Finite Element Modeling Methods for Photonics (Artech House Applied Photonics)

The term photonics can be used loosely to refer to a vast array of components, devices, and technologies that in some way involve manipulation of light. One of the most powerful numerical approaches available to engineers developing photonic components and devices is the Finite Element Method (FEM), which can be used to model and simulate such components/devices and analyze how they will behave in response to various outside influences. This resource provides a comprehensive description of the formulation and applications of FEM in photonics applications ranging from telecommunications, astronomy, and sensing, to chemistry, imaging, and biomedical R&D. This book emphasizes practical, problem-solving applications and includes real-world examples to assist readers in understanding how mathematical concepts translate to computer code for finite element-based methods applicable to a range of photonic structures. In addition, this is the perfect support to anyone using the COMSOL Multiphysics© RF Module. Photonics Library Finite Element Modeling Methods for Photonics 1 Contents 8 Preface 14 Acknowledgments 16 1 Introduction 18 1.1 Significance of Numerical Methods 18 1.2 Numerical Methods 19 1.3 Maxwell’s Equations and Boundary Conditions 19 1.3.1 Maxwell’s Equations 19 1.3.2 Boundary Conditions across Material Interfaces 21 1.3.3 Boundary Conditions: Natural and Forced 22 1.3.4 Boundary Conditions: Truncation of Domains 23 1.4 Basic Assumptions of Numerical Methods and Their Applicability 24 1.4.1 Time Harmonic and Time-Dependent Solutions 24 1.4.2 The Wave Equations 25 1.4.3 Scalar and Vector Nature of the Equations/Solutions 26 1.4.4 Modal Solutions 27 1.4.5 Beam Propagation Methods 28 1.5 Choosing a Modeling Method 31 1.6 Finite-Element-Based Methods 32 References 33 2 The Finite-Element Method 38 2.1 Basic Concept of FEM: Essence of FEM-based Formulations 38 2.2 Setting up the FEM 41 2.2.1 The Variational Approach 41 2.2.2 The Galerkin Method 44 2.3 Scalar and Vector FEM Formulations 46 2.3.1 The Scalar Formulation 46 2.3.2 The Vector Formulation 48 2.4 Implementation of FEM 52 2.4.1 Flowchart of Main Steps in FEM 52 2.4.2 Meshing and Shape Functions 52 2.4.3 Shape Functions 57 2.4.4 Examples of Meshing 58 2.5 Formation of Element and Global Matrices 72 2.5.1 Mass and Stiffness Matrix Evaluation for First-order Triangular Elements 75 2.5.2 Mass and Stiffness Matrix Evaluation for Second-order Triangular Elements 77 2.5.3 Assembly of Global Matrices: Bandwidth and Sparsity of Matrices 79 2.5.4 Penalty Function Method for Elimination of Spurious Modes 81 2.6 Solution of the System of Equations 82 2.7 Implementation of Boundary Conditions 84 2.7.1 Natural Boundary Condition and Symmetry: Electric and Magnetic Wall 84 2.7.2 Absorbing Boundary Condition and Perfectly Matched Layer (PML) Boundary Condition 86 2.7.3 Periodic Boundary Conditions (PBC) 92 2.8 Practical Illustrations of FEM Applied to Photonic Structures/devices 95 2.8.1 The Rectangular Waveguide: Si Nanowire 95 2.8.2 Waveguide with a Circular Cross Section: Photonic Crystal Fiber (PCF) 101 2.8.3 Plasmonic Waveguides 105 2.8.4 Photonic Crystal Waveguide and Periodic Boundary Conditions 109 2.9 FEM Analysis of Bent Waveguides 110 2.10 Perturbation Analysis for Loss/Gain in Optical Waveguides 114 2.10.1 Perturbation Method with the Scalar FEM 116 2.10.2 Perturbation Method with the Vector FEM 118 2.11 Accuracy and Convergence in FEM 120 2.11.1 Discretisation and Interpolation Errors in FEM Analysis 120 2.11.2 Element Shape Quality and the Stiffness Matrix 121 2.11.3 Error Dependence on Element Size, Order and Arrangement 121 2.11.4 Adaptive Mesh Refinement 125 2.12 Computer Systems and FEM Implementation 125 References 127 3 Finite-Element Beam Propagation Methods 136 3.1 Introduction 136 3.2 Setting up BPM Methods 139 3.3 Vector FE-BPM with PML Boundary Conditions 139 3.3.1 Semi-Vector and Scalar FE-BPM 149 3.3.2 Wide-angle FE-BPM 150 3.3.3 Paraxial FE-BPM 150 3.3.4 Implementation of the BPM and Stability 152 3.3.5 Practical Illustrations of FE-BPM applied to Photonic Structures/Devices 154 3.4 Junction Analysis with FEM: The LSBR Method 165 3.4.1 Analysis of High Index Contrast Bent Waveguide 167 3.5 Bi-directional BPM 172 3.6 Imaginary Axis/distance BPM 175 3.6.1 Analysis of 3D Leaky Waveguide by the Imaginary Axis BPM 177 References 179 4 Finite-Element Time Domain Method 184 4.1 Time Domain Numerical Methods 184 4.2 Finite-Element Time Domain (FETD) BPM Method 186 4.2.1 Wide Band and Narrow Band Approximations 189 4.2.2 Implementation of the FETD BPM Method: Implicit and Explicit Schemes 189 4.3 Practical Illustrations of FETD BPM Applied to Photonic Structures/Devices 192 4.3.1 Optical Grating 192 4.3.2 90° Sharp Bends 193 References 196 5 Incorporating Physical Effects within the Finite-Element Method 200 5.1 Introduction 200 5.2 The Thermal Model 201 5.2.1 Thermal Modeling of a VCSEL 204 5.3 The Stress Model 207 5.3.1 Stress Analysis of a Polarization Maintaining Bow-tie Fiber 207 5.4 The Acoustic Model 211 5.4.1 Acousto-optic Analysis of a Silica Waveguide 212 5.4.2 SBS Analysis of a Silica Nanowire 213 5.5 The Electro-optic Model 215 5.5.1 Analysis of a Lithium Niobate (LN) Electro-optic Modulator 217 5.6 Nonlinear Photonic Devices 221 5.6.1 Analysis of a Strip-loaded Nonlinear Waveguide 222 5.6.2 Analysis of a Nonlinear Directional Coupler 224 5.6.3 Analysis of Second Harmonic Generation in an Optical Waveguide 228 References 235 6 FE-based Methods: The Present and Future Directions 238 6.1 Introduction 238 6.2 Salient Features of FE-based Methods 239 6.3 Future Trends and Challenges for FE-based Methods 240 Appendix A Scalar FEM with Perturbation 244 TE Modes 244 TM Modes 246 Appendix B Vector FEM with Perturbation 248 Appendix C Green’s Theorem 254 About the Authors 256 Index 258 Element,modeling;,Photonics;,Artech,House;,978-1-60807-531-7 Element modeling,Photonics,Artech House,978-1-60807-531-7 Machine generated contents note: 1.Introduction 1.1.Significance of Numerical Methods 1.2.Numerical Methods 1.3.Maxwell's Equations and Boundary Conditions 1.3.1.Maxwell's Equations 1.3.2.Boundary Conditions across Material Interfaces 1.3.3.Boundary Conditions: Natural and Forced 1.3.4.Boundary Conditions: Truncation of Domains 1.4.Basic Assumptions of Numerical Methods and Their Applicability 1.4.1.Time Harmonic and Time-Dependent Solutions 1.4.2.The Wave Equations 1.4.3.Scalar and Vector Nature of the Equations/Solutions 1.4.4.Modal Solutions 1.4.5.Beam Propagation Methods 1.5.Choosing a Modeling Method 1.6.Finite-Element-Based Methods References 2.The Finite-Element Method 2.1.Basic Concept of FEM: Essence of FEM-based Formulations 2.2.Setting up the FEM 2.2.1.The Variational Approach 2.2.2.The Galerkin Method 2.3.Scalar and Vector FEM Formulations 2.3.1.The Scalar Formulation 2.3.2.The Vector Formulation 2.4.Implementation of FEM 2.4.1.Flowchart of Main Steps in FEM 2.4.2.Meshing and Shape Functions 2.4.3.Shape Functions 2.4.4.Examples of Meshing 2.5.Formation of Element and Global Matrices 2.5.1.Mass and Stiffness Matrix Evaluation for First-order Triangular Elements 2.5.2.Mass and Stiffness Matrix Evaluation for Second-order Triangular Elements 2.5.3.Assembly of Global Matrices: Bandwidth and Sparsity of Matrices 2.5.4.Penalty Function Method for Elimination of Spurious Modes 2.6.Solution of the System of Equations 2.7.Implementation of Boundary Conditions 2.7.1.Natural Boundary Condition and Symmetry: Electric and Magnetic Wall 2.7.2.Absorbing Boundary Condition and Perfectly Matched Layer (PML) Boundary Condition 2.7.3.Periodic Boundary Conditions (PBC) 2.8.Practical Illustrations of FEM Applied to Photonic Structures/devices 2.8.1.The Rectangular Waveguide: Si Nanowire 2.8.2.Waveguide with a Circular Cross Section: Photonic Crystal Fiber (PCF) 2.8.3.Plasmonic Waveguides 2.8.4.Photonic Crystal Waveguide and Periodic Boundary Conditions 2.9.FEM Analysis of Bent Waveguides 2.10.Perturbation Analysis for Loss/gain in Optical Waveguides 2.10.1.Perturbation Method with the Scalar FEM 2.10.2.Perturbation Method with the Vector FEM 2.11.Accuracy and Convergence in FEM 2.11.1.Discretisation and Interpolation Errors in FEM Analysis 2.11.2.Element Shape Quality and the Stiffness Matrix 2.11.3.Error Dependence on Element Size, Order and Arrangement 2.11.4.Adaptive Mesh Refinement 2.12.Computer Systems and FEM Implementation References 3.Finite-Element Beam Propagation Methods 3.1.Introduction 3.2.Setting up BPM Methods 3.3.Vector FE-BPM with PML Boundary Conditions 3.3.1.Semi-vector and Scalar FE-BPM 3.3.2.Wide-angle FE-BPM 3.3.3.Paraxial FE-BPM 3.3.4.Implementation of the BPM and Stability 3.3.5.Practical Illustrations of FE-BPM applied to Photonic Structures/devices 3.4.Junction Analysis with FEM: The LSBR Method 3.4.1.Analysis of High Index Contrast Bent Waveguide 3.5.Bi-directional BPM 3.6.Imaginary Axis/distance BPM 3.6.1.Analysis of 3D Leaky Waveguide by the Imaginary Axis BPM References 4.Finite-Element Time Domain Method 4.1.Time Domain Numerical Methods 4.2.Finite-Element Time Domain (FETD) BPM Method 4.2.1.Wide Band and Narrow Band Approximations 4.2.2.Implementation of the FETD BPM Method: Implicit and Explicit Schemes 4.3.Practical Illustrations of FETD BPM Applied to Photonic Structures/devices 4.3.1.Optical Grating 4.3.2.90° Sharp Bends References 5.Incorporating Physical Effects within the Finite-Element Method 5.1.Introduction 5.2.The Thermal Model 5.2.1.Thermal Modeling of a VCSEL 5.3.The Stress Model 5.3.1.Stress Analysis of a Polarization Maintaining Bow-tie Fiber 5.4.The Acoustic Model 5.4.1.Acousto-optic Analysis of a Silica Waveguide 5.4.2.SBS Analysis of a Silica Nanowire 5.5.The Electro-optic Model 5.5.1.Analysis of a Lithium Niobate (LN) Electro-optic Modulator 5.6.Nonlinear Photonic Devices 5.6.1.Analysis of a Strip-loaded Nonlinear Waveguide 5.6.2.Analysis of a Nonlinear Directional Coupler 5.6.3.Analysis of Second Harmonic Generation in an Optical Waveguide References 6.FE-based Methods: The Present and Future Directions 6.1.Introduction 6.2.Salient Features of FE-based Methods 6.3.Future Trends and Challenges for FE-based Methods Appendix A Scalar FEM with Perturbation TE Modes TM Modes Appendix B Vector FEM with Perturbation Appendix C Green's Theorem. The term photonics can be used loosely to refer to a vast array of components, devices, and technologies that in some way involve manipulation of light. One of the most powerful numerical approaches available to engineers developing photonic components and devices is the finite element method (FEM), which can be used to model and simulate such components/devices and analyze how they will behave in response to various outside influences. This resource provides a comprehensive description of the formulation and applications of FEM in photonics devices that can be used in a wide range of applications. This book emphasizes practical, problem-solving applications and includes real-world examples to assist readers in understanding how mathematical concepts translate to computer code for finite-element-based methods applicable to a range of photonic structures. In addition, this is the perfect support to anyone using the COMSOL Multiphysics© RF Module. Contents Overview: Numerical Methods, Finite Element Methods, Beam Propagation Methods, Time Domain Methods, Physical Effects within the Finite Element Method, Present and Future Directions Methods Book jacket