Finite Crystal Research Articles

Density of optical states in rolled-up photonic crystals and quasi crystals

We describe the local density of optical states (LDOS) and the density of optical states (DOS) in three dimensional (3D) finite size rolled-up photonic crystals and quasi crystals calculated using optimized code based on finite difference time domain (FDTD) techniques. Because the Bloch–Floquet theorem is not used in these calculations, the code can be applied to any dielectric structures that lack discrete translational symmetry, making it ideal for experimentally fabricated structures that are finite in size and include defects. Our software can be used for either LDOS or DOS calculations, is parallelized using standard message passing interface (MPI) protocols, and is freely available at www.fpspackage.com. Program summaryProgram title: FPSCatalogue identifier: AFBP_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AFBP_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU GPL v3No. of lines in distributed program, including test data, etc.: 118342No. of bytes in distributed program, including test data, etc.: 2385803Distribution format: tar.gzProgramming language:C++.Computer: PCs and distributed memory machines.Operating system: Linux, Unix.Has the code been vectorised or parallelized?: Parallelized using message passing interface.RAM: Depending on the problem; gigabytes.Classification: 10.External routines: Minimum: MPI; [1] fftw3; [2] optional: hdf5-tools. [3]Nature of problem: The aim of FPS is providing a fast and easy to use package for studying local density of optical states and density of optical states in arbitrary shaped finite size photonic structures.Solution method: Parallel finite difference time domain method.Restrictions: The post processing codes are not parallelized. At least two processes should be used for the calculations.Unusual features: FPS provides non-experts in computational physics an intuitive tool to obtain local density of optical states and density of optical states calculations. Inserting parameters that define the geometry and the spectrums of interest into a “txt” or “h5” file are all a user needs to do.Additional comments: The package includes a tutorial describing the package and its installation, using an example.Running time: It was found that the average time to compute a single LDOS was 21 minutes. Runtime in general depends on mesh size and the number of points that local density of optical states should be calculated.

Characterization of Few-Layer 1T' MoTe2 by Polarization-Resolved Second Harmonic Generation and Raman Scattering.

We study the crystal symmetry of few-layer 1T' MoTe2 using the polarization dependence of the second harmonic generation (SHG) and Raman scattering. Bulk 1T' MoTe2 is known to be inversion symmetric; however, we find that the inversion symmetry is broken for finite crystals with even numbers of layers, resulting in strong SHG comparable to other transition-metal dichalcogenides. Group theory analysis of the polarization dependence of the Raman signals allows for the definitive assignment of all the Raman modes in 1T' MoTe2 and clears up a discrepancy in the literature. The Raman results were also compared with density functional theory simulations and are in excellent agreement with the layer-dependent variations of the Raman modes. The experimental measurements also determine the relationship between the crystal axes and the polarization dependence of the SHG and Raman scattering, which now allows the anisotropy of polarized SHG or Raman signal to independently determine the crystal orientation.

Electromagnetic-field amplification in finite one-dimensional photonic crystals

The electromagnetic-field distribution in a finite one-dimensional photonic crystal is studied using the numerical solution of Maxwell’s equations by the transfer-matrix method. The dependence of the transmission coefficient T on the period d (or the wavelength λ) has the characteristic form with M–1 (M is the number of periods in the structure) maxima with T = 1 in the allowed band of an infinite crystal and zero values in the forbidden band. The field-modulus distribution E(x) in the structure for parameters that correspond to the transmission maxima closest to the boundaries of forbidden bands has maxima at the center of the structure; the value at the maximum considerably exceeds the incident-field strength. For the number of periods M ~ 50, more than an order of magnitude increase in the field amplification is observed. The numerical results are interpreted with an analytic theory constructed by representing the solution in the form of a linear combination of counterpropagating Floquet modes in a periodic structure.

Destruction of vertical transitions and increase in the matrix elements of the electron-photon interaction in small crystals

Abstractauthoren The matrix elements of electron–photon interaction were investigated in a one-dimensional model of a finite length crystal lattice. Its electronic stationary states belong both to quasi-discrete and to continuous energy band spectra. We analyze formation and degree of applicability for the law of “momentum” conservation and for the vertical transitions approximation as the lattice length increases. The values of matrix elements and their dependence on lattice length are different for the cases when the final state is lying under the vacuum level and when the final state is lying above this level. In the latter case, the resonant penetration of electrons in the lattice may increase the matrix elements by many orders of magnitude. The results are different in the middle and near the thresholds of the energy bands, for the cases of tight and weak binding of electrons in the lattice cell. The selection rules of the “momentum” conservation and the approximation of vertical transitions disappear at extremely weak binding.

Mode Excitation in Finite Dust Clusters Using an Optical Trap

With the ability to trap a single particle in a dusty plasma, it becomes possible to alter the structural properties of the particle assembly or to manipulate its dynamical behavior. Here, we report on a first experiment to excite the eigenmodes of a finite dusty plasma crystal by means of a controlled oscillation of a single particle being trapped in a modified optical tweezer. The eigenmode structure and the excitation process are analyzed.

On beam shaping of the field radiated by a line source coupled to finite or infinite photonic crystals.

Comparison of the beam-shaping effect of a field radiated by a line source, when an ideal infinite structure constituted by two photonic crystals and an actual finite one are considered, has been carried out by means of two different methods. The lattice sums technique combined with the generalized reflection matrix method is used to rigorously investigate the radiation from the infinite photonic crystals, whereas radiation from crystals composed of a finite number of rods along the layers is analyzed using the cylindrical-wave approach. A directive radiation is observed with the line source embedded in the structure. With an increased separation distance between the crystals, a significant edge diffraction appears that provides the main radiation mechanism in the finite layout. Suitable absorbers are implemented to reduce the above-mentioned diffraction and the reflections at the boundaries, thus obtaining good agreement between radiation patterns of a localized line source coupled to finite and infinite photonic crystals, when the number of periods of the finite structure is properly chosen.

Magnetic-field tunable multichannel filter in a plasma photonic crystal at microwave frequencies.

The microwave magnetic-field tunable filtering properties in a multichannel filter based on use of a one-dimensional finite magnetized plasma photonic crystal (PPC) are theoretically investigated. The considered PPC has a structure of air/(AB)^N/air, where A is a dielectric layer, B is a plasma layer, and N is the stack number. First, in the absence of an externally applied magnetic field, the structure can work as a multichannel filter whose channel number is equal to N-1 for N>1. Next, in the presence of an externally applied field, the filtering properties become tunable, i.e., the channel frequencies can be shifted as a function of the applied magnetic field. We find that the effect of the magnetic field will cause the channel frequencies to be blue-shifted or red-shifted depending on the orientation of the applied magnetic field.

Finite element analysis of true and pseudo surface acoustic waves in one-dimensional phononic crystals

In this paper, we report a theoretical investigation of surface acoustic waves propagating in one-dimensional phononic crystal. Using finite element method eigenfrequency and frequency response studies, we develop two model geometries suitable to distinguish true and pseudo (or leaky) surface acoustic waves and determine their propagation through finite size phononic crystals, respectively. The novelty of the first model comes from the application of a surface-like criterion and, additionally, functional damping domain. Exemplary calculated band diagrams show sorted branches of true and pseudo surface acoustic waves and their quantified surface confinement. The second model gives a complementary study of transmission, reflection, and surface-to-bulk losses of Rayleigh surface waves in the case of a phononic crystal with a finite number of periods. Here, we demonstrate that a non-zero transmission within non-radiative band gaps can be carried via leaky modes originating from the coupling of local resonances with propagating waves in the substrate. Finally, we show that the transmission, reflection, and surface-to-bulk losses can be effectively optimised by tuning the geometrical properties of a stripe.

Simple models for disequilibrium fractional melting and batch melting with application to REE fractionation in abyssal peridotites

Disequilibrium melting arises when the kinetics of chemical exchange between a residual mineral and partial melt is sluggish compare to the rate of melting. To better understand the role of a finite crystal–melt exchange rate on trace element fractionation during mantle melting, we have developed a disequilibrium melting model for partial melting in an upwelling steady-state column. We use linear kinetics to approximate crystal–melt mass exchange rate and obtain simple analytical solutions for cases of perfect fractional melting and batch melting. A key parameter determining the extent of chemical disequilibrium during partial melting is an element specific dimensionless ratio (ε) defined as the melting rate relative to the solid–melt chemical exchange rate for the trace element of interest. In the case of diffusion in mineral limited chemical exchange, ε is inversely proportional to diffusivity of the element of interest. Disequilibrium melting is important for the trace element when ε is comparable to or greater than the bulk solid–melt partition coefficient for the trace element (k). The disequilibrium fractional melting model is reduced to the equilibrium perfect fractional melting model when ε is much smaller than k. Hence highly incompatible trace elements with smaller mobilities in minerals are more susceptible to disequilibrium melting than moderately incompatible and compatible trace elements. Effect of chemical disequilibrium is to hinder the extent of fractionation between residual solid and partial melt, making the residual solid less depleted and the accumulated melt more depleted in incompatible trace element abundances relative the case of equilibrium melting.Application of the disequilibrium fractional melting model to REE and Y abundances in clinopyroxene in abyssal peridotites from the Central Indian Ridge and the Vema Lithospheric Section, Mid-Atlantic Ridge revealed a positive correlation between the disequilibrium parameter ε and the degree of melting, which can be explained by an increase in melting rate and a decrease in REE diffusion rate in the upper part of the melting column. Small extent of disequilibrium melting for LREE and equilibrium melting for HREE in the upper part of the melting column can explain the elevated LREE abundances or spoon-shaped REE patterns in clinopyroxene in more refractory abyssal peridotites. The latter has often been attributed to melt refertilization.

Boundary effects in finite size plasmonic crystals: focusing and routing of plasmonic beams for optical communications

Plasmonic crystals, which consist of periodic arrangements of surface features at a metal–dielectric interface, allow the manipulation of optical information in the form of surface plasmon polaritons. Here we investigate the excitation and propagation of plasmonic beams in and around finite size plasmonic crystals at telecom wavelengths, highlighting the effects of the crystal boundary shape and illumination conditions. Significant differences in broad plasmonic beam generation by crystals of different shapes are demonstrated, while for narrow beams, the propagation from a crystal onto the smooth metal film is less sensitive to the crystal boundary shape. We show that by controlling the boundary shape, the size and the excitation beam parameters, directional control of propagating plasmonic modes and their behaviour such as angular beam splitting, focusing power and beam width can be efficiently achieved. This provides a promising route for robust and alignment-independent integration of plasmonic crystals with optical communication components.

Propagation of a magnetostatic surface spin wave through a finite magnonic crystal

A new method of theoretical investigation of magnetostatic spin waves propagation in a 1D finite bi-component magnonic crystals is developed. The transmission and reflection coefficients for a magnonic crystal with low and high magnetization contrast are obtained. Using the method we define the boundaries of the infinite medium approximation. This method can be used for the investigation of non-periodic bi-component structures. Applications of said approach are presented.

Phase retrieval for randomly terminated finite crystals

Phase retrieval for randomly terminated finite crystals

A new approach for modelling lattice energy in finite crystal domains

Evaluation of internal energy in a crystal lattice requires precise calculation of lattice sums. Such evaluation is a problem in the case of small (nano) particles because the traditional methods are usually effective only for infinite lattices and are adapted to certain specific potentials. In this work, a new method has been developed for calculation of lattice energy. The method is a generalisation of conventional geometric probability techniques for arbitrary fixed lattices in a finite crystal domain. In our model, the lattice energy for wide range of two- body central interaction potentials (including long-range Coulomb potential) has been constructed using absolutely convergent sums. No artificial cut-off potential or periodical extension of the domain (which usually involved for such calculations) have been made for calculation of the lattice energy under this approach. To exemplify the applications of these techniques, the energy of Coulomb potential has been plotted as the function of the domain size.

A three-dimensional Bloch wave expansion to determine external scattering from finite phononic crystals.

External scattering from a finite phononic crystal (PC) is studied using the Helmholtz-Kirchhoff integral theorem integrated with a Bloch wave expansion (BWE). The BWE technique is used to describe the internal pressure field of a semi-infinite or layered PC subject to an incident monochromatic plane wave. Following the BWE solution, the Helmholtz-Kirchhoff integral is used to determine the external scattered field. For cubic PCs, the scattered results are compared to numerical treatments in both the frequency and time domain. The presented approach is expected to be valid when the PC size is larger than the acoustic wavelength. However, very good agreement in the spatial beam pattern is also documented for both large and small (with respect to the wavelength) PCs. The result of this work is a fully-analytical, efficient, and verified approach for accurately predicting external scattering from finite, three-dimensional PCs.

Guiding airborne sound through surface modes of a two-dimensional phononic crystal

Existence and guiding properties of surface modes bound to the interface between a finite two-dimensional phononic crystal and the host medium are experimentally and numerically demonstrated. Surface modes can be observed on both (1 0) and (1 1) surfaces of a square phononic crystal of steel cylinders in air. Numerical investigations of band properties and simulations of mode excitation are carried out through the finite-element method. Excited by the far field of a speaker, existence of surface modes is investigated by recording the sound field in the vicinity of the respective crystal surfaces. Both surface bands of the square phononic crystal depart from bulk bands and extend into the band gap for sufficiently high filling fractions. While such a surface band can be obtained for considerably smaller scatterer radii for the (1 0) surface, significantly higher radii around 0.49 of the lattice constant are required to obtain propagating surface modes on the (1 1) surface. Persistence of the guided surface mode along the (1 0) surface, where it diminishes in a length scale of the lattice constant in the transverse direction is demonstrated. The modes of the (1 1) surface decay faster into the air in the transverse direction. Guided modes on both surfaces propagate in a beating manner where the beat length can be determined by the wave number of the mode.

Elliptical optical solitary waves in a finite nematic liquid crystal cell

The addition of orbital angular momentum has been previously shown to stabilise beams of elliptic cross-section. In this article the evolution of such elliptical beams is explored through the use of an approximate methodology based on modulation theory. An approximate method is used as the equations that govern the optical system have no known exact solitary wave solution. This study brings to light two distinct phases in the evolution of a beam carrying orbital angular momentum. The two phases are determined by the shedding of radiation in the form of mass loss and angular momentum loss. The first phase is dominated by the shedding of angular momentum loss through spiral waves. The second phase is dominated by diffractive radiation loss which drives the elliptical solitary wave to a steady state. In addition to modulation theory, the “chirp” variational method is also used to study this evolution. Due to the significant role radiation loss plays in the evolution of an elliptical solitary wave, an attempt is made to couple radiation loss to the chirp variational method. This attempt furthers understanding as to why radiation loss cannot be coupled to the chirp method. The basic reason for this is that there is no consistent manner to match the chirp trial function to the generated radiating waves which is uniformly valid in time. Finally, full numerical solutions of the governing equations are compared with solutions obtained using the various variational approximations, with the best agreement achieved with modulation theory due to its ability to include both mass and angular momentum losses to shed diffractive radiation.

Thermal tuning of omnidirectional reflection bands in one-dimensional finite phononic crystals

This study investigates the temperature-tuned omnidirectional reflection (ODR) bands in a one-dimensional (1D) finite phononic crystal (PnC), formed by alternating layers of nitinol and epoxy. An analytical model, based on the transfer matrix method, is developed to study reflection and transmission characteristics of the acoustic waves including shear and compressional waves in a 1D PnC. Existence criteria and the sensitive and continuous temperature-tunability of ODR bands in the nitinol/epoxy PnC are demonstrated using the analyses of projected-band structures and reflection spectra. The width and location of the ODR bands shift markedly with temperature variations of nitinol across the phase transition from martensite to austenite. The effects of temperature, filling fraction of nitinol layers, and the Si clad layer on ODR bands are considered. The results will be of benefit in the design and optimization of thermal tuning of omnidirectional acoustic mirrors.

Direct Phasing of Finite Crystals Illuminated with a Free-Electron Laser

Imaging biological macromolecules allows scientists to accurately probe dynamic molecular structures and engineer drugs. Brief and intense coherent x-ray pulses can yield new forms of data, enabling structure determination without prior-known structural information.

Selective mode excitation in finite size plasma crystals by diffusely reflected laser light

The possibility to use diffuse reflections of a laser beam to exert a force on levitating dust particles is studied experimentally. Measurements and theoretical predictions are found to be in good agreement. Further, the method is applied to test the selective excitation of breathing-like modes in finite dust clusters.

Design and Optimization of Omnidirectional Band Gap for One-Dimensional Periodic and Quasiperiodic Phononic Heterostructures

A new kind of one-dimensional multilayer phononic heterostructure is constructed to obtain a broad acoustic omnidirectional reflection (ODR) band. The heterostructure is formed by combining finite periodic phononic crystals (PnCs) and Fibonacci (or Thue—Morse) quasiperiodic PnCs. From the numerical results performed by the transfer matrix method, it is found that the ODR bands can be enlarged obviously by using the combination of periodic and quasi-periodic PnCs. Moreover, an application of particle swarm optimization in designing and optimizing acoustic ODR bands is reported. With regards to different thickness ratios and periodic numbers in the heterostructure, we give some optimization examples and finally achieve phononic heterostructure with a very broad ODR bandwidth. The result provides a new approach to achieve broad acoustic ODR bandwidth, and will be applied in design of omnidirectional acoustic mirrors.