Introduction to Optical Waveguide Analysis: Solving Maxwell's Equation and the Schrödinger Equation

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

3 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 Fade 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 233

6.1 Discretization of Electromagnetic Fields 233

6.2 Stability Condition 239

6.3 Absorbing Boundary Conditions 241

Problems 245

References 249

7 Schrodinger 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