Periodic Materials and Interference Lithography for Photonics, Phononics and Mechanics

Periodic Materials and Interference Lithography......Page 5
Contents......Page 7
Preface......Page 13
Introduction......Page 15
Theory......Page 21
1 Structural Periodicity......Page 23
1.1 Nonperiodic versus Periodic Structures......Page 24
1.2 Two-dimensional Point Lattices......Page 26
1.3 Three-dimensional Point Lattices......Page 30
1.3.1 Primitive and Nonprimitive Unit Cells......Page 34
1.4 Mathematical Description of Periodic Structures......Page 36
1.5.1 Fourier Series for Two-dimensional Periodic Functions......Page 40
1.5.2 Fourier Series for Three-dimensional Periodic Functions......Page 43
1.5.3 Arbitrary Unit Cells......Page 45
Problems......Page 46
2 Periodic Functions and Structures......Page 49
2.1 Introduction......Page 50
2.2.1 The Square Lattice......Page 51
2.2.2 The Triangular Lattice......Page 58
2.3 Creating Simple Periodic Functions in Three Dimensions......Page 61
2.3.1 The Simple Cubic Lattice......Page 64
2.3.2 The Face-centered-cubic Lattice......Page 67
2.3.3 The Body-centered-cubic Lattice......Page 71
2.4 Combination of Simple Periodic Functions......Page 79
Problems......Page 81
3 Interference of Waves and Interference Lithography......Page 83
3.1 Electromagnetic Waves......Page 84
3.2 The Wave Equation......Page 85
3.3 Electromagnetic Plane Waves......Page 88
3.4 The Transverse Character of Electromagnetic Plane Waves......Page 89
3.5 Polarization......Page 92
3.5.1 Linearly Polarized Electromagnetic Plane Waves......Page 93
3.5.2 Circularly Polarized Electromagnetic Plane Waves......Page 94
3.6 Electromagnetic Energy......Page 95
3.6.2 Time-averaged Values......Page 97
3.6.3 Intensity......Page 100
3.7 Interference of Electromagnetic Plane Waves......Page 101
3.7.1 Three-dimensional Interference Patterns......Page 106
3.8.1 Photoresist Materials......Page 109
3.8.2 The Interference Lithography Technique......Page 112
3.8.3 Designing Periodic Structures......Page 113
Problems......Page 114
4 Periodic Structures and Interference Lithography......Page 117
4.1 The Connection between the Interference of Plane Waves and Fourier Series......Page 118
4.2 Simple Periodic Structures in Two Dimensions Via Interference Lithography......Page 120
4.3 Simple Periodic Structures in Three Dimensions Via Interference Lithography......Page 124
Further Reading......Page 130
Problems......Page 131
Experimental......Page 133
5 Fabrication of Periodic Structures......Page 135
5.2 Light Beams......Page 136
5.3 Multiple Gratings and the Registration Challenge......Page 138
5.4.1 Using Four Beams......Page 139
5.4.2 Using a Single Beam (Phase Mask Lithography)......Page 140
5.5 Pattern Transfer: Material Platforms and Photoresists......Page 142
5.5.1 Negative Photoresists......Page 144
5.5.2 Positive Photoresists......Page 146
5.6.1 Preserving Polarizations and Directions......Page 148
5.6.2 Contrast......Page 151
5.6.3 Drying......Page 152
5.6.5 Backfilling – Creating Inverse Periodic Structures......Page 153
5.6.6 Volume Fraction Control......Page 154
5.7 Closing Remarks......Page 155
Further Reading......Page 156
Applications......Page 159
6 Photonic Crystals......Page 161
6.1 Introduction......Page 162
6.2.1 Finite Periodic Structures......Page 163
6.2.2 Infinite Periodic Structures......Page 167
6.2.3 Finite versus Infinite Periodic Structures......Page 170
6.3 Two-dimensional Photonic Crystals......Page 171
6.3.1 Reciprocal Lattices and Brillouin Zones in Two Dimensions......Page 172
6.3.2 Band Diagrams and Photonic Band Gaps in Two Dimensions......Page 177
6.3.3 Photonic Band Gaps in Two-dimensional Simple Periodic Structures......Page 180
6.4 Three-dimensional Photonic Crystals......Page 182
6.4.1 Reciprocal Lattices and Brillouin Zones in Three Dimensions......Page 184
6.4.2 Band Diagrams and Photonic Band Gaps in Three Dimensions......Page 188
6.4.3 Photonic Band Gaps in Three-dimensional Simple Periodic Structures......Page 190
Further Reading......Page 196
Problems......Page 199
7 Phononic Crystals......Page 203
7.1.1 Elastic Waves in Homogeneous Solid Materials......Page 204
7.1.2 Acoustic Waves in Homogeneous Fluid Materials......Page 207
7.2 Phononic Crystals......Page 208
7.3.1 Finite Periodic Structures......Page 210
7.3.2 Infinite Periodic Structures......Page 214
7.4.1 Vacuum Cylinders in a Solid Background......Page 218
7.4.2 Solid Cylinders in Air......Page 222
7.4.3 Phononic Band Gaps in Two-dimensional Simple Periodic Structures......Page 225
7.5 Three-dimensional Phononic Crystals......Page 227
7.5.1 Solid Spheres in a Solid Background Material......Page 228
Further Reading......Page 230
Problems......Page 233
8 Periodic Cellular Solids......Page 235
8.1 Introduction......Page 236
8.2 One-dimensional Hooke’s Law......Page 238
8.3 The Stress Tensor......Page 239
8.4 The Strain Tensor......Page 241
8.4.1 Expansion......Page 245
8.4.2 General Deformation......Page 246
8.4.3 Resolving a General Deformation as Strain Plus Rotation......Page 247
8.5 Stress–Strain Relationship: The Generalized Hooke’s Law......Page 249
8.6 The Generalized Hooke’s Law in Matrix Notation......Page 250
8.7 The Elastic Constants of Cubic Crystals......Page 252
8.7.1 Young’s Modulus and Poisson’s Ratio......Page 253
8.7.2 The Shear Modulus......Page 255
8.7.3 The Bulk Modulus......Page 257
8.8 Topological Design of Periodic Cellular Solids......Page 258
8.10 Linear Elastic Mechanical Properties of Periodic Cellular Solids......Page 263
8.11 Twelve-connected Stretch-dominated Periodic Cellular Solids via Interference Lithography......Page 267
8.12 Fabrication of a Simple Cubic Cellular Solid via Interference Lithography......Page 269
8.13 Plastic Deformation of Microframes......Page 270
Further Reading......Page 272
9 Further Applications......Page 275
9.1 Controlling the Spontaneous Emission of Light......Page 276
9.2 Localization of Light: Microcavities and Waveguides......Page 279
9.3 Simultaneous Localization of Light and Sound in Photonic–Phononic Crystals: Novel Acoustic–Optical Devices......Page 284
9.4 Negative Refraction and Superlenses......Page 288
9.5 Multifunctional Periodic Structures: Maximum Transport of Heat and Electricity......Page 292
9.6 Microfluidics......Page 293
9.7.1 Peltier Effect......Page 295
9.7.2 Thomson Effect......Page 296
9.7.3 Seebeck Effect......Page 297
Further Reading......Page 298
Appendix A MATLAB Program to Calculate the Optimal Electric Field Amplitude Vectors for the Interfering Light Beams......Page 301
Appendix B MATLAB Program to Calculate Reflectance versus Frequency for One-dimensional Photonic Crystals......Page 309
Appendix C MATLAB Program to Calculate Reflectance versus Frequency for One-dimensional Phononic Crystals......Page 317
Index......Page 325