Periodic Materials and Interference

Written by the department head of materials science and engineering at MIT, this concise and stringent introduction takes readers from the fundamental theory to in-depth knowledge.It sets out with a theoretical scheme for the design of desirable periodic structures, then presents the experimental techniques that allow for fabrication of the periodic structure and exemplary experimental data. Subsequently, theory and numerical data are used to demonstrate how these periodic structures control the photonic, acoustic, and mechanical properties of materials, concluding with examples from these three important fields of applications.The result is must-have knowledge for both beginners and veterans in the field.

Martin Maldovan is currently a Post Doctoral Associate in the Department of Materials Science and Engineering at the Massachusetts Institute of Technology in Cambridge, Massachusetts, USA. He received his B.S. in Physics from the University of Buenos Aires, Argentina , and his M.S. and Ph.D. in Materials Science from the Massachusetts Institute of Technology. Dr. Martin Maldovan has authored numerous scientific publications in the fields of photonics, phononics and mechanics and obtained the 2006 Scientific Writing Award to Professionals in Acoustics from the Acoustical Society of America.

Edwin L. Thomas is currently the Department Head of Materials Science and Engineering at the Massachusetts Institute of Technology in Cambridge, Massachusetts, USA. He received his B.S. in Mechanical Engineering and Engineering Science from the University of Massachusetts and his Ph.D. in Materials Science from Cornell University. Professor Thomas has authored over 300 scientific publications and has received numerous scientific awards, including the Special Creativity Award of the National Science Foundation (1996 and 1988), the High Polymer Physics Prize of the American Physical Society (1991), and the American Chemical Society Creative Polymer Chemist Award (1985).

Preface	p. XI
Introduction	p. XIII
Theory	p. 1
Structural Periodicity	p. 3
Nonperiodic versus Periodic Structures	p. 4
Two-dimensional Point Lattices	p. 6
Three-dimensional Point Lattices	p. 10
Primitive and Nonprimitive Unit Cells	p. 14
Mathematical Description of Periodic Structures	p. 16
Fourier Series	p. 20
Fourier Series for Two-dimensional Periodic Functions	p. 20
Fourier Series for Three-dimensional Periodic Functions	p. 23
Arbitrary Unit Cells	p. 25
Further Reading	p. 26
Problems	p. 26
Periodic Functions and Structures	p. 29
Introduction	p. 30
Creating Simple Periodic Functions in Two Dimensions	p. 31
The Square Lattice	p. 31
The Triangular Lattice	p. 38
Creating Simple Periodic Functions in Three Dimensions	p. 41
The Simple Cubic Lattice	p. 44
The Face-centered-cubic Lattice	p. 47
The Body-centered-cubic Lattice	p. 51
Combination of Simple Periodic Functions	p. 59
Problems	p. 61
Interference of Waves and Interference Lithography	p. 63
Electromagnetic Waves	p. 64
The Wave Equation	p. 65
Electromagnetic Plane Waves	p. 68
The Transverse Character of Electromagnetic Plane Waves	p. 69
Polarization	p. 72
Linearly Polarized Electromagnetic Plane Waves	p. 73
Circularly Polarized Electromagnetic Plane Waves	p. 74
Elliptically Polarized Electromagnetic Plane Waves	p. 75
Electromagnetic Energy	p. 75
Energy Density and Energy Flux for Electromagnetic Plane Waves	p. 77
Time-averaged Values	p. 77
Intensity	p. 80
Interference of Electromagnetic Plane Waves	p. 81
Three-dimensional Interference Patterns	p. 86
Interference Lithography	p. 89
Photoresist Materials	p. 89
The Interference Lithography Technique	p. 92
Designing Periodic Structures	p. 93
Further Reading	p. 94
Problems	p. 94
Periodic Structures and Interference Lithography	p. 97
The Connection between the Interference of Plane Waves and Fourier Series	p. 98
Simple Periodic Structures in Two Dimensions Via Interference Lithography	p. 100
Simple Periodic Structures in Three Dimensions Via Interference Lithography	p. 104
Further Reading	p. 110
Problems	p. 111
Experimental	p. 113
Fabrication of Periodic Structures	p. 115
Introduction	p. 116
Light Beams	p. 116
Multiple Gratings and the Registration Challenge	p. 118
Beam Configuration	p. 119
Using Four Beams	p. 119
Using a Single Beam (Phase Mask Lithography)	p. 120
Pattern Transfer: Material Platforms and Photoresists	p. 122
Negative Photoresists	p. 124
Positive Photoresists	p. 126
Organic-Inorganic Hybrids Resists	p. 128
Practical Considerations for Interference Lithography	p. 128
Preserving Polarizations and Directions	p. 128
Contrast	p. 131
Drying	p. 132
Shrinkage	p. 133
Backfilling - Creating Inverse Periodic Structures	p. 133
Volume Fraction Control	p. 134
Closing Remarks	p. 135
Further Reading	p. 136
Applications	p. 139
Photonic Crystals	p. 141
Introduction	p. 142
One-dimensional Photonic Crystals	p. 143
Finite Periodic Structures	p. 143
Infinite Periodic Structures	p. 147
Finite versus Infinite Periodic Structures	p. 150
Two-dimensional Photonic Crystals	p. 151
Reciprocal Lattices and Brillouin Zones in Two Dimensions	p. 152
Band Diagrams and Photonic Band Gaps in Two Dimensions	p. 157
Photonic Band Gaps in Two-dimensional Simple Periodic Structures	p. 160
Three-dimensional Photonic Crystals	p. 162
Reciprocal Lattices and Brillouin Zones in Three Dimensions	p. 164
Band Diagrams and Photonic Band Gaps in Three Dimensions	p. 168
Photonic Band Gaps in Three-dimensional Simple Periodic Structures	p. 170
Further Reading	p. 176
Problems	p. 179
Photonic Crystals	p. 183
Introduction	p. 184
Elastic Waves in Homogeneous Solid Materials	p. 184
Acoustic Waves in Homogeneous Fluid Materials	p. 187
Phononic Crystals	p. 188
One-dimensional Phononic Crystals	p. 190
Finite Periodic Structures	p. 190
Infinite Periodic Structures	p. 194
Two-dimensional Phononic Crystals	p. 198
Vacuum Cylinders in a Solid Background	p. 198
Solid Cylinders in Air	p. 202
Phononic Band Gaps in Two-dimensional Simple Periodic Structures	p. 205
Three-dimensional Phononic Crystals	p. 207
Solid Spheres in a Solid Background Material	p. 208
Further Reading	p. 210
Problems	p. 213
Periodic Cellular Solids	p. 215
Introduction	p. 216
One-dimensional Hooke's Law	p. 218
The Stress Tensor	p. 219
The Strain Tensor	p. 221
Expansion	p. 225
General Deformation	p. 226
Resolving a General Deformation as Strain Plus Rotation	p. 227
Stress-Strain Relationship: The Generalized Hooke's Law	p. 229
The Generalized Hooke's Law in Matrix Notation	p. 230
The Elastic Constants of Cubic Crystals	p. 232
Young's Modulus and Poisson's Ratio	p. 233
The Shear Modulus	p. 235
The Bulk Modulus	p. 237
Topological Design of Periodic Cellular Solids	p. 238
Finite Element Program to Calculate Linear Elastic Mechanical Properties	p. 243
Linear Elastic Mechanical Properties of Periodic Cellular Solids	p. 243
Twelve-connected Stretch-dominated Periodic Cellular Solids via Interference Lithography	p. 247
Fabrication of a Simple Cubic Cellular Solid via Interference Lithography	p. 249
Plastic Deformation of Microframes	p. 250
Further Reading	p. 252
Further Applications	p. 255
Controlling the Spontaneous Emission of Light	p. 256
Localization of Light: Microcavities and Waveguides	p. 259
Simultaneous Localization of Light and Sound in Photonic-Phononic Crystals: Novel Acoustic-Optical Devices	p. 264
Negative Refraction and Superlenses	p. 268
Multifunctional Periodic Structures: Maximum Transport of Heat and Electricity	p. 272
Microfluidics	p. 273
Thermoelectric Energy	p. 275
Peltier Effect	p. 275
Thomson Effect	p. 276
Seebeck Effect	p. 277
Further Reading	p. 278
MATLAB Program to Calculate the Optimal Electric Field Amplitude Vectors for the Interfering Light Beams	p. 281
MATLAB Program to Calculate Reflectance versus Frequency for One-dimensional Photonic Crystals	p. 289
MATLAB Program to Calculate Reflectance versus Frequency for One-dimensional Phononic Crystals	p. 297
Index	p. 305
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Periodic Materials and Interference Lithography For Photonics, Phononics and Mechanics

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