By Martin Maldovan
Conceal -- Contents -- Preface -- creation -- concept -- 1 Structural Periodicity -- 1.1 Nonperiodic as opposed to Periodic buildings -- 1.2 Two-dimensional element Lattices -- 1.3 third-dimensional aspect Lattices -- 1.4 Mathematical Description of Periodic constructions -- 1.5 Fourier sequence -- additional interpreting -- difficulties -- 2 Periodic capabilities and buildings -- 2.1 creation -- 2.2 developing easy Periodic features in Dimensions -- 2.3 growing uncomplicated Periodic capabilities in 3 Dimensions -- 2.4 mix of straightforward Periodic services -- difficulties -- three Interference of Waves and Interference Lithography -- 3.1 Electromagnetic Waves -- 3.2 The Wave Equation -- 3.3 Electromagnetic aircraft Waves -- 3.4 The Transverse personality of Electromagnetic airplane Waves -- 3.5 Polarization -- 3.6 Electromagnetic power -- 3.7 Interference of Electromagnetic aircraft Waves -- 3.8 Interference Lithography -- extra studying -- difficulties -- four Periodic constructions and Interference Lithography -- 4.1 the relationship among the Interference of aircraft Waves and Fourier sequence -- 4.2 uncomplicated Periodic constructions in Dimensions through Interference Lithography -- 4.3 uncomplicated Periodic constructions in 3 Dimensions through Interference Lithography -- additional studying -- difficulties -- Experimental -- five Fabrication of Periodic buildings -- 5.1 advent -- 5.2 gentle Beams -- 5.3 a number of Gratings and the Registration problem -- 5.4 Beam Configuration -- 5.5 development move: fabric structures and Photoresists -- 5.6 functional issues for Interference Lithography -- 5.7 last comments -- additional studying -- purposes -- 6 Photonic Crystals -- 6.1 advent -- 6.2 One-dimensional Photonic Crystals -- 6.3 Two-dimensional Photonic Crystals -- 6.4 3-dimensional Photonic Crystals -- extra studying -- difficulties -- 7 Phononic Crystals -- 7.1 creation -- 7.2 Phononic Crystals -- 7.3 One-dimensional Phononic Crystals -- 7.4 Two-dimensional Phononic Crystals -- 7.5 three-d Phononic Crystals -- extra studying -- difficulties -- eight Periodic mobile Solids -- 8.1 creation -- 8.2 One-dimensional Hookes legislation -- 8.3 the tension Tensor -- 8.4 the stress Tensor -- 8.5 StressStrain courting: The Generalized Hookes legislation -- 8.6 The Generalized Hookes legislations in Matrix Notation -- 8.7 The Elastic Constants of Cubic Crystals -- 8.8 Topological layout of Periodic mobile Solids -- 8.9 Finite point software to Calculate Linear Elastic Mechanical houses -- 8.10 Linear Elastic Mechanical homes of Periodic mobile Solids -- 8.11 Twelve-connected Stretch-dominated Periodic mobile Solids through Interference Lithography -- 8.12 Fabrication of an easy Cubic mobile stable through Interference Lithography -- 8.13 Plastic Deformation of Microframes -- extra analyzing -- nine extra purposes -- 9.1 Controlling the Spontaneous Emission of sunshine -- 9.2 Localization of sunshine: Microcavities and Waveguides -- 9.3 Simultaneous Localization of sunshine and Sound in PhotonicPhononic Crystals: Novel AcousticOptical units -- 9.4 damaging Refraction and Superlenses -- 9.5 Multifunctional Periodic constructions: greatest delivery of warmth and electrical energy -- 9.6 Microfluidics -- 9.7 Thermoelectric power -- extra examining -- Appendix A MATLAB application
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Additional resources for Periodic materials and interference lithography for photonics, phononics and mechanics
By using this technique, each periodic function has an associated distinctive set of weighted coefﬁcients (or Fourier coefﬁcients) that univocally represent the periodic function. In the next section, we deﬁne the Fourier coefﬁcients and show how to calculate them for arbitrary two- and three-dimensional periodic functions. 1 Fourier Series for Two-dimensional Periodic Functions Suppose we have a two-dimensional real function f (x, y) deﬁned within a rectangular unit cell of sides a and b. 13), the function f (x, y) also repeats itself over and over along the x and y directions at intervals of a and b, respectively.
6a. 6b). The set of parallel planes identiﬁes the set of objects that repeat regularly in space, each of which is identical in size and shape. 6b). Note that the equally spaced planes also generate a set of three-dimensional cells within which the objects are enclosed. 6d–f show the same scheme, but applied to a slightly different three-dimensional periodic structure made of rods connected at right angles. 6b is arbitrary. Therefore, the deﬁnition of the object that repeats regularly in space is arbitrary and the periodic structure can alternatively be described by different choices for the repeating object.
As in the f1 (x, y) case, no additional structures are obtained by considering inverse structures. 1a. 1a. As a result, the simple periodic functions f1 (x, y) and f2 (x, y) represent the same two-dimensional periodic structure. 1a. In fact, the periodic structures given by the f4 (x, y) functions are the same as the structures given by the f1 (x, y) functions, but compressed by a factor of 2. That is, in the case of the square lattice, by considering the groups d = 2 and d = 4, we obtain additional simple periodic functions, but the corresponding periodic structures are the same as the f1 (x, y) structures except for a scaling factor.