Beam Propagation Method for Design of Optical Waveguide Devices

The basic of the BPM technique in the frequency domain relies on treating the slowly varying envelope of the monochromatic electromagnetic field under paraxial propagation, thus allowing efficient numerical computation in terms of speed and allocated memory. In addition, the BPM based on finite differences is an easy way to implement robust and efficient computer codes. This book presents several approaches for treating the light: wide-angle, scalar approach, semivectorial treatment, and full vectorial treatment of the electromagnetic fields. Also, special topics in BPM cover the simulation of light propagation in anisotropic media, non-linear materials, electro-optic materials, and media with gain/losses, and describe how BPM can deal with strong index discontinuities or waveguide gratings, by introducing the bidirectional-BPM. BPM in the time domain is also described, and the book includes the powerful technique of finite difference time domain method, which fills the gap when the standard BPM is no longer applicable.  Once the description of these numerical techniques have been detailed, the last chapter includes examples of passive, active and functional integrated photonic devices, such as waveguide reflectors, demultiplexers, polarization converters, electro-optic modulators, lasers or frequency converters.  The book will help readers to understand several BPM approaches, to build their own codes, or to properly use the existing commercial software based on these numerical techniques.

Table of Contents:
Preface xii List of Acronyms xiv List of Symbols xvi 1 Electromagnetic Theory of Light 1 Introduction 1 1.1 Electromagnetic Waves 21.1.1 Maxwell’s Equations 2 1.1.2 Wave Equations in Inhomogeneous Media 5 1.1.3 Wave Equations in Homogeneous Media: Refractive Index 6 1.2 Monochromatic Waves 7 1.2.1 Homogeneous Media: Helmholtz’s Equation 9 1.2.2 Light Propagation in Absorbing Media 9 1.2.3 Light Propagation in Anisotropic Media 11 1.2.4 Light Propagation in Second-Order Non-Linear Media 13 1.3 Wave Equation Formulation in Terms of the Transverse Field Components 16 1.3.1 Electric Field Formulation 16 1.3.2 Magnetic Field Formulation 18 1.3.3 Wave Equation in Anisotropic Media 19 1.3.4 Second Order Non-Linear Media 20 References 21 2 The Beam-Propagation Method 22 Introduction 22 2.1 Paraxial Propagation: The Slowly Varying Envelope Approximation (SVEA).Full Vectorial BPM Equations 23 2.2 Semi-Vectorial and Scalar Beam Propagation Equations 29 2.2.1 Scalar Beam Propagation Equation 30 2.3 BPM Based on the Finite Difference Approach 31 2.4 FD-Two-Dimensional Scalar BPM 32 2.5 Von Neumann Analysis of FD-BPM 37 2.5.1 Stability 38 2.5.2 Numerical Dissipation 39 2.5.3 Numerical Dispersion 40 2.6 Boundary Conditions 44 2.6.1 Energy Conservation in the Difference Equations 45 2.6.2 Absorbing Boundary Conditions (ABCs) 47 2.6.3 Transparent Boundary Conditions (TBC) 49 2.6.4 Perfectly Matched Layers (PMLs) 51 2.7 Obtaining the Eigenmodes Using BPM 56 2.7.1 The Correlation Function Method 58 2.7.2 The Imaginary Distance Beam Propagation Method 64 References 68 3 Vectorial and Three-Dimensional Beam Propagation Techniques 71 Introduction 71 3.1 Two-Dimensional Vectorial Beam Propagation Method 72 3.1.1 Formulation Based on the Electric Field 72 3.1.2 Formulation Based on the Magnetic Field 81 3.2 Three-Dimensional BPM Based on the Electric Field 84 3.2.1 Semi-Vectorial Formulation 88 3.2.2 Scalar Approach 96 3.2.3 Full Vectorial BPM 102 3.3 Three-Dimensional BPM Based on the Magnetic Field 113 3.3.1 Semi-Vectorial Formulation 116 3.3.2 Full Vectorial BPM 120 References 129 4 Special Topics on BPM 130 Introduction 130 4.1 Wide-Angle Beam Propagation Method 130 4.1.1 Formalism of Wide-Angle-BPM Based on Padé Approximants 131 4.1.2 Multi-step Method Applied to Wide-Angle BPM 133 4.1.3 Numerical Implementation of Wide-Angle BPM 135 4.2 Treatment of Discontinuities in BPM 140 4.2.1 Reflection and Transmission at an Interface 140 4.2.2 Implementation Using First-Order Approximation to the Square Root 144 4.3 Bidirectional BPM 148 4.3.1 Formulation of Iterative Bi-BPM 148 4.3.2 Finite-Difference Approach of the Bi-BPM 151 4.3.3 Example of Bidirectional BPM: Index Modulation Waveguide Grating 154 4.4 Active Waveguides 157 4.4.1 Rate Equations in a Three-Level System 158 4.4.2 Optical Attenuation/Amplification 160 4.4.3 Channel Waveguide Optical Amplifier 161 4.5 Second-Order Non-Linear Beam Propagation Techniques 165 4.5.1 Paraxial Approximation of Second-Order Non-Linear Wave Equations 166 4.5.2 Second-Harmonic Generation in Waveguide Structures 169 4.6 BPM in Anisotropic Waveguides 173 4.6.1 TE TM Mode Conversion 175 4.7 Time Domain BPM 177 4.7.1 Time-Domain Beam Propagation Method (TD-BPM) 178 4.7.2 Narrow-Band 1D-TD-BPM 179 4.7.3 Wide-Band 1D-TD-BPM 180 4.7.4 Narrow-Band 2D-TD-BPM 187 4.8 Finite-Difference Time-Domain Method (FD-TD) 193 4.8.1 Finite-Difference Expressions for Maxwell’s Equations in Three Dimensions 194 4.8.2 Truncation of the Computational Domain 198 4.8.3 Two-Dimensional FDTD: TM Case 199 4.8.4 Setting the Field Source 208 4.8.5 Total-Field/Scattered-Field Formulation 209 4.8.6 Two-Dimensional FDTD: TE Case 212 References 219 5 BPM Analysis of Integrated Photonic Devices 222 Introduction 222 5.1 Curved Waveguides 222 5.2 Tapers: Y-Junctions 228 5.2.1 Taper as Mode-Size Converter 228 5.2.2 Y-Junction as 1 × 2 Power Splitter 230 5.3 Directional Couplers 231 5.3.1 Polarization Beam-Splitter 232 5.3.2 Wavelength Filter 235 5.4 Multimode Interference Devices 237 5.4.1 Multimode Interference Couplers 237 5.4.2 Multimode Interference and Self-Imaging 239 5.4.3 1×N Power Splitter Based on MMI Devices 243 5.4.4 Demultiplexer Based on MMI 244 5.5 Waveguide Gratings 248 5.5.1 Modal Conversion Using Corrugated Waveguide Grating 249 5.5.2 Injecting Light Using Relief Gratings 250 5.5.3 Waveguide Reflector Using Modulation Index Grating 252 5.6 Arrayed Waveguide Grating Demultiplexer 257 5.6.1 Description of the AWG Demultiplexer 257 5.6.2 Simulation of the AWG 263 5.7 Mach-Zehnder Interferometer as Intensity Modulator 270 5.8 TE-TM Converters 276 5.8.1 Electro-Optical TE-TM Converter 277 5.8.2 Rib Loaded Waveguide as Polarization Converter 280 5.9 Waveguide Laser 282 5.9.1 Simulation of Waveguide Lasers by Active-BPM 283 5.9.2 Performance of a Nd3+-Doped LiNbO3 Waveguide Laser 286 5.10 SHG Using QPM in Waveguides 293 References 297 Appendix A: Finite Difference Approximations of Derivatives 300 A.1 FD-Approximations of First-Order Derivatives 300 A.2 FD-Approximation of Second-Order Derivatives 301 Appendix B: Tridiagonal System: The Thomas Method Algorithm 304 Reference 306 Appendix C: Correlation and Relative Power between Optical Fields 307 C.1 Correlation between Two Optical Fields 307 C.2 Power Contribution of a Waveguide Mode 307 References 309 Appendix D: Poynting Vector Associated to an Electromagnetic Wave Using the SVE Fields 310 D.1 Poynting Vector in 2D-Structures 310 D.1.1 TE Propagation in Two-Dimensional Structures 310 D.1.2 TM Propagation in Two-Dimensional Structures 312 D.2 Poynting Vector in 3D-Structures 314 D.2.1 Expression as a Function of the Transverse Electric Field 315 D.2.2 Expression as Function of the Transverse Magnetic Field 319 Reference 322 Appendix E: Finite Difference FV-BPM Based on the Electric Field Using the Scheme Parameter Control 323 E.1 First Component of the First Step 325 E.2 Second Component of the First Step 326 E.3 Second Component of the Second Step 327 E.4 First Component of the Second Step 328 Appendix F: Linear Electro-Optic Effect 330 Reference 332 Appendix G: Electro-Optic Effect in GaAs Crystal 333 References 339 Appendix H: Electro-Optic Effect in LiNbO3 Crystal 340 References 345 Appendix I: Padé Polynomials for Wide-Band TD-BPM 346 Appendix J: Obtaining the Dispersion Relation for a Monomode Waveguide Using FDTD 349 Reference 350 Appendix K: Electric Field Distribution in Coplanar Electrodes 351 K.1 Symmetric Coplanar Strip Configuration 351 K.2 Symmetric Complementary Coplanar Strip Configuration 356 References 359 Appendix L: Three-Dimensional Anisotropic BPM Based on the Electric Field Formulation 360 L.1 Numerical Implementation 365 L.1.1 First Component of the First Step 365 L.1.2 Second Component of the First Step 366 L.1.3 Second Component of the Second Step 367 L.1.4 First Component of the Second Step 368 References 369 Appendix M: Rate Equations in a Four-Level Atomic System 370 References 372 Appendix N: Overlap Integrals Method 373 References 376 Index 377