Periodic Structures

Ruey-Bing (Raybeam) Hwang, National Chiao Tung University, Taiwan

• Preface ix1 Introduction 11.1 Historical Perspective on the Research in Periodic Structures 11.2 From 1D Periodic Stratified Medium to 3D Photonic Crystals: An Overview of this Book 31.2.1 Chapter 2: Wave Propagation in Multiple Dielectric Layers 31.2.2 Chapter 3: One-Dimensional Periodic Medium 41.2.3 Chapter 4: Two- and Three-Dimensional Periodic Structures 61.2.4 Chapter 5: Introducing Defects into Periodic Structures 91.2.5 Chapter 6: Periodic Impedance Surface 111.2.6 Chapter 7: Exotic Dielectrics Made of Periodic Structures 13References 14Further Readings 152 Wave Propagation in Multiple Dielectric Layers 172.1 Plane-Wave Solutions in a Uniform Dielectric Medium 172.2 Transmission-Line Network Representation of a Dielectric Layer of Finite Thickness 212.2.1 Wave Propagating in Regular and Exotic Mediums 252.3 Scattering Characteristics of Plane Wave by Multiple Dielectric Layers 282.3.1 Recursive-Impedance Method 302.3.2 Transfer-Matrix Method 322.3.3 Scattering-Matrix Method 372.4 Transverse Resonance Technique for Determining the Guiding Characteristics of Waves in Multiple Dielectric Layers 452.4.1 Transverse Resonance Technique 452.4.2 Will Surface Waves be Supported in a Single Interface Environment? 472.4.3 Single Dielectric Layer Backed with a PEC or PMC 492.4.4 Mode Dispersion Relation of a Closed Structure Consisting of Dielectric Layers 53Appendix: Dyadic Definition and Properties 61References 62Further Reading 633 One-Dimensional Periodic Medium 653.1 Bloch–Floquet Theorem 653.2 Eigenwave in a 1D Holographic Grating 663.2.1 Two Space-Harmonic Approximation 683.2.2 Single Interface between a Semi-infinite Uniform and a1D Periodic Medium 763.3 Eigenwave in 1D Dielectric Gratings: Modal Transmission-Line Approach 813.3.1 In-Plane Incidence: ky = 0 883.3.2 Out-of-Plane Incidence: ky ≠ 0 893.3.3 Eigenwave in a Two-Tone Periodic Medium 943.3.4 Sturm–Liouville Differential Equation with Periodic Boundary Condition 963.4 Eigenwave in a 1D Metallic Periodic Medium 983.4.1 Generalized Scattering Matrix at the Interface between a1D Metallic Periodic Medium and Uniform Medium 993.5 Hybrid-Mode Analysis of a 1D Dielectric Grating: Fourier-Modal Approach 1023.6 Input–Output Relation of a 1D Periodic Medium of Finite Thickness 1083.7 Scattering Characteristics of a Grating Consisting of Multiple 1D Periodic Layers 1113.7.1 Building-Block Approach 1113.7.2 Scattering Analysis of 1D Diffraction Gratings 1123.8 Guiding Characteristics of Waveguides Consisting of Multiple 1D Periodic Layers 1193.8.1 Transverse Resonance Technique 1193.8.2 Dispersion Relation of a 1D Grating Waveguide 119References 129Further Readings 1304 Two- and Three-Dimensional Periodic Structures 1314.1 Modal Transmission-Line Approach for a 2D Periodic Metallic Medium: In-Plane Propagation 1314.1.1 Generalized Scattering Matrix at the Interface between a 1D Periodic Metallic Medium and Uniform Medium 1334.1.2 Periodic Boundary Condition on the Unit Cell along the y-axis 1374.1.3 A Simple Graphical Method 1384.1.4 Phase Relation: the Relationship Among KX, KY, and KO 1384.1.5 Dispersion Relation: the Relationship Between KO and Kx (or ky) 1434.1.6 Brillouin Zone and Band Structure 1464.2 Modal Transmission Line Approach for a 2D Periodic Dielectric Medium: In-Plane Propagation 1524.2.1 Input–Output Relation at the Interface: Generalized Scattering Matrix Representation 1564.2.2 Brillouin Diagram and Phase Relation 1584.3 Double Fourier-Modal Approach for a 2D Dielectric Periodic Structure: Out-of-Plane Propagation 1664.3.1 Scattering Analysis of a 2D Grating: Out-of-Plane Propagation 1714.4 Three-Dimensional Periodic Structures 1724.4.1 Scattering Analysis of a 3D Periodic Structure 1744.4.2 Eigenwave Analysis of a 3D Periodic Medium 180Appendix: Closed-Form Solution of εpq,mn and μpq,mn 189References 1905 Introducing Defects into Periodic Structures 1915.1 A Parallel-Plane Waveguide having a Pair of 1D Semi-Infinite Periodic Structures as its Side Walls 1915.1.1 Bloch Impedance 1925.1.2 Surface States Supported at the Interface of a Semi-Infinite 1D Periodic Structure 1935.1.3 A Semi-Infinite 1D Periodic Structure Consisting of Symmetric Dielectric Waveguides 2005.2 Dispersion Relation of a Parallel-Plane Waveguide with Semi-Infinite 1D Periodic Structures as Waveguide Side Walls 2035.2.1 Numerical Example 2045.3 A Parallel-Plane Waveguide with 2D Dielectric Periodic Structures as its Side Walls 2085.3.1 Method of Mathematical Analysis 2115.3.2 Dispersion Relation of a Channel with a Pair of 2D Periodic Structures as its Waveguide Side Walls 2145.4 Scattering Characteristics of a Periodic Structure with Defects 2235.4.1 Fabry–Perot Etalon 2295.4.2 The Correlation between the Scattering and Guiding Characteristics 2315.5 A Parallel-Plane Waveguide with 2D Metallic Periodic Structures as its Side Walls 2365.6 Other Applications in Microwave Engineering 240References 2436 Periodic Impedance Surface 2456.1 Scattering Characteristics of Plane Wave by a 1D Periodic Structure Consisting of a Cavities Array 2466.1.1 An AMC Surface Made of Corrugated Metal Surface with Quarter-Wavelength Depth 2566.2 Periodic Impedance Surface Approach (PISA) 2646.3 Scattering of Plane Wave by 1D Periodic Impedance Surface: Non-Principal Plane Propagation 2686.3.1 Guiding Characteristics of Waves Supported by a 1D Periodic Impedance Surface 2776.4 Scattering of Plane Wave by a Dyadic 2D Periodic Impedance Surface 277References 2807 Exotic Dielectrics Made of Periodic Structures 2837.1 Synthetic Dielectrics Using a 2D Dielectric Columns Array 2837.1.1 Description of the Example 2847.1.2 Phase-Relation Diagram of a Uniform Dielectric Medium 2857.2 Refractive Index of a 2D Periodic Medium 2877.2.1 Conclusion 2917.3 An Artificial Dielectric Made of 1D Periodic Dielectric Layers 2927.3.1 Effective Refractive Index of the 1D Dielectric Periodic Medium 2937.3.2 Effective Wave-Impedance of the 1D Dielectric Periodic Medium 2937.4 Conclusion 295References 295Index 297