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E-grāmata: Finite Element Modeling Methods for Photonics

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  • Formāts: 268 pages
  • Izdošanas datums: 31-Jan-2013
  • Izdevniecība: Artech House Publishers
  • ISBN-13: 9781608075324
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Finite Element Modeling Methods for PhotonicsPalielināt
  • Formāts: 268 pages
  • Izdošanas datums: 31-Jan-2013
  • Izdevniecība: Artech House Publishers
  • ISBN-13: 9781608075324
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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 MultiphysicsA(c) RF Module.
Preface xiii
Acknowledgments xv
1 Introduction
1(20)
1.1 Significance of Numerical Methods
1(1)
1.2 Numerical Methods
2(1)
1.3 Maxwell's Equations and Boundary Conditions
2(5)
1.3.1 Maxwell's Equations
2(2)
1.3.2 Boundary Conditions across Material Interfaces
4(1)
1.3.3 Boundary Conditions: Natural and Forced
5(1)
1.3.4 Boundary Conditions: Truncation of Domains
6(1)
1.4 Basic Assumptions of Numerical Methods and Their Applicability
7(7)
1.4.1 Time Harmonic and Time-Dependent Solutions
7(1)
1.4.2 The Wave Equations
8(1)
1.4.3 Scalar and Vector Nature of the Equations/Solutions
9(1)
1.4.4 Modal Solutions
10(1)
1.4.5 Beam Propagation Methods
11(3)
1.5 Choosing a Modeling Method
14(1)
1.6 Finite-Element-Based Methods
15(6)
References
16(5)
2 The Finite-Element Method
21(98)
2.1 Basic Concept of FEM: Essence of FEM-based Formulations
21(3)
2.2 Setting up the FEM
24(5)
2.2.1 The Variational Approach
24(3)
2.2.2 The Galerkin Method
27(2)
2.3 Scalar and Vector FEM Formulations
29(6)
2.3.1 The Scalar Formulation
29(2)
2.3.2 The Vector Formulation
31(4)
2.4 Implementation of FEM
35(20)
2.4.1 Flowchart of Main Steps in FEM
35(1)
2.4.2 Meshing and Shape Functions
35(5)
2.4.3 Shape Functions
40(1)
2.4.4 Examples of Meshing
41(14)
2.5 Formation of Element and Global Matrices
55(10)
2.5.1 Mass and Stiffness Matrix Evaluation for First-order Triangular Elements
58(2)
2.5.2 Mass and Stiffness Matrix Evaluation for Second-order Triangular Elements
60(2)
2.5.3 Assembly of Global Matrices: Bandwidth and Sparsity of Matrices
62(2)
2.5.4 Penalty Function Method for Elimination of Spurious Modes
64(1)
2.6 Solution of the System of Equations
65(2)
2.7 Implementation of Boundary Conditions
67(11)
2.7.1 Natural Boundary Condition and Symmetry: Electric and Magnetic Wall
67(2)
2.7.2 Absorbing Boundary Condition and Perfectly Matched Layer (PML) Boundary Condition
69(6)
2.7.3 Periodic Boundary Conditions (PBC)
75(3)
2.8 Practical Illustrations of FEM Applied to Photonic Structures/devices
78(15)
2.8.1 The Rectangular Waveguide: Si Nanowire
78(6)
2.8.2 Waveguide with a Circular Cross Section: Photonic Crystal Fiber (PCF)
84(4)
2.8.3 Plasmonic Waveguides
88(4)
2.8.4 Photonic Crystal Waveguide and Periodic Boundary Conditions
92(1)
2.9 FEM Analysis of Bent Waveguides
93(4)
2.10 Perturbation Analysis for Loss/gain in Optical Waveguides
97(6)
2.10.1 Perturbation Method with the Scalar FEM
99(2)
2.10.2 Perturbation Method with the Vector FEM
101(2)
2.11 Accuracy and Convergence in FEM
103(5)
2.11.1 Discretisation and Interpolation Errors in FEM Analysis
103(1)
2.11.2 Element Shape Quality and the Stiffness Matrix
104(1)
2.11.3 Error Dependence on Element Size, Order and Arrangement
104(4)
2.11.4 Adaptive Mesh Refinement
108(1)
2.12 Computer Systems and FEM Implementation
108(11)
References
110(9)
3 Finite-Element Beam Propagation Methods
119(48)
3.1 Introduction
119(3)
3.2 Setting up BPM Methods
122(1)
3.3 Vector FE-BPM with PML Boundary Conditions
122(26)
3.3.1 Semi-vector and Scalar FE-BPM
132(1)
3.3.2 Wide-angle FE-BPM
133(1)
3.3.3 Paraxial FE-BPM
133(2)
3.3.4 Implementation of the BPM and Stability
135(2)
3.3.5 Practical Illustrations of FE-BPM applied to Photonic Structures/devices
137(11)
3.4 Junction Analysis with FEM: The LSBR Method
148(7)
3.4.1 Analysis of High Index Contrast Bent Waveguide
150(5)
3.5 Bi-directional BPM
155(3)
3.6 Imaginary Axis/distance BPM
158(9)
3.6.1 Analysis of 3D Leaky Waveguide by the Imaginary Axis BPM
160(2)
References
162(5)
4 Finite-Element Time Domain Method
167(16)
4.1 Time Domain Numerical Methods
167(2)
4.2 Finite-Element Time Domain (FETD) BPM Method
169(6)
4.2.1 Wide Band and Narrow Band Approximations
172(1)
4.2.2 Implementation of the FETD BPM Method: Implicit and Explicit Schemes
172(3)
4.3 Practical Illustrations of FETD BPM Applied to Photonic Structures/devices
175(8)
4.3.1 Optical Grating
175(1)
4.3.2 90° Sharp Bends
176(3)
References
179(4)
5 Incorporating Physical Effects within the Finite-Element Method
183(38)
5.1 Introduction
183(1)
5.2 The Thermal Model
184(6)
5.2.1 Thermal Modeling of a VCSEL
187(3)
5.3 The Stress Model
190(4)
5.3.1 Stress Analysis of a Polarization Maintaining Bow-tie Fiber
190(4)
5.4 The Acoustic Model
194(4)
5.4.1 Acousto-optic Analysis of a Silica Waveguide
195(1)
5.4.2 SBS Analysis of a Silica Nanowire
196(2)
5.5 The Electro-optic Model
198(6)
5.5.1 Analysis of a Lithium Niobate (LN) Electro-optic Modulator
200(4)
5.6 Nonlinear Photonic Devices
204(17)
5.6.1 Analysis of a Strip-loaded Nonlinear Waveguide
205(2)
5.6.2 Analysis of a Nonlinear Directional Coupler
207(4)
5.6.3 Analysis of Second Harmonic Generation in an Optical Waveguide
211(7)
References
218(3)
6 FE-based Methods: The Present and Future Directions
221(6)
6.1 Introduction
221(1)
6.2 Salient Features of FE-based Methods
222(1)
6.3 Future Trends and Challenges for FE-based Methods
223(4)
Appendix A Scalar FEM with Perturbation
227(4)
TE Modes
227(2)
TM Modes
229(2)
Appendix B Vector FEM with Perturbation
231(6)
Appendix C Green's Theorem
237(2)
About the Authors 239(2)
Index 241
B. M. Azizur Rahman is a professor of photonics in the Department of Electrical, Electronic and Information Engineering at City University London. He earned his PhD in Electronics at University College London. Arti Agrawal is a lecturer of photonics in the department of electrical, electronic and information engineering at City University London. She earned her PhD in Physics from Indian Institute of Technology, Delhi.
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