Electronic transport properties of quasicrystals: a Review

We present a review of some results concerning electronic transport properties of quasicrystals. After a short introduction to the basic concepts of quasiperiodicity, we consider the experimental transport properties of electrical conductivity with particular focus on the effect of temperature, magnetic field and defects. Then, we present some heuristic approaches that tend to give a coherent view of different, and to some extent complementary, transport mechanisms in quasicrystals. Numerical results are also presented and in particular the evaluation of the linear response Kubo-Greenwood formula of conductivity in quasiperiodic systems in presence of disorder.Comment: Latex, 28 pages, Journ. of Math. Phys., Vol38 April 199

Transmission Resonance in an Infinite Strip of Phason-Defects of a Penrose Approximant Network

An exact method that analytically provides transfer matrices in finite networks of quasicrystalline approximants of any dimensionality is discussed. We use these matrices in two ways: a) to exactly determine the band structure of an infinite approximant network in analytical form; b) to determine, also analytically, the quantum resistance of a finite strip of a network under appropriate boundary conditions. As a result of a subtle interplay between topology and phase interferences, we find that a strip of phason-defects along a special symmetry direction of a low 2-d Penrose approximant, leads to the rigorous vanishing of the reflection coefficient for certain energies. A similar behavior appears in a low 3-d approximant. This type of ``resonance" is discussed in connection with the gap structure of the corresponding ordered (undefected) system.Comment: 18 pages special macros jnl.tex,reforder.tex, eqnorder.te

Solving Inverse Obstacle Scattering Problem with Latent Surface Representations

We propose a novel iterative numerical method to solve the three-dimensional inverse obstacle scattering problem of recovering the shape of the obstacle from far-field measurements. To address the inherent ill-posed nature of the inverse problem, we advocate the use of a trained latent representation of surfaces as the generative prior. This prior enjoys excellent expressivity within the given class of shapes, and meanwhile, the latent dimensionality is low, which greatly facilitates the computation. Thus, the admissible manifold of surfaces is realistic and the resulting optimization problem is less ill-posed. We employ the shape derivative to evolve the latent surface representation, by minimizing the loss, and we provide a local convergence analysis of a gradient descent type algorithm to a stationary point of the loss. We present several numerical examples, including also backscattered and phaseless data, to showcase the effectiveness of the proposed algorithm

Deep Point Correlation Design

Designing point patterns with desired properties can require substantial effort, both in hand-crafting coding and mathematical derivation. Retaining these properties in multiple dimensions or for a substantial number of points can be challenging and computationally expensive. Tackling those two issues, we suggest to automatically generate scalable point patterns from design goals using deep learning. We phrase pattern generation as a deep composition of weighted distance-based unstructured filters. Deep point pattern design means to optimize over the space of all such compositions according to a user-provided point correlation loss, a small program which measures a pattern’s fidelity in respect to its spatial or spectral statistics, linear or non-linear (e. g., radial) projections, or any arbitrary combination thereof. Our analysis shows that we can emulate a large set of existing patterns (blue, green, step, projective, stair, etc.-noise), generalize them to countless new combinations in a systematic way and leverage existing error estimation formulations to generate novel point patterns for a user-provided class of integrand functions. Our point patterns scale favorably to multiple dimensions and numbers of points: we demonstrate nearly 10 k points in 10-D produced in one second on one GPU. All the resources (source code and the pre-trained networks) can be found at https://sampling.mpi-inf.mpg.de/deepsampling.html

Engineering aperiodic spiral order for photonic-plasmonic device applications

Thesis (Ph.D.)--Boston UniversityDeterministic arrays of metal (i.e., Au) nanoparticles and dielectric nanopillars (i.e., Si and SiN) arranged in aperiodic spiral geometries (Vogel's spirals) are proposed as a novel platform for engineering enhanced photonic-plasmonic coupling and increased light-matter interaction over broad frequency and angular spectra for planar optical devices. Vogel's spirals lack both translational and orientational symmetry in real space, while displaying continuous circular symmetry (i.e., rotational symmetry of infinite order) in reciprocal Fourier space. The novel regime of "circular multiple light scattering" in finite-size deterministic structures will be investigated. The distinctive geometrical structure of Vogel spirals will be studied by a multifractal analysis, Fourier-Bessel decomposition, and Delaunay tessellation methods, leading to spiral structure optimization for novel localized optical states with broadband fluctuations in their photonic mode density. Experimentally, a number of designed passive and active spiral structures will be fabricated and characterized using dark-field optical spectroscopy, ellipsometry, and Fourier space imaging. Polarization-insensitive planar omnidirectional diffraction will be demonstrated and engineered over a large and controllable range of frequencies. Device applications to enhanced LEDs, novel lasers, and thin-film solar cells with enhanced absorption will be specifically targeted. Additionally, using Vogel spirals we investigate the direct (i.e. free space) generation of optical vortices, with well-defined and controllable values of orbital angular momentum, paving the way to the engineering and control of novel types of phase discontinuities (i.e., phase dislocation loops) in compact, chip-scale optical devices. Finally, we report on the design, modeling, and experimental demonstration of array-enhanced nanoantennas for polarization-controlled multispectral nanofocusing, nanoantennas for resonant near-field optical concentration of radiation to individual nanowires, and aperiodic double resonance surface enhanced Raman scattering substrates

Engineering optical nonlinearities in metal nanoparticle arrays

Thesis (Ph.D.)--Boston UniversityMetal nanostructures supporting localized surface plasmon (LSP) resonances are an emerging technology for sensing, optical switching, radiative engineering, and solar energy harvesting, among other applications. The unique property of LSP resonances that enable these technologies is their ability to localize and enhance the optical field near the surface of metal nanoparticles. However, many questions still remain regarding the effects of nanoparticle coupling on the linear and nonlinear optical properties of these structures. In this thesis, I investigate the role of long-range photonic and near-field plasmonic coupling on the linear and nonlinear optical properties of metal nanoparticles in periodic and deterministic aperiodic arrays within a combined experimental and theoretical framework. In particular, I have developed optical characterization techniques to study various properties of planar metal nano-cylinder arrays fabricated by electron beam lithography (EBL). These include the effect of Fano-type coupling between structural grating modes and LSP resonances on linear diffraction and second harmonic generation (SHG), the influence of near-field coupling on the efficiency of plasmon enhanced metal photoluminescence (PL), the dependence of two-photon PL (TPPL) on nanoparticle size, and the multi-polar nature of SHG from planar plasmonic arrays. Experimental results are fully supported by linear scattering theory of the near and far-field properties of particle arrays based on a range of analytical, semi-analytical, and fully numerical techniques. The breadth of computational methods used allows the investigation of a wide range of structures including large aperiodic arrays with hundreds of discrete particles and periodic arrays with realistic particle shapes, substrates, and excitation conditions. The technological potential of engineered plasmonic structures is demonstrated by enhanced vibrational sum frequency generation (VSFG) spectroscopy, a novel nonlinear sensing technique. These studies have revealed design principles for engineering the interplay of photonic and plasmonic coupling for future linear and nonlinear plasmonic devices for sensing, switching, and modulation. The optical characterization techniques developed in this thesis may additionally be used across a wide range of devices in photonics and nano-optics

Optical Properties of Quasiperiodically Arranged Semiconductor Nanostructures

This work consists of two parts which are entitled "One-Dimensional Resonant Fibonacci Quasicrystals" and "Resonant Tunneling of Light in Silicon Nanostructures". A microscopic theory has been applied to investigate the optical properties of the respective semiconductor nanostructures. The studied one-dimensional resonant Fibonacci quasicrystals consist of GaAs quantum wells (QW) that are separated by either a large spacer L or a small one S. These spacers are arranged according to the Fibonacci sequence LSLLSLSL... The average spacing satisfies a generalized Bragg condition with respect to the 1s-exciton resonance of the QWs. A theory, that makes use of the transfer-matrix method and that allows for the microscopic description of many-body effects such as excitation-induced dephasing caused by the Coulomb scattering of carriers, has been applied to compute the optical spectra of such structures. Based on an appropriate single set of fixed sample parameters, the theory provides reflectance spectra that are in excellent agreement with the corresponding measured linear and nonlinear spectra. A pronounced sharp reflectivity minimum is found in the vicinity of the heavy-hole resonance both in the measured as well as in the calculated linear 54-QW spectra. Such sharp spectral features are suitable for application as optical switches or for slow-light effects. Hence, their properties have been studied in detail. Specifically, the influence of the carrier density, of the QW arrangement, of a detuning away from the exact Bragg condition, of the average spacing as well as of the ratio of the optical path lengths of the large and small spacers L and S, respectively, and of the QW number on the optical properties of the samples have been studied. The features of measured spectra could have been attributed to different sample properties related to the sample setup. Additionally, self-similarity among reflection spectra corresponding to different QW numbers that exceed a Fibonacci number by one is observed, which identifies certain spectral features as true fingerprints of the Fibonacci spacing. In the second part, resonant tunneling of light in stacked structures consisting of alternating parallel layers of silicon and air have been studied theoretically. While usually total internal reflection is expected for light shined on a silicon-air interface under an angle larger than the critical angle, light may tunnel through the air barrier due to the existence of evanescent waves inside the air layers if the neighboring silicon layer is close enough. This tunneling of light is in analogy to the well-known tunneling of a quantum particle through a potential barrier. In particular, the wave equation and the stationary Schrödinger equation are of the same form. Hence, the resonant tunneling of light can be understood in analogy to the resonant tunneling of e.g. electrons as well. The characteristic feature of resonant tunneling is a complete transmission through the barrier at certain resonance energies. The transmission, reflection, and propagation properties of the samples have been determined numerically using a transfer-matrix method. Analytical expressions for the energetic resonance positions have been derived and are in excellent agreement with the numerical simulations. Special attention has been drawn to the lowest resonance out of a series of resonant-tunneling resonances. There, light has been observed to be concentrated within silicon layers the extension of which is smaller than the corresponding wavelength of the light. Specifically, the quality factor is large at the resonance energies, so that the resonant light leaves the sample delayed, which allows for the study of slow light. A detailed investigation of how the sample geometry influences the optical properties of the sample has been presented. In particular, it has been outlined how to design a sample to obtain certain desired optical properties. The optical properties that are related to the resonant tunneling strongly rely on the (mirror-)symmetry of the samples. If asymmetries - especially of the silicon wells inside the air barrier - are present in the sample setup, the resonant-tunneling efficiency is diminished. Such asymmetries are unavoidable in the production of the samples. Therefore, a parameter range has been identified in which reasonable transmission above a transmission probability of 50% can be expected taking typical fluctuations caused by the production process into account. Silicon-based resonant-tunneling structures of a setup proposed by the presented theory have already been fabricated and first experiments are under way. This will allow for theory-experiment comparisons