Ey Pxy Ey
er Ey ; @x er @y @x er @y @x @y @2 E~ y @ 1 @ ~ ~
e E
k02 er b2 E~ y ; Pyy Ey 2 @y er @y r y @x @ 1 @er ~ @ 1 @ @2 E~ x ~ ~ Ex Pyx Ex
er Ex : @y er @x @y er @x @y @x
5:281
5:282
5:283
5:284
Using the SVEAs Hx
x; y; z H~ x
x; y; z exp
jbz; Hy
x; y; z H~ y
x; y; z exp
jbz;
5:285
5:286
we get the wide-angle wave equation for the magnetic ®eld representation: 2jb where
@ @ H~ x Pxx ~ Pyx H @z @z y
Pxy Pyy
H~ x ; H~ y
! 2 ~ ~ H @ @ 1 @ H x Pxx H~ x 2x er
k02 er @y er @y @x
b2 H~ x ;
! ~ y @2 H~ y ~y @ H @ H 1 @e @ 1 r er ; Pxy H~ y er @y @x @y er @x @y @x ! @H~ y @2 H~ y @ 1 Pyy H~ y er 2
k02 er b2 H~ y ; @x er @x @y ! 2 ~ ~ ~ H 1 @e @ H @ @ 1 @ H r x x x Pyx H~ x er : er @x @y @x er @y @x @y
5:287
5:288
5:289
5:290
5:291
Neglecting the interaction terms Pxy and Pyx in Eqs. (5.280) and (5.287), we get the semivectorial wave equations. Neglecting the second terms on
5.5 THREE-DIMENSIONAL FULLY VECTORIAL ANALYSIS
225
the left-hand sides of Eqs. (5.280) and (5.287), we get the vectorial wave equations based on the Fresnel approximation.
5.5.2 Finite-Difference Expressions Here, we discuss the fully vectorial FD-BPM using as an example the vectorial wave equations (5.287)±(5.291) for the magnetic ®eld representation. For simplicity, the equidistant discretizations Dx and Dy are assumed in the x and y directions. Since Pxx is an operator corresponding to the semivectorial analysis for H~ x, ~ Pxx H~ x axx w Hx
p
~ 1; q axx e Hx
p 1; q
~ axx n Hx
p; q
~ 1 axx s Hx
p; q 1
xx ~ 2
axx x ay Hx
p; q fk0 er
p; q
b2 gH~ x
p; q;
5:292
where, from Eqs. (4.143)±(4.149), we get axx w
1 ;
Dx2
5:293
axx e
1 ;
Dx2
5:294
axx n
1 2er
p; q 2 e
p; q e
p; q
Dy r r
;
5:295
axx s
1 2er
p; q ; 2 e
p; q e
p; q 1
Dy r r
5:296
axx x
2
Dx2
axx y
1 2er
p; q 2 e
p; q e
p; q
Dy r r
axx n
axx s :
axx e
1
axx w;
5:297
1
1 2er
p; q 2 e
p; q e
p; q 1
Dy r r
5:298
226
BEAM PROPAGATION METHODS
On the other hand,
~ Pyy H~ y ayy w Hy
p
~ 1; q ayy e Hy
p 1; q
~ ayy n Hy
p; q
~ 1 ayy s Hy
p; q 1
yy ~ 2
ayy x ay Hy
p; q fk0 er
p; q
b2 gH~ y
p; q;
5:299
where, from Eqs (4.123) to (4.129), we get
ayy w
1 2er
p; q ; 2 e
p; q e
p 1; q
Dx r r
5:300
ayy e
1 2er
p; q ; 2 e
p; q e
p 1; q
Dx r r
5:301
ayy n
1 ;
Dy2
5:302
ayy s
1 ;
Dy2
5:303
ayy x
1 2er
p; q 2 e
p; q e
p 1; q
Dx r r 1 2er
p; q 2 e
p; q e
p 1; q
Dx r r
ayy y
2
Dy2
ayy n
ayy s :
axx w
axx e ;
5:304
5:305
Although the derivation is not shown here, the FD expressions for the interaction terms Pxy H~ y and Pyx H~ x (for the interaction between
PROBLEMS
227
H~ x and H~ y ) are
1 er
p; q 1 H~
p 1; q 1 4 Dx Dy er
p; q 1 y er
p; q H~
p 1; q 1 1 er
p; q 1 y er
p; q H~
p 1; q 1 1 er
p; q 1 y er
p; q H~
p 1; q 1; 1 er
p; q 1 y 1 e
p; q r 1 H~
p 1; q 1 Pyx H~ x 4 Dx Dy er
p 1; q x er
p; q H~
p 1; q 1 1 er
p 1; q x er
p; q H~
p 1; q 1 1 er
p 1; q x er
p; q H~
p 1; q 1: 1 er
p 1; q x Pxy H~ y
5:306
5:307
The ®nal FD beam propagation equations to be solved can be obtained by applying the FD procedure to the derivative with respect to the z coordinate.
PROBLEMS 1. Using a plane wave, evaluate the difference error for the FD approximation used in the FD-BPM.
ANSWER When we use the center difference scheme, the second derivative of the wave function f
x with respect to x is d 2 fi fi1 2fi fi dx2
Dx2
1
;
P5:1
228
BEAM PROPAGATION METHODS
where Dx is the discretization width and i is the number of the node at the difference center. If we assume that wave function f
x is a plane wave f
x f0 exp
jkx x f0 exp
jkx Dx i;
P5:2
the left- and right-hand sides of Eq. (5.1) can be respectively rewritten as kx2 fi
P5:3
and 1 fi exp
jkx Dx
Dx2
2 exp
jkx Dx
1 2 cos
kx Dx
Dx2
2fi
2
cos
kx Dx 1fi
Dx2 2 2 kx Dx 2 sin fi : 2
Dx2
Using these two factors, we get the relative error e: e
fkx2
( 4=
Dx2 sin2
kx Dx=2gfi 1 2 kx2 kx kx2 fi
2 ) 2 kx Dx sin : Dx 2
P5:4
When the discretization width Dx is very small, sin
kx Dx=2 can be approximated as kx Dx=2. Thus, the relative error e becomes zero as Dx ! 0. 2. Figure P5.1 shows a simple example of an analysis region for a 1D FDBPM. Nodes 1±4 are inside the analysis window, and nodes 0±5 are outside the window. Show the form of the matrix equation for this example.
229
PROBLEMS
FIGURE P5.1. Simple example of a 1D FD-BPM.
ANSWER 0
B0
1
B B A
2 B B @
10
C
1
fl1 1
0
1
D
1
1
B
2
C
2
A
3
B
3
C CB l1 C B CB f2 C B D
2 C C CB CB B l1 C B D
3 C; C
3 C f A A@ 3 A @
A
4
B0
4
fl1 4
P5:5
D
4
where B0
1 and B0
4 include the boundary conditions due to nodes 0 and 5. For explicit forms of the coef®cients on the left-hand side and of D
1 to D
4 on the right-hand side, see Eqs. (5.178)±(5.181) and Eqs. (5.186) and (5.193). 3. The 3D semivectorial analysis shown in Section 5.3 was based on the ADI method. Discuss the procedure for a 3D analysis not using the ADI method. ANSWER The starting equation is Eq. (5.256). Using a procedure similar to the one speci®ed in Eqs. (5.131)±(5.133), we get 2jb
fl1 p;q
flp;q
Dz 1
al1 fl1 fal1 k02 el1 x r
p; q 2 w p 1;q
n2eff gfl1 p;q
l1 l1 l1 l1 l1 l1 l1 al1 e fp1;q an fp;q 1 as fp;q1 ay fp;q
1
alw flp 2
1;q
aln flp;q
als flp;q1 aly flp;q ;
1
falx k02 elr
p; q
n2eff gflp;q ale flp1;q
where the superscripts l 1 and l respectively correspond to the unknown and the known quantities.
230
BEAM PROPAGATION METHODS
Thus, we get the 3D FD-BPM expression l1 al1 w fp 1;q
al1 x
al1 y
4jb Dz
k02 el1 r
p; q
l1 l1 l1 al1 al1 al1 e fp1;q n fp;q 1 s fp;q1 4jb l l aw fp 1;q alx aly k02 elr
p; q Dz
ale flp1;q aln flp;q
1
als flp;q1 :
n2eff
n2eff
fl1 p;q
flp;q
P5:6
REFERENCES [1] M. D. Feit and J. A. Freck, Jr., ``Light propagation in graded-index optical ®bers,'' Appl. Opt., vol. 17, pp. 3990±3998, 1978. [2] L. ThyleÂn, ``The beam propagation method: An analysis of its applicability,'' Opt. Quantum Electron, vol. 15, pp. 433±439, 1983. [3] J. Yamauchi, J. Shibayama, and H. Nakano, ``Beam propagation method using Pade approximant operators,'' Trans. IEICE Jpn., vol. J77-C-I, pp. 490±494, 1994. [4] Y. Chung and N. Dagli, ``Assessment of ®nite difference beam propagation,'' IEEE J. Quantum Electron., vol. 26, pp. 1335±1339, 1990. [5] G. R. Hadley, ``Wide-angle beam propagation using Pade approximant operators,'' Opt. Lett., vol. 17, pp. 1426±1428, 1992. [6] G. R. Hadley, ``A multistep method for wide angle beam propagation,'' Integrated Photon. Res., vol. ITu I5-1, pp. 387±391, 1993. [7] J. Yamauchi, J. Shibayama, and H. Nakano, ``Modi®ed ®nite-difference beam propagation method on the generalized Douglas scheme for variable coef®cients,'' IEEE Photon. Technol. Lett., vol. 7, pp. 661±663, 1995. [8] J. Yamauchi, T. Ando, and H. Nakano, ``Beam-propagation analysis of optical ®bres by alternating direction implicit method,'' Electron. Lett., vol. 27, pp. 1663±1665, 1991. [9] J. Yamauchi, T. Ando, and H. Nakano, ``Propagating beam analysis by alternating-direction implicit ®nite-difference method,'' Trans. IEICE Jpn., vol. J75-C-I, pp. 148±154, 1992 (in Japanese). [10] P. L. Liu and B. J. Li, ``Study of form birefringence in waveguide devices using the semivectorial beam propagation method,'' IEEE Photon. Technol. Lett., vol. 3, pp. 913±915, 1991.
REFERENCES
231
[11] W. P. Huang and C. L. Xu, ``Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,'' IEEE J. Quantum Electron., vol. 29, pp. 2639±2649, 1993. [12] M. Koshiba and Y. Tsuji, ``A wide-angle ®nite element beam propagation method,'' IEEE Photon. Technol. Lett., vol. 8, pp. 1208±1210, 1996. [13] G. R. Hadley, ``Transparent boundary condition for beam propagation,'' Opt. Lett., vol. 16, pp. 624±626, 1992. [14] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, Cambridge University Press, New York, 1992. [15] K. Kawano, M. Kohtoku, M. Wada, H. Okamoto, Y. Itaya, and M. Naganuma, ``Design of spotsize-converter-integrated laser diode (SS-LD) with a lateral taper, thin-®lm core and ridge in the 1.3 mm wavelength region based on the 3-D BPM,'' IEEE J. Select. Top. Quantum Electron., vol. 2, pp. 348±354, 1996. [16] K. Kawano, M. Kohtoku, H. Okamoto, Y. Itaya, and M. Naganuma, ``Coupling and conversion characteristics of spot-size converter integrated laser diodes,'' IEEE J. Select. Top. Quantum Electron., vol. 3, pp. 1351± 1360, 1997. [17] K. Kawano, M. Kohtoku, N. Yoshimoto, S. Sekine, and Y. Noguchi, ``2 2 InGaAlAs=InAlAs multiple quantum well (MQW) directional coupler waveguide switch modules integrated with spot-size converters,'' Electron. Lett., vol. 30, pp. 353±354, 1994.
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 6
FINITE-DIFFERENCE TIME-DOMAIN METHOD In the preceding chapters, steady-state wave equations were solved in which the derivative with respect to the time t (i.e., @=@t) was replaced by jo. In this chapter, we discuss the ®nite-difference time-domain method (FD-TDM), which was developed by Yee [1] and which directly solves time-dependent Maxwell equations. The FD-TDM was originally proposed for electromagnetic waves with long wavelengths, such as 1 1 ± 20 microwaves, because the spatial discretization it requires is small (10 of the wavelength). As the FD-TDM is an explicit scheme, the time step in the calculation is de®ned by the spatial discretization width. Thus, the time step in the optical waveguide analysis is extremely short when wavelengths are of micrometer order. The amount of required memory is enormous for 3D structures, but the method is readly applicable to 2D structures. Finite-difference TDM CAD software suitable for microwave wavelengths as well as optical wavelengths is available on the market.
6.1 DISCRETIZATION OF ELECTROMAGNETIC WAVES The 3D formulation is shown here because it is more versatile than the 2D formulation, which can be easily obtained from the 3D formulation. 233
234
FINITE-DIFFERENCE TIME-DOMAIN METHOD
The time-dependent Maxwell equations are @H =3E; @t @E e0 er =3H; @t m0
6:1
6:2
and using Eqs. (2.5)±(2.10), we can write their component representations as follows: @Hx @Ez @t @y
@Ey ; @z
6:3
@Hy @Ex @t @z @Ey @H m0 z @t @x
@Ez ; @x
6:4
@Ex ; @y
6:5
@Ex @Hz @t @y
@Hy ; @z
6:6
@Ey @Hx @t @z @Hy @E e0 er z @t @x
@Hz ; @x
6:7
@Hx : @y
6:8
m0 m0
e0 er e0 er
When we assume that Dx, Dy, and Dz are spatial discretizations and that Dt is a time step, the function F
x; y; z; t is discretized as F n
i; j; k F
iDx; jDy; kDz; nDt F
x; y; z; t:
6:9
Figure 6.1 shows what we call the Yee lattice [1]. Using a to represent a spatial coordinate such as x, y, and z, we de®ne 8 > < spatial coordinate a: half -integer; Ea the other spatial ones: integer;
6:10 > : time: integer; 8 > < spatial coordinate a: integer;
6:11 Ha the other spatial ones: half -integer; > : time: half -integer; Difference centers play important roles when Eqs. (6.3)±(6.8) are discretized, and here we investigate Eq. (6.3) as an example. Since the spatial difference centers of the left-hand side of the equation are the same
6.1
DISCRETIZATION OF ELECTROMAGNETIC WAVES
235
FIGURE 6.1. Yee lattice.
as those for Hx, the spatial difference centers in the x, y, and z directions are respectively found to be x i Dx, y
j 12 Dy, and z
k 12 Dz. Since the time difference center of the right-hand side of the equation is the same as that for the electric ®elds Ey and Ez, we can write t n Dt. Here, i, j, k, and n are integers. In a similar manner, we can also obtain the following difference centers of the spatial coordinates and the time for Eqs. (6.3)±(6.8). Eq:
6:3:
y
j 12 Dy;
x i Dx; t n Dt;
Eq:
6:4:
x
i
1 2
Dx;
y j Dy;
z
k 12 Dz; z
k
t n Dt; Eq:
6:5:
x
i
1 2
Dx;
y
j
t n Dt; Eq:
6:6:
x t
Eq:
6:7: Eq:
6:8:
i 12 Dx;
n 12 Dt;
x i Dx;
t
n 12 Dt;
x i Dx; t
n 12 Dt:
1 2
Dy;
1 2
6:12
Dz;
6:13
z k Dz;
6:14
y j Dy;
z k Dz;
6:15
y
j y j Dy;
1 2
Dy;
z k Dz;
6:16 z
k
1 2
Dz;
6:17
236
FINITE-DIFFERENCE TIME-DOMAIN METHOD
The 3D ®nite-difference time-domain expressions for Eqs. (6.3)±(6.8) can be obtained by discretizing them on the basis of the difference centers (6.12)±(6.17). Again, we investigate Eq. (6.3) as an example. The difference centers x and t are both integers and the difference centers y and z are both half-integers. Thus, for the left-hand side of Eq. (6.3), we get m0 n1=2
i; j 12 ; k 12 H Dt x
Hxn
1=2
i; j 12 ; k 12:
6:18
For the right-hand side, we get 1 n E
i; j 1; k 12 Ezn
i; j; k 12 Dy z 1 n E
i; j 12 ; k 1 Eyn
i; j 12 ; k: Dz y
6:19
Using expressions (6.18) and (6.19), we get the following ®nite-difference time-domain expression for Eq. (6.3): Hxn1=2
i; j 12 ; k 12 Hxn
1=2
i; j 12 ; k 12
Dt 1 n E
i; j 1; k 12 m0 Dy z
1 n E
i; j 12 ; k 1 Dz y
Ezn
i; j; k 12
Eyn
i; j
1 2 ; k
:
6:20 Through the same procedure, we get the following ®nite-difference time-domain expressions for the y and z components of the magnetic ®elds: Hyn1=2
i 12 ; j; k 12 Hyn
1=2
i 12 ; j; k 12
Dt 1 n E
i 12 ; j; k 1 m0 Dz x 1 n E
i 1; j; k 12 Dx z
Exn
i 12 ; j; k
Ezn
i; j; k
1 2
:
6:21
6.1
237
DISCRETIZATION OF ELECTROMAGNETIC WAVES
and Hzn1=2
i 12 ; j 12 ; k Hzn
1=2
i 12 ; j 12 ; k
Dt 1 n E
i 1; j 12 ; k m0 Dx y 1 n E
i 12 ; j 1; k Dy x
Exn
i
Eyn
i; j 12 ; k
1 2 ; j; k
:
6:22 Next, we discretize Eq. (6.6). According to expression (6.15), the difference centers of x and t are both half-integers and the difference centers y and z are both integers. Thus, for the left-hand side of Eq. (6.6), we get e0 er n1 E
i 12 ; j; k Dt x
Exn
i 12 ; j; k:
6:23
For the right-hand side, we get 1 H n1=2
i 12 ; j 12 ; k Dy z
Hzn1=2
i 12 ; j
1 H n1=2
i 12 ; j; k 12 Dz y
1 2 ; k
Hyn1=2
i 12 ; j; k
1 2:
6:24
Using expressions (6.23) and (6.24), we get the following ®nite-difference time-domain expression for Eq. (6.6): Exn1
i 12 ; j; k Exn
i 12 ; j; k Dt 1 H n1=2
i 12 ; j 12 ; k e0 er Dy z Hzn1=2
i 12 ; j
1 2 ; k
1 H n1=2
i 12 ; j; k 12 Dz y n1=2 1 1
i 2 ; j; k 2 : Hy
6:25
238
FINITE-DIFFERENCE TIME-DOMAIN METHOD
Through the same procedure, we get the following ®nite-difference time-domain expressions for the y and z components of the electric ®elds: Eyn1
i; j 12 ; k Eyn
i; j 12 ; k Dt 1 H n1=2
i; j 12 ; k 12 e0 er Dz x Hxn1=2
i; j 12 ; k
1 2
1 H n1=2
i 12 ; j 12 ; k Dx z Hzn1=2
i 12 ; j 12 ; k
6:26
and Ezn1
i; j; k 12 Ezn
i; j; k 12 Dt 1 H n1=2
i 12 ; j; k 12 e0 er Dx y Hyn1=2
i
1 2 ; j; k
12
1 H n1=2
i; j 12 ; k 12 Dy x n1=2 1 1
i; j 2 ; k 2 : Hx
6:27
Magnetic ®elds Han1=2 with the half-integer time step
n 1=2 Dt are calculated ®rst from Eqs. (6.20)±(6.22) by using the electric ®elds with the integer time step n Dt. Then those ®elds are used to calculate the electric ®elds Ean1 with the integer time step
n 1 Dt by using Eqs. (6.25)± (6.27). Repeating these two steps, we can calculate the time evolution of the electric and magnetic ®elds directly. It should be noted that the relative permittivity at the interface between two media is approximated better by using
er1 er2 =2 than by using only er1 or er2.
6.2 STABILITY CONDITION
239
6.2 STABILITY CONDITION In an explicit scheme such as the FD-TDM, the time step Dt in the calculation is restricted by the spatial discretization. For simplicity in discussing the stability condition here, we will use the 1D scalar Helmholtz equation @2 f @x2
em0
@2 f 0; @t 2
6:28
where f is a 1D wave function that designates the time-dependent ®eld. Using bx as the x-directed propagation constant, we express this wave function as f
x; t exp
jbx x exp
at exp
jbx p Dx exp
an Dt exp
jbx p Dxxn ;
6:29
where x exp
a Dt. Thus, if the ®eld is to be stable, x has to satisfy the condition jxj 1:
6:30
Substituting Eq. (6.29) into (6.28), we get 1 fexp jbx
p 1 Dxxn 2 exp
jbx p Dxxn exp jbx
p 1 Dxxn g 2
Dx em0 fexp
jbx p Dxxn1 2 exp
jbx p Dxxn exp
jbx p Dxxn 1 g 0: 2
Dt Dividing this equation by exp
jbx p Dxxn, we reduce it to 1 fexp
jbx Dx
Dx2
2 exp
jbx Dxg
em0
x
Dt2
2 x 1 0:
6:31
240
FINITE-DIFFERENCE TIME-DOMAIN METHOD
Dividing Eq. (6.31) by em0 =
Dt2 x and considering that the ®rst term of Eq. (6.31) can be rewritten as fexp
jbx Dx
2 exp
jbx Dxg 2
cos
bx Dx 1 Dx 2 ; 4 sin bx 2
we get
x
2
Dt2 em0
2x 1
4 Dx 2 x0 sin bx 2
Dx2
and therefore x2
2A 1 0;
6:32
where the parameter A is de®ned by A
2
Dt2 1 Dx 2 1: sin bx 2 em0
Dx2
6:33
The roots of Eq. (6.32) are x1 A x2 A
p A2 1; p A2 1:
6:34
6:35
Because jxj 1 and 0 sin2 y, we get the relation A
2
Dt2 1 Dx 2 sin b 1 1: x 2 em0
Dx2
6:36
We can thus specify the stability condition in terms of A: Case A < 1. Since, according to Eq. (6.35), 1 < jx2 j, the ®eld is unstable.
6.3
Case
ABSORBING BOUNDARY CONDITIONS
241
1 A 1. Since x1 and x2 can be expressed as x1 A x2 A
p p A2 1 A j 1 A2 ; p p A2 1 A j 1 A2 ;
6:37
6:38
their absolute values can be expressed as jx1 j jx2 j A2
1
A2 1:
6:39
Thus, the ®eld is stable when 1 A 1:
6:40
Relation (6.40) can be interpreted as imposing the following restriction on the time step (see Problem 1): p 1 Dt em0
Dx2
1=2
p er 1 c0
Dx2
1=2
1 1 n
Dx2
1=2
;
6:41
where Dx is the spatial discretization width and Dt is the time step and p where c0 , er , and n c0 = er are respectively the velocity of the light in a vacuum, the relative permittivity of the medium, and the velocity of the light in the medium. Equation (6.41) is for a 1D structure, and the corresponding restriction for a 3D structure is 1 1 1 1 Dt n
Dx2
Dy2
Dz2
1=2
:
6:42
6.3 ABSORBING BOUNDARY CONDITIONS Since the FD-TDM, like the BPM in Chapter 5, has ®nite analysis windows, an arti®cial boundary condition suppressing re¯ections at the analysis windows is required. Mur's absorbing boundary condition (ABC) [2] is often used for this purpose, though Berenger's perfectly matched layer (PML) scheme [3] has also come into use recently. The PML scheme suppresses re¯ections better than Mur's ABC does, but Mur's condition is easier to use. Here, we discuss Mur's ®rst-order ABC.
242
FINITE-DIFFERENCE TIME-DOMAIN METHOD
As shown in Fig. 6.2, the analysis window is de®ned in ranges of
0; Lx in the x direction,
0; Ly in the y direction, and
0; Lz in the z direction. 1. x 0: Ey and Ez . The electric ®elds Ey and Ez are on the boundary x 0. The wave function W for the left-traveling wave incident perpendicular to the boundary is 1 W expj
ot bx x exp jo t x ; nx
6:43
where nx is the velocity of the wave. Thus, the derivatives of the wave function with respect to x and t are @W 1 1 1 jo exp jo t x joW ; @x nx nx nx
6:44
@W joW : @t
6:45
Substituting Eq. (6.45) into (6.44), we get @W 1 @W @x nx @t
FIGURE 6.2. Analysis region.
6.3
ABSORBING BOUNDARY CONDITIONS
243
and therefore @W @x
1 @W 0: nx @t
Thus, the wave equation for Mur's ®rst-order ABC is
@ @x
1 @ 0: W nx @t x0
6:46
Next, we discretize the wave equation (6.46) for the case that the wave function is that for the y-directed electric ®eld Ey . Assuming that the node number of the node on the boundary is 0, we discretize the derivative of the electric ®eld Ey with respect to the coordinate x as @Ey 1 1 n Ey 1; j 12 ; k 2 Dx @x
Eyn 0; j 12 ; k
1 n1 Ey 1; j 12 ; k Dx
Eyn1 0; j 12 ; k
;
6:47
where the time average between n and n 1 was taken on the right-hand. On the other hand, we discretize the derivative of the electric ®eld Ey with respect to time t as @Ey 1 1 n1 Ey 1; j 12 ; k 2 Dt @x
1 n1 Ey 0; j 12 ; k Dt
Eyn 1; j 12 ; k Eyn 0; j 12 ; k
;
6:48
where the spatial average between i 0 and i 1 was taken. Substituting Eqs. (6.47) and (6.48) into Eq. (6.46), we can derive the following ®nitedifference time-domain expression for the electric ®eld Ey : Eyn1
0; j 12 ; k Eyn
1; j 12 ; k
nx Dt Dx n1 E
1; j 12 ; k nx Dt Dx y
Eyn
0; j 12 ; k:
6:49
244
FINITE-DIFFERENCE TIME-DOMAIN METHOD
We can similarly derive the ®nite-difference time-domain expression for the electric ®eld Ez : Ezn1
0; j; k 12 Ezn
1; j; k 12
nx Dt Dx n1 E
1; j; k 12 nx Dt Dx z
Ezn
0; j; k 12:
6:50
2. x Lx : Ey and Ez . The electric ®elds Ey and Ez are on the boundary x Lx . The wave function W for the right-traveling wave incident perpendicular to the boundary is W exp j
ot
bx x exp jo t
1 x ; nx
6:51
where nx is a velocity of the wave. Thus, the derivatives of the wave function with respect to x and t are @W @x
1 joW ; nx
6:52
@W joW : @t
6:53
Substituting Eq. (6.53) into (6.52), we get the wave equation @ 1 @ 0: W @x nx @t xLx
6:54
Next, we discretize the wave equation (6.46) for the case that the wave function is the y-directed electric ®eld Ey . Assuming that the node number of the node on the boundary is Nx , we discretize the derivative of the electric ®eld Ey with respect to x as @Ey 1 1 n Ey Nx ; j 12 ; k 2 Dx @x
1 n1 Ey Nx ; j 12 ; k Dx
Eyn Nx
1; j 12 ; k
Eyn1 Nx
1; j 12 ; k
;
6:55
PROBLEMS
245
where the time average between n and n 1 was taken on the right-hand side. On the other hand, we discretize the derivative of the electric ®eld Ey with respect to time t as @Ey 1 1 n1 Ey Nx ; j 12 ; k 2 Dt @t 1 n1 Nx E Dt y
1; j
Eyn Nx ; j 12 ; k
1 2;k
Eyn
Nx
1; j
1 2;k
;
6:56
where the spatial average between i Nx and Nx 1 was taken. Substituting Eqs. (6.55) and (6.56) into Eq. (6.54), we can derive the ®nitedifference time-domain expression for the electric ®eld Ey : Eyn1
Nx ; j 12 ; k Eyn
Nx
1; j 12 ; k
nx Dt Dx n1 E
Nx nx Dt Dx y
1; j 12 ; k
Eyn
Nx ; j 12 ; k:
6:57
We can similarly derive the ®nite-difference time-domain expression for the electric ®eld Ez : Ezn1
Nx ; j; k 12 Ezn
Nx
1; j; k 12
nx Dt Dx n1 E
Nx nx Dt Dx z Ezn
Nx ; j; k 12:
1; j; k 12
6:58
For the ABCs on y 0 and y Ly and on z 0 and z Lz, see Problem 2.
PROBLEMS 1. Derive the restriction on the time step Dt speci®ed in Eq. (6.41) by using the stability condition 1 A 1 in relation (6.40).
246
FINITE-DIFFERENCE TIME-DOMAIN METHOD
ANSWER Using Eq. (6.33), we can rewrite the stability condition in relation (6.40) as 1
2
Dt2 1 Dx 2 sin bx 1 1: 2 em0
Dx2
P6:1
Since the relation between the center term and right-hand term is always satis®ed, we have to consider only the left-hand term and center term: 1
2
Dt2 1 Dx 2 sin bx 1: 2 em0
Dx2
Multiplying both sides by 1 and considering the case in which the righthand side reaches its maximum, when sin2
1, we can rewrite the above relation as 1
2
Dt2 1 em0
Dx2
1:
Thus, we get 2
Dt2 1 ; 2 em0
Dx2 and therefore 2
Dt em0
1
Dx2
1
:
And this relation can be rewritten as the restriction on the time step Dt: p 1 Dt em0
Dx2
1=2
p er 1 c0
Dx2
1=2
1 1 n
Dx2
1=2
:
P6:2
2. Derive the ABC ®elds for y 0 and y Ly and z 0 and z Lz .
247
PROBLEMS
ANSWER a. y 0. The wave equation on this boundary is ! @ 1 @ 0: W @y ny @t y0
P6:3
The ®nite-difference time-domain expressions for the electric ®elds Ex and Ez on this boundary are Exn1
i 12 ; 0; k Exn
i 12 ; 1; k
ny Dt Dy n1 E
i 12 ; 1; k ny Dt Dy x
Exn
i 12 ; 0; k;
P6:4
Ezn1
i; 0; k 12 Ezn
i; 1; k 12
ny Dt Dy n1 E
i; 1; k 12 ny Dt Dy z
Exn
i; 0; k 12:
P6:5
b. y Ly . The wave equation on this boundary is ! @ 1 @ 0: W @y ny @t yLy
P6:6
The ®nite-difference time-domain expressions for the electric ®elds Ex and Ez on this boundary are Exn1
i 12 ; Ny ; k Exn
i 12 ; Ny
1; k
ny Dt Dy n1 E
i 12 ; Ny ny Dt Dy x
1; k
Exn
i 12 ; Ny ; k;
P6:7
Ezn1
i; Ny ; k 12 Ezn
i; Ny
1; k 12
ny Dt Dy n1 E
i; Ny ny Dt Dy z
1; k 12
Ezn
i; Ny ; k 12:
P6:8
248
FINITE-DIFFERENCE TIME-DOMAIN METHOD
c. z 0. The wave equation on this boundary is
@ @z
1 @ W 0: nz @t z0
The ®nite-difference time-domain expressions for the electric ®elds Ex and Ey on this boundary are Exn1
i 12 ; j; 0 Exn
i 12 ; j; 1
nz Dt Dz n1 E
i 12 ; j; 1 nz Dt Dz x
Exn
i 12 ; j; 0;
P6:9
Eyn1
i; j 12 ; 0 Eyn
i; j 12 ; 1
nz Dt Dy n1 E
i; j 12 ; 1 nz Dt Dy y
Eyn
i; j 12 ; 0:
P6:10
d. z Lz . The wave equation on this boundary is
@ 1 @ W 0: @z nz @t zLz
P6:11
The ®nite-difference time-domain expressions for the electric ®elds Ex and Ey on this boundary are Exn1
i 12 ; j; Nz Exn
i 12 ; j; Nz
1
nz Dt Dz n1 E
i 12 ; j; Nz nz Dt Dz x
1
Exn
i 12 ; j; Nz ;
P6:12
Eyn1
i; j 12 ; Nz Eyn
i; j 12 ; Nz
1
nz Dt Dy n1 E
i; j 12 ; Nz nz Dt Dy y
1
Eyn
i; j 12 ; Nz :
P6:13
REFERENCES
249
REFERENCES [1] K. S. Yee, ``Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,'' IEEE Trans. Antennas Propagat., vol. AP-14, pp. 302±307, 1966. [2] G. Mur, ``Absorbing boundary conditions for the ®nite-difference timedomain approximation of the time domain electromagnetic ®eld equations,'' IEEE Trans. Electromagn. Compat., vol. EMC-23, pp. 377±382, 1981. [3] J.-P. Berenger, ``A perfectly matched layer for the absorption of electromagnetic waves,'' J. Computat. Phys., vol. 114, pp. 185±220, 1994.
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 7
È DINGER EQUATION SCHRO In this chapter, we will investigate ways to solve the SchroÈdinger equation, which must be solved when quantum wells for semiconductor optical waveguide devices are designed [1, 2]. The time-dependent SchroÈdinger equation resembles the Fresnel wave equation, and the time-independent SchroÈdinger equation resembles the wave equation for the cross-sectional analysis. Thus, the analysis techniques given earlier for optical waveguides can be applied to the analysis of these SchroÈdinger equations. Here, the analysis of the time-dependent SchroÈdinger equation will be based on the 2D FD-BPM and the analysis of the time-independent SchroÈdinger equation will be based on the 1D FDM and the 1D FEM.
7.1 TIME-DEPENDENT STATE Let us solve the time-dependent SchroÈdinger equation by using the 2D FD-BPM. The only major difference between the time-dependent SchroÈdinger equation and the BPM wave equation based on the Fresnel approximation is that the derivative in the SchroÈdinger equation is with respect to time, whereas the derivative in the BPM wave equation is with respect to position. 251
252
È DINGER EQUATION SCHRO
Under the effective mass approximation, the time-dependent SchroÈdinger equation is expressed as @c
x; t h2 @ 1 @c
x; t U
xc
x; t;
7:1 jh @t @x 2 @x m
x where U
x, m
x, and h are respectively an arbitrary potential shown in Fig. 7.1, an effective mass, and the Plank constant [3]. First, the time t and the space x are discretized. The nonequidistant discretization shown in Fig. 5.5 is assumed, and the wave function c
x; t, the potential U
x, and the effective mass m
x are expressed as c
x; t c
xp ; tn cnp ;
7:2
U
x U
xp U
p;
7:3
m
x m
xp m
p:
7:4
The 2D Fresnel wave equation for the TM mode of an optical waveguide that was derived in Chapter 5 [Eq. (5.118)] is @f
x; z @ 1 @f
x; z 2jb er
x k02 er
x n2eff f
x; z:
7:5 @z @x er
x @x Comparing this with the time-dependent SchroÈdinger equation (7.1), we ®nd the following correspondences: z $ t; @ 1 @ h2 @ 1 @ $ ; er
x @x er
x @x 2 @x m
x @x 2jb $ jh; k02 er
x
n2eff $ U
x:
7:6
7:7
7:8
7:9
FIGURE 7.1. Potential distribution for a semiconductor quantum well.
7.2
FINITE-DIFFERENCE ANALYSIS OF TIME-INDEPENDENT STATE
253
Thus, from Eq. (5.126) and Eqs. (5.142)±(5.144), we can get the ®nitedifference expression for the right-hand side of correspondence (7.7): h2 @ 1 @f aw cp 2 @x m
x @x
1
ax cp ae cp1 ;
7:10
where h2 2 2 2 w
e w m
p m
p
aw
1
;
h2 2 2 ; 2 e
e w m
p m
p 1 h2 2 2 ax 2 w
e w m
p m
p 1 2 2 e
e w m
p m
p 1 ae
ae
aw :
7:11
7:12
7:13
Considering the relations shown in correspondences (7.6)±(7.9) and in Eq. (5.135), we get the following ®nite-difference expression for the timedependent SchroÈdinger equation:
2jh ax Dt
U
p cpn1
aw cn1 p 1
ae cn1 p1
aw clp 1
2jh ax U
p clp ae clp1 : Dt
7:14
It is easily understood that this equation can be solved in the same way that the 2D FD-BPM is solved and that the transparent boundary condition can also be used with this equation.
7.2 FINITE-DIFFERENCE ANALYSIS OF TIME-INDEPENDENT STATE The wave function c
x; t that satis®es the time-dependent SchroÈdinger equation (7.1) is divided into the space-dependent term and the time-
254
È DINGER EQUATION SCHRO
dependent term as follows: c
x; t f
x exp
jot f
x exp
jE
t ; h
7:15
where E is the eigenenergy to be calculated and corresponds to the effective index in the optical waveguide problems. Substituting Eq. (7.15) into Eq. (7.1) and dividing the resultant equation by exp
jEt=h, we get h2 d 1 df
x U
x Ef
x 0:
7:16 2 dx m
x dx This is the time-independent SchroÈdinger equation. Discretizing it by using Eqs. (7.10)±(7.13), we get the following ®nite-difference expression: a w fp
1
ae fp1 ax fp U
p
Efp 0
aw fp
1
ax U
pfp ae fp1
Efp 0:
and therefore
7:17
The remaining task is to construct a matrix equation for Eq. (7.17).
7.3 FINITE-ELEMENT ANALYSIS OF TIME-INDEPENDENT STATE 7.3.1 Eigenvalue Equation To solve the time-independent SchroÈdinger equation, we ®rst derive the eigenvalue equation based on the FEM [4, 5]. The following normalizations are introduced: x
coordinate; W U
x U
x 1
potential; E1 x
E E 1 E1 m
m
x m0
eigenenergy;
effective mass;
7:18
7:19
7:20
7:21
7.3
FINITE-ELEMENT ANALYSIS OF TIME-INDEPENDENT STATE
255
where E11 h 2 p2 =
2m0 W 2 , m0 , and W are respectively the electron energy of the ground state in an in®nitely deep well, the static mass of an electron, and the width of a quantum well. When we substitute these normalized parameters into Eq. (7.16), we get the normalized SchroÈdinger equation d 1 df
x x 0:
7:22 p2 U
x Ef
d x m
x d x Next, we use the Galerkin method to solve this equation. For simplicity, the overbar, denoting normalized quantities in Eqs. (7.18) to (7.22), is omitted. Figure 7.2 shows nodes used in the 1D FEM. Using shape functions, we expand the wave function f
x for an element e: fe
Me P i1
Nei fei Ne T ffe g:
7:23
Substituting Eq. (7.23) into Eq. (7.22), we get d 1 d Ne T ffe g p2 U
x ENe T ffe g 0: dx me dx Multiplying the left-hand side of this equation by the shape function and integrating the resultant equation in element e, we get
d 1 d Ne Ne T dxffe g dx m dx e e
p2 Ne U
x ENe T dxffe g f0g:
7:24 e
Applying the partial integration to the ®rst term of Eq. (7.24) and denoting the node numbers of the left- and right-hand nodes in element e as i and i 1 , we rewrite Eq. (7.24) as i1
1 dNe T dNe 1 dNe T Ne ffe g dxffe g me dx dx e dx me i
p2 Ne
U
x ENe T dxffe g f0g: e
FIGURE 7.2. Nodes in the 1D ®nite-element method.
256
È DINGER EQUATION SCHRO
Furthermore, assuming that both the effective mass m and the potential U
x
Ue are constant in the element, we can reduce this equation to i1
1 dNe T 1 dNe dNe T ffe g dxffe g Ne me dx dx dx e me i
2 p
Ue E Ne Ne T dxffe g f0g: e
7:25
Next, we sum Eq. (7.25) for all elements. The ®rst term of Eq. (7.25) becomes P e
Ne
i1 1 dNe T ffe g m dx i
1 df2 1 df1 1 df3 1 df2 f1 f3 f2 f2 m2 dx m1 dx m3 dx m2 dx 1 dfM 1 1 dfM 2 fM 1 fM 2 mM 1 dx mM 2 dx 1 dfM 1 dfM 1 fM fM 1 mM dx mM 1 dx
f1
1 df1 1 dfM fM ; m1 dx mM dx
where M is the total number of nodes. It should be noted that the following continuity conditions for the wave function and its derivative are assumed at adjacent elements e and e 1: 1 @f 1 @f :
7:26 fje fje1 ; m @x e m @x e1 Thus, after summing Eq. (7.25) for all elements, we get
P 1 dNe dNe T 1 dfM fM dxffe g mM dx dx e me e dx
P T 2 P 2 Ue Ne Ne dxffe g p E Ne Ne T dxffe g f0g: p
1 df1 f1 m1 dx
e
e
e
e
7:27
7.3
FINITE-ELEMENT ANALYSIS OF TIME-INDEPENDENT STATE
This equation can be reduced to 1 df1 1 dfM Pffg p2 Qffg f1 fM m1 dx mM dx
257
p2 ERffg f0g;
7:28
where P 1 P e me Q R ffg
P e
P e
Ue
e
dNe dNe T dx; dx e dx e
Ne Ne T dx;
Ne Ne T dx;
P ffe g:
7:29
7:30
7:31
7:32
e
Assuming the Dirichlet condition or the Neumann conditionÐthat is, assuming f0
7:33
df 0 dx
7:34
or
at the leftmost node 1 and the rightmost node MÐwe can obtain from Eq. (7.28) the simple eigenvalue equation
P p2 Qffg
E
p2 Rffg f0g:
7:35
To solve this equation, we have to transform Eqs. (7.28) and (7.35) into eigenvalue matrix equations. To this end, in the following, the ®rst- and second-order shape functions will be obtained and the explicit expressions for the matrixes will be shown. 7.3.2 Matrix Elements A. First-Order Line Element The matrixes for the eigenvalue equation will be calculated by using the ®rst-order line element. Figure 7.3 shows the ®rst-order line element. The node numbers i and j and the coordinates xi and xj are assumed to correspond to the local coordinates 1
258
È DINGER EQUATION SCHRO
and 2. An arbitrary coordinate x in element e is de®ned using the parameter x, which takes a value between 0 and 1: x
1
xxi xxj xi
xj
xi x xi Le x;
7:36
where Le is the length of the element
xj xi . Since the ®rst-order line element has two nodes, the wave function fe
x in element e is expanded as fe
x
2 P i1
Ni
xfi N T ffe g
7:37
by using the shape function N T . The shape functions N1 and N2 for the line elements are expressed by the linear functions N1
x a1 x b1 ;
N2
x a2 x b2 ;
7:38
and the conditions that must be met by the shape functions are x 0:
N1
0 1;
N2
0 0;
7:39
x 1:
N1
1 0;
N2
1 1:
7:40
Thus, the following shape functions can be obtained for the ®rst-order line element: N1
x 1
x;
N2
x x:
7:41
7:42
Next, we calculate the matrix elements shown in Eqs. (7.29)±(7.32).
FIGURE 7.3. First-order line element.
7.3
1.
e
259
FINITE-ELEMENT ANALYSIS OF TIME-INDEPENDENT STATE
dNe =dx
dNe T =dx dx From Eq. (7.36), we get dx 1 dx Le
and therefore, dx Le dx:
7:43
dN1 dx
7:44
Since the relations 1;
dN2 1; dx
7:45
hold, we get dNe dx dNe 1 dx dx dx Le Thus, we get
dNe dNe T 1 dx 2 Le dx e dx 1 Le 1 Le
2.
R
1" 1 # 1
0
1 : 1
"
1
1
0
1Le dx
1
1" 1 1
1
1
1
1 #
7:46
# dx
:
Ne Ne T dx Through a similar procedure, we get
1 "
1 x #
1 x xL dx
e Ne Ne T dx e 0 x " # "
1
1 x2 x
1 x Le 2 Le dx 6 1 0 x
1 x x2
7:47
e
1 2
# :
7:48
260
È DINGER EQUATION SCHRO
Since Eqs. (7.47) and (7.48) can be used to construct the matrixes P, Q, and R, the eigenenergy can be calculated from the eigenvalue matrix equation (7.35). B. Second-Order Line Element Next, we discuss the second-order line element, which is more accurate than the ®rst-order line element. Figure 7.4 shows the second-order line element. The node numbers i, j, and k and the coordinates xi , xj , and xk are assumed to correspond to the local coordinates 1, 2, and 3. An arbitrary coordinate x in element e is de®ned using the parameter x, which takes a value between 1 and 1: x xj 12
xk
xi x:
7:49
Since the second-order line element has three nodes, the wave function fe
x in element e is expanded as fe
x
3 P i1
Ni
xfi Ne T ffe g
7:50
by using the shape function N T . The shape functions N1 , N2 , and N3 for the line elements are expressed by the quadratic polynomials N1
x a1 x2 b1 x c1 ; N2
x a2 x2 b2 x c2 ;
7:51
N3
x a3 x2 b3 x c3 ; and the conditions that must be met by the shape functions are x 0:
N1
0 0;
N2
0 1;
N3 0;
7:52
x 1:
N1
1 0;
N2
1 0;
N3
1 1;
7:53
N1
1 1;
N2
1 0;
N3
1 0:
7:54
x
1:
Thus, the following shape functions can be obtained for the secondorder line element: N1
x
1 2 x
1
x;
N2
x
1 x
1 N3
x 12 x
1 x:
x;
7:55
7:56
7:57
Next, we calculate the matrix elements shown in Eqs. (7.29)±(7.32).
7.3
FINITE-ELEMENT ANALYSIS OF TIME-INDEPENDENT STATE
261
FIGURE 7.4. Second-order line element.
1.
e
dNe =dx
dNe T =dx dx From Eq. (7.49), we get dx 2 dx Le
and therefore, dx 12 Le dx:
7:58
dN1 x dx
7:59
Since the relations
dN2 dx
1 2;
2x;
7:60
dN3 x 12 dx
7:61
hold, we get 2 dNe dx dNe 2 6 4 dx dx dx xk xi
x
13 2
2
2x
1
3
1 6 7 7 2x 5 4 4x 5: xk xi x 12 2x 1
7:62
262
È DINGER EQUATION SCHRO
Thus, we get
dNe dNe T dx dx e dx
x xi 3 2 4x 2x 1 k dx
1 2x 1 2x 1 2 1 7 6 4 4x 5
xk xi 2 1 2x 1 1 2
xk xi 2 3
2x 12
2x 1
4x
2x 1
2x 1
1 6 7 4 4x
2x 1
4x2
4x
2x 1 5 dx 1
2x
2x 12
1
2x 1
2x 1
4x 3 14 16 2 1 16 7 32 16 5: 4 16 2
xk xi 3 2 16 14 2
This equation is summarized as
2.
e
2
14 dNe dNe 1 4 16 dx 6Le dx e dx 2 T
16 32 16
3 2 16 5: 14
7:63
Ne Ne T dx Through a similar procedure, we get
Ne Ne T dx 2 1 3
1 2 x
1 x 6 7 4
1 x
1 x 5 1 1 2 x
1 x e
12
xk
L e 2
1 1
2 6 6 4
2
1 2 x
1
x
1 x
1
x
1 2 x
1 x
xi dx 1 2 4 x
1
x2
2 1 2 x
1 x
1 x 1 2 4 x
1 x
1 x
4 2 Le 1 6 4 2 16 2 15 1 2
1
3
2
1 2 x
1 x
1 2
1 x
1
x2 x2
2 1 2 x
1 x
1
4 2 7 Le 6 2 5 4 2 16 30 4 1 2
1
3
7 2 5: 4
x
1 4
1 x
1 2 1 2 x
1 x
1 2 1 2 4 x
1 x
x
3
7 x 7 5 dx
7:64
REFERENCES
263
Since Eqs. (7.63) and (7.64) can be used to construct the matrixes P, Q, and R, the eigenenergy can be calculated from the eigenvalue matrix equation (7.35). REFERENCES [1] K. Kawano, S. Sekine, H. Takeuchi, M. Wada, M. Kohtoku, N. Yoshimoto, T. Ito, M. Yanagibashi, S. Kondo, and Y. Noguchi, ``4 4 InGaAlAs=InAlAs MQW directional coupler waveguide switch modules integrated with spot-size converters and their 10 Gbit=s operation,'' Electron. Lett., vol. 31, pp. 96±97, 1995. [2] K. Kawano, K. Wakita, O. Mitomi, I. Kotaka, and M. Naganuma, ``Design of InGaAs=InAlAs multiple-quantum well (MQW) optical modulators,'' IEEE J. Quantum Electron., vol. QE-28, pp. 228±230, 1992. [3] L. I. Schiff, Quantum Mechanics, McGraw-Hill, New York, 1968. [4] K. Nakamura, A. Shimizu, M. Koshiba, and K. Hayata, ``Finite-element analysis of quantum wells of arbitray semiconductors with arbitrary potential potential pro®les,'' IEEE J. Quantum Electron., vol. 25, pp. 889±895, 1989. [5] M. Koshiba, H. Saitoh, M. Eguchi, and K. Hirayama, `Simple scalar ®niteelement approach to optical waveguides,'' IEE Proc. J., vol. 139, pp. 166± 171, 1992.
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)
APPENDIX A
VECTORIAL FORMULAS In the following, i, j and k are respectively unit vectors in the x, y, and z directions and f and A are respectively a scalar and a vector: A Ax i Ay j Az k;
A:1
@ @ @ i j k; @x @y @z
A:2
=
= ? =3A 0; =3
=3A =
= ? A
A:3 H2 A;
A:4
H2
@2 @2 @2 2 2; 2 @x @y @z
A:5
H2?
@2 @2 ; @x2 @y2
A:6
=
fA =f ? A f= ? A; i j k @ @ @ =3A @x @y @z Ax Ay Az @Az @Ay @Ax i @y @z @z
A:7
@Ay @Az j @x @x
@Ax k: @y
A:8 265
266
VECTORIAL FORMULAS
If r, u, and z are respectively unit vectors in the radial, azimuthal, and longitudinal directions, the rotation formula for a vector A Ar r Ay u Az z is expressed as 1 @Az =3A r @y
@Ay @Ar r @z @z
@Az 1 @
rAy u r @r @r
1 @Ar z: r @y
A:9
A Laplacian H2 for a cylindrical coordinate is given as @2 @z2 1 @ @ 1 @2 @2 r 2 2 2 r @r @r r @y @z
H2 H2?
@2 1 @ 1 @2 @2 : @r2 r @r r2 @y2 @z2
A:10
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)
APPENDIX B
INTEGRATION FORMULA FOR AREA COORDINATES The integration formula shown in Eq. (3.184) is derived here by calculating the following integration for a triangular element e shown in Fig. 3.4:
Ie
i; j; k
e
Li1 Lj2 Lk3 dx dy;
B:1
where i, j, and k are integers and the spatial coordinates of nodes 1, 2, and 3 are respectively
x1 ; y1 ,
x2 ; y2 , and
x3 ; y3 . As shown in Eq. (3.72), we have the following relation between the spatial coordinates and the area coordinates: 0 1 0 x x1 B C B @ y A @ y1 1 1
x2 y2 1
10 1 L1 x3 CB C y 3 A@ L 2 A: 1 L3
B:2
As shown by the bottom row of Eq. (B.2), L1 L2 L3 1:
B:3 267
268
INTEGRATION FORMULA FOR AREA COORDINATES
According to Eq. (B.2), x and y can be expressed using L1 and L2 as follows: x x1 L1 x2 L2 x3 L3 x1 L1 x2 L2 x3
1 x3 L1
x2
x1
L1
L2
x3 L2 ;
B:4
y y1 L1 y2 L2 y3 L3 y1 L1 y2 L2 y3
1
y1
y3 L1
y2
L1
L2
y3 L2 :
B:5
Thus, we get @x @L @
x; y 1 @
L1 ; L2 @y @L 1
@x @L2 x1 @y y1 @L2
x3
x2
y3
y2
x3 2Se ; y3
B:6
where Se is the area of element e. Using Eq. (B.6) to transform x and y to L1 and L2, we can rewrite Eq. (B.1) as follows:
Ie
i; j; k
e
e
Li1 Lj2 Lk3 dx dy
2Se 2Se 2Se
@
x; y dx dy @
L1 ; L2
Li1 Lj2 Lk3
1 0
1 0
e
Li1 Lj2 Lk3 dL1 dL2
dL1
1
Li1 dL1
L1
0
1 0
Li1 Lj2
1 L1
Lj2
1
L1
L2 k dL2
L1
L2 k dL2 :
B:7
INTEGRATION FORMULA FOR AREA COORDINATES
269
The second integral of Eq. (B.7) is calculated as I0
j; k
1
L1
0
Lj2
1
L1
L2 k dL2
1 L1 1 j1 k L
1 L1 L2 j1 2 0
1 L1 k Lj1 L1 L2 k 1 dL2 2
1 j1 0
1 L1 k 0 Lj1 L1 L2 k 1 dL2 2
1 j1 0 k I
j 1; k 1 j1 0 1 L1 1 j2 k 1 L
1 L1 L2 j2 2 0
1 L1 k
k 1 Lj2 L1 2
1
j 1
j 2 0
0
k
k 1 I
j 2; k
j 1
j 2 0
L2 k
2
dL2
2
k
k 1 1 I
j k; 0
j 1
j 2
j k 0
k!j! I
j k; 0;
j k! 0
B:8
where I0
j k; 0
1
L1
0
Ljk 2
dL2
1
1 jk 1
L1
1 Ljk1 jk 1 2
jk1
:
1
L1
0
B:9
Substituting Eq. (B.9) into Eq. (B.8), we get I0
j; k
j!k!
1
j k 1!
L1 jk1 :
B:10
270
INTEGRATION FORMULA FOR AREA COORDINATES
And substituting Eq. (B.10) into Eq. (B.7), we get j!k! Ie
i; j; k 2Se
j k 1!
1 0
L1 jk1 dL1 :
Li1
1
B:11
The remaining calculation is
I1
i; j k 1
1 0
Li1
1
L1 jk1 dL1
1 1 i1 jk1 L
1 L1 i1 1 0
1 jk1 Li1 L1 jk dL1 1
1 i1 0 0
jk 1 I
i 1; j k i1 1
j k 1
j k 2 1 I
i j k 1; 0
i 1
i 2
i j k 1 1
j k 1!i! I
i j k 1; 0:
i j k 1! 1
B:12
Substituting I1
i j k 1; 0, where I1
i j k 1; 0 can be rewitten as
I1
i j k 1; 0
1 0
Lijk1 1
1 dL1 Lijk2 ijk2 1
1 ; ijk2
1 0
B:13
into Eq. (B.12), we get I1
i; j k 1
j k 1!i! :
i j k 2!
B:14
INTEGRATION FORMULA FOR AREA COORDINATES
271
Then substituting this equation into Eq. (B.11), we ®nally get the formula Ie
i; j; k 2Se
i! j!k! :
i j k 2!
B:15