Periodic Structures

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

Preface ix

1 Introduction 1

1.1 Historical Perspective on the Research in Periodic Structures 1

1.2 From 1D Periodic Stratified Medium to 3D Photonic Crystals: An Overview of this Book 3
1.2.1 Chapter 2: Wave Propagation in Multiple Dielectric Layers 3
1.2.2 Chapter 3: One-Dimensional Periodic Medium 4
1.2.3 Chapter 4: Two- and Three-Dimensional Periodic Structures 6
1.2.4 Chapter 5: Introducing Defects into Periodic Structures 9
1.2.5 Chapter 6: Periodic Impedance Surface 11
1.2.6 Chapter 7: Exotic Dielectrics Made of Periodic Structures 13

References 14

Further Readings 15

2 Wave Propagation in Multiple Dielectric Layers 17

2.1 Plane-Wave Solutions in a Uniform Dielectric Medium 17

2.2 Transmission-Line Network Representation of a Dielectric Layer of Finite Thickness 21
2.2.1 Wave Propagating in Regular and Exotic Mediums 25

2.3 Scattering Characteristics of Plane Wave by Multiple Dielectric Layers 28
2.3.1 Recursive-Impedance Method 30
2.3.2 Transfer-Matrix Method 32
2.3.3 Scattering-Matrix Method 37

2.4 Transverse Resonance Technique for Determining the Guiding Characteristics of Waves in Multiple Dielectric Layers 45
2.4.1 Transverse Resonance Technique 45
2.4.2 Will Surface Waves be Supported in a Single Interface Environment? 47
2.4.3 Single Dielectric Layer Backed with a PEC or PMC 49
2.4.4 Mode Dispersion Relation of a Closed Structure Consisting of Dielectric Layers 53

Appendix: Dyadic Definition and Properties 61

References 62

Further Reading 63

3 One-Dimensional Periodic Medium 65

3.1 Bloch–Floquet Theorem 65

3.2 Eigenwave in a 1D Holographic Grating 66
3.2.1 Two Space-Harmonic Approximation 68
3.2.2 Single Interface between a Semi-infinite Uniform and a
1D Periodic Medium 76

3.3 Eigenwave in 1D Dielectric Gratings: Modal Transmission-Line Approach 81
3.3.1 In-Plane Incidence: ky = 0 88
3.3.2 Out-of-Plane Incidence: ky ≠ 0 89
3.3.3 Eigenwave in a Two-Tone Periodic Medium 94
3.3.4 Sturm–Liouville Differential Equation with Periodic Boundary Condition 96

3.4 Eigenwave in a 1D Metallic Periodic Medium 98
3.4.1 Generalized Scattering Matrix at the Interface between a
1D Metallic Periodic Medium and Uniform Medium 99

3.5 Hybrid-Mode Analysis of a 1D Dielectric Grating: Fourier-Modal Approach 102

3.6 Input–Output Relation of a 1D Periodic Medium of Finite Thickness 108

3.7 Scattering Characteristics of a Grating Consisting of Multiple 1D Periodic Layers 111
3.7.1 Building-Block Approach 111
3.7.2 Scattering Analysis of 1D Diffraction Gratings 112

3.8 Guiding Characteristics of Waveguides Consisting of Multiple 1D Periodic Layers 119
3.8.1 Transverse Resonance Technique 119
3.8.2 Dispersion Relation of a 1D Grating Waveguide 119

References 129

Further Readings 130

4 Two- and Three-Dimensional Periodic Structures 131

4.1 Modal Transmission-Line Approach for a 2D Periodic Metallic Medium: In-Plane Propagation 131
4.1.1 Generalized Scattering Matrix at the Interface between a 1D Periodic Metallic Medium and Uniform Medium 133
4.1.2 Periodic Boundary Condition on the Unit Cell along the y-axis 137
4.1.3 A Simple Graphical Method 138
4.1.4 Phase Relation: the Relationship Among KX, KY, and KO 138
4.1.5 Dispersion Relation: the Relationship Between KO and Kx (or ky) 143
4.1.6 Brillouin Zone and Band Structure 146

4.2 Modal Transmission Line Approach for a 2D Periodic Dielectric Medium: In-Plane Propagation 152
4.2.1 Input–Output Relation at the Interface: Generalized Scattering Matrix Representation 156
4.2.2 Brillouin Diagram and Phase Relation 158

4.3 Double Fourier-Modal Approach for a 2D Dielectric Periodic Structure: Out-of-Plane Propagation 166
4.3.1 Scattering Analysis of a 2D Grating: Out-of-Plane Propagation 171

4.4 Three-Dimensional Periodic Structures 172
4.4.1 Scattering Analysis of a 3D Periodic Structure 174
4.4.2 Eigenwave Analysis of a 3D Periodic Medium 180

Appendix: Closed-Form Solution of εpq,mn and μpq,mn 189

References 190

5 Introducing Defects into Periodic Structures 191

5.1 A Parallel-Plane Waveguide having a Pair of 1D Semi-Infinite Periodic Structures as its Side Walls 191
5.1.1 Bloch Impedance 192
5.1.2 Surface States Supported at the Interface of a Semi-Infinite 1D Periodic Structure 193
5.1.3 A Semi-Infinite 1D Periodic Structure Consisting of Symmetric Dielectric Waveguides 200

5.2 Dispersion Relation of a Parallel-Plane Waveguide with Semi-Infinite 1D Periodic Structures as Waveguide Side Walls 203
5.2.1 Numerical Example 204

5.3 A Parallel-Plane Waveguide with 2D Dielectric Periodic Structures as its Side Walls 208
5.3.1 Method of Mathematical Analysis 211
5.3.2 Dispersion Relation of a Channel with a Pair of 2D Periodic Structures as its Waveguide Side Walls 214

5.4 Scattering Characteristics of a Periodic Structure with Defects 223
5.4.1 Fabry–Perot Etalon 229
5.4.2 The Correlation between the Scattering and Guiding Characteristics 231

5.5 A Parallel-Plane Waveguide with 2D Metallic Periodic Structures as its Side Walls 236

5.6 Other Applications in Microwave Engineering 240

References 243

6 Periodic Impedance Surface 245

6.1 Scattering Characteristics of Plane Wave by a 1D Periodic Structure Consisting of a Cavities Array 246
6.1.1 An AMC Surface Made of Corrugated Metal Surface with Quarter-Wavelength Depth 256

6.2 Periodic Impedance Surface Approach (PISA) 264

6.3 Scattering of Plane Wave by 1D Periodic Impedance Surface: Non-Principal Plane Propagation 268
6.3.1 Guiding Characteristics of Waves Supported by a 1D Periodic Impedance Surface 277

6.4 Scattering of Plane Wave by a Dyadic 2D Periodic Impedance Surface 277

References 280

7 Exotic Dielectrics Made of Periodic Structures 283

7.1 Synthetic Dielectrics Using a 2D Dielectric Columns Array 283
7.1.1 Description of the Example 284
7.1.2 Phase-Relation Diagram of a Uniform Dielectric Medium 285

7.2 Refractive Index of a 2D Periodic Medium 287
7.2.1 Conclusion 291

7.3 An Artificial Dielectric Made of 1D Periodic Dielectric Layers 292
7.3.1 Effective Refractive Index of the 1D Dielectric Periodic Medium 293
7.3.2 Effective Wave-Impedance of the 1D Dielectric Periodic Medium 293

7.4 Conclusion 295

References 295

Index 297