Photonic Band Structure Calculations Research Articles

The Influence of Geometric Parameters for Training an Artificial Neural Network to Predict the Band Structure of 1-D Fishbone Photonic Crystal

With the rising demand for the transmission of large amounts of information over long distances, the development of integrated light circuits is the key to improving this technology, and silicon photonics have been developed with low absorption in the near-infrared range and with sophisticated fabrication techniques. To build devices that work in different functionalities, photonic crystals are one of the most used structures due to their ability to manipulate light. The investigation of photonic crystals requires the calculation of photonic band structures and is usually time-consuming work. To reduce the time spent on calculations, a trained ANN is introduced in this study to directly predict the band structures using only a minimal amount of pre-calculated band structure data. A well-used 1-D fishbone-like photonic crystal in the form of a nanobeam is used as the training target, and the influence of adjusting the geometric parameters is discussed, especially the lattice constant and the thickness of the nanobeam. To train the ANN with very few band structures, each of the mode points in the band structure is considered as a single datapoint to increase the amount of training data. The datasets are composed of various raw band structure data. The optimized ANN is introduced at the end of this manuscript.

Accessing new avenues of photonic bandgaps using two-dimensional non-Moiré geometries

Photonic crystals (PhC) formed by 2-D non-Moiré geometries are realized in this work. Non-Moiré (NM) tiles are the contours of trigonometric functions that generate exciting shapes and geometries. Photonic bandstructure calculations reveal that 2-D NM geometries exhibit new avenues of photonic bandgaps compared to the regular circular rod-based PhCs. The band structures are anisotropic and show, intriguing orientation-dependent partial bandgaps. A few of the orientation-dependent frequency selective properties of the realized NM geometry-based PhCs are demonstrated using full-wave electromagnetic simulations. The proposed geometries are practically realizable, and in this work, we experimentally demonstrate the fabrication process using the 3-D printing technique for microwave frequencies.

Photonic band structure calculation of 3D-finite nanostructured supercrystals.

Computational modeling of plasmonic periodic structures are challenging due to their multiscale nature. On one hand, nanoscale building blocks require very fine spatial discretization of the computation domain to describe the near-field nature of the localized surface plasmons. On the other hand, the microscale supercrystals require large simulation domains. To tackle this challenge, two approaches are generally taken: (i) an effective medium approach, neglecting the nanoscale effects and (ii) the use of a unit cell with periodic boundary conditions, neglecting the overall habit of the supercrystal. The latter, which is used to calculate the photonic band structure of these supercrystals, fails to describe the photonic properties arising from their finite-size such as Fabry-Pérot modes (FPMs), whispering gallery modes (WGMs), and decrease of the photonic mode lifetime. Here, we developed a computational approach, based on the finite-difference time-domain method to accurately calculate the photonic band structures of finite supercrystals. We applied this new approach to 3D periodic microstructures of Au nanoparticles with cubic, spherical, and rhombic dodecahedral habits and discuss how their photonic band structures differ from those of infinite structures. Finally, we compared the photonic band structures to reflectance spectra and describe phenomena such as FPMs, WGMs, and polaritonic bandgaps.

Colloidal diamond.

Self-assembling colloidal particles in the cubic diamond crystal structure could potentially be used to make materials with a photonic bandgap1-3. Such materials are beneficial because they suppress spontaneous emission of light1 and are valued for their applications as optical waveguides, filters and laser resonators4, for improving light-harvesting technologies5-7 and for other applications4,8. Cubic diamond is preferred for these applications over more easily self-assembled structures, such as face-centred-cubic structures9,10, because diamond has a much wider bandgap and is less sensitive to imperfections11,12. In addition, the bandgap in diamond crystals appears at a refractive index contrast of about 2, which means that a photonic bandgap could be achieved using known materials at optical frequencies; this does not seem to be possible for face-centred-cubic crystals3,13. However, self-assembly of colloidal diamond is challenging. Because particles in a diamond lattice are tetrahedrally coordinated, one approach has been to self-assemble spherical particles with tetrahedral sticky patches14-16. But this approach lacks a mechanism to ensure that the patchy spheres select the staggered orientation of tetrahedral bonds on nearest-neighbour particles, which is required for cubic diamond15,17. Here we show that by using partially compressed tetrahedral clusters with retracted sticky patches, colloidal cubic diamond can be self-assembled using patch-patch adhesion in combination with a steric interlock mechanism that selects the required staggered bond orientation. Photonic bandstructure calculations reveal that the resulting lattices (direct and inverse) have promising optical properties, including a wide and complete photonic bandgap. The colloidal particles in the self-assembled cubic diamond structure are highly constrained and mechanically stable, which makes it possible to dry the suspension and retain the diamond structure. This makes these structures suitable templates for forming high-dielectric-contrast photonic crystals with cubic diamond symmetry.

Achievement of unidirectional air waveguide with extra-broad operation bandwidth in magneto-optical photonic crystals with a triangle lattice

We have presented a coupling structure of two-dimensional (2D) magneto-optical photonic crystals (MOPCs) with a triangular lattice. The numerical calculation of photonic band structure is performed based on the finite element method. The result shows that photonic band gap of 2D MOPCs with a triangular lattice is much larger than that of 2D MOPCs with a square lattice or a honeycomb lattice under the same applied external magnetic field, i.e., working bandwidth of 2D MOPCs with a triangular lattice is much larger than that of the other two MOPCs. We have studied the coupling effect of one-way edge modes in 2D MOPCs with a triangle lattice, and achieved a unidirectional air waveguide with high-transmission, strong nonreciprocity and extra-broad operation bandwidth. Furthermore, we have qualitatively analyzed waveguide transmission characteristics based on S-parameter method, and illustrated the effective operating bandwidth and the optimum operating frequency for optical waveguide nonreciprocity devices according to different transmission losses in application.

Photonic crystal micro-pixelation and additive color mixing in weevil scales

The origin of the brilliant near angle-independent coloration of the weevil Eupholus chevrolati was investigated by a combination of optical and electron microscopy tools, photonic band structure calculations, and color mixing analysis. Optical microscopy and scanning micro-spectroscopy revealed the presence of micrometer-sized red, yellow, green, and blue reflective pixels covering the entire exoskeleton of the weevil. Scanning electron microscopy in combination with focused ion beam milling showed that each micro-pixel consisted of a diamond-based photonic crystal structure and the different reflective colors were the result of different orientations of the photonic crystal. Color mixing analysis was used to study the collective behavior of the reflective micro-pixels. A pointillist, additive color-mixing scheme of the reflective photonic crystal micro-pixels was determined as the origin of the weevil’s bright and near angle-independent yellow-green coloration.

Topological photonic crystal with equifrequency Weyl points

Weyl points in three-dimensional photonic crystals behave as monopoles of Berry flux in momentum space. Here, based on general symmetry analysis, we show that a minimal number of four symmetry-related (consequently equifrequency) Weyl points can be realized in time-reversal invariant photonic crystals. We further propose an experimentally-feasible way to modify double-gyroid photonic crystals to realize four equifrequency Weyl points, which is explicitly confirmed by our first-principle photonic band-structure calculations. Remarkably, photonic crystals with equifrequency Weyl points are qualitatively advantageous in applications including angular selectivity, frequency selectivity, invisibility cloaking, and three dimensional imaging.

Theory and design of quantum light sources from quantum dots embedded in semiconductor-nanowire photonic-crystal systems

We introduce a new platform for realizing on-chip quantum electrodynamics using photonic crystal waveguide structures comprised of periodic nanowire arrays with embedded semiconductor quantum dots to act as quantum light sources. These nanowire-based structures, which can now be fabricated with excellent precision, are found to produce waveguide Purcell factors exceeding 100 and on-chip beta factors up to 99%. We investigate the fundamental optical properties of photonic crystal waveguides and finite-size structures using both photonic band structure calculations and rigorous Green function computations which allows us to obtain the modal properties and the local density of photon states. A comparison with slab-based photonic crystals is also made and we a highlight key advantages in the nanowire system, including the potential to minimize extrinsic scattering losses and produce high theoretical Purcell factors and beta-factors on-chip. We also demonstrate that these structures exhibit rich photonic Lamb shifts over broadband frequencies.

An Efficient Calculation of Photonic Crystal Band Structures Using Taylor Expansions

AbstractIn this paper we present an efficient algorithm for the calculation of photonic crystal band structures and band structures of photonic crystal waveguides. Our method relies on the fact that the dispersion curves of the band structure are smooth functions of the quasi-momentum in the one-dimensional Brillouin zone. We show the derivation and computation of the group velocity, the group velocity dispersion, and any higher derivative of the dispersion curves. These derivatives are then employed in a Taylor expansion of the dispersion curves. We control the error of the Taylor expansion with the help of a residual estimate and introduce an adaptive scheme for the selection of nodes in the one-dimensional Brillouin zone at which we solve the underlying eigenvalue problem and compute the derivatives of the dispersion curves. These derivatives are then employed in a Taylor expansion of the dispersion curves. We control the error of the Taylor expansion with the help of a residual estimate and introduce an adaptive scheme for the selection of nodes in the one-dimensional Brillouin zone at which we solve the underlying eigenvalue problem and compute the derivatives of the dispersion curves. The proposed algorithm is not only advantageous as it decreases the computational effort to compute the band structure but also because it allows for the identification of crossings and anti-crossings of dispersion curves, respectively. This identification is not possible with the standard approach of solving the underlying eigenvalue problem at a discrete set of values of the quasi-momentum without taking the mode parity into account.

Non-reciprocal transmission of terahertz waves through a photonic crystal cavity with graphene

In this work, we theoretically investigate non-reciprocal propagation of terahertz (THz) waves through a photonic crystal cavity integrated with graphene under external magnetic field. The magnetic field applied perpendicular to the graphene plane introduces an asymmetric conductivity tensor into the system, thus the Jones matrix of THz waves propagating in one direction is not the transpose of the one in the reversal direction. As a consequence, non-reciprocal propagation is achieved in the THz regime, which has been verified by the calculation of non-reciprocal photonic band structures, transmission spectra, and also electrical-field distributions. Further, by tuning the magnetic field or the Fermi level of graphene, such non-reciprocal transmission can be drastically tuned. This tunable non-reciprocity at the level of material response enables us to design photonic structures which can let a specific mode pass but its time-reversal counterpart stop. Moreover, high-performance non-reciprocal features have been found in periodic photonic crystal stacks with graphene. Our investigations may provide a unique approach to guide THz waves and achieve potential applications on one-way devices (e.g. isolators) in the THz regime.

Transfer-matrix approach for finite-difference time-domain simulation of periodic structures

Optical properties of periodic structures can be calculated using the transfer-matrix approach, which establishes a relation between amplitudes of the wave incident on a structure with transmitted or reflected waves. The transfer matrix can be used to obtain transmittance and reflectance spectra of finite periodic structures as well as eigenmodes of infinite structures. Traditionally, calculation of the transfer matrix is performed in the frequency domain and involves linear algebra. In this work, we present a technique for calculation of the transfer matrix using the finite-difference time-domain (FDTD) method and show the way of its implementation in FDTD code. To illustrate the performance of our technique we calculate the transmittance spectra for opal photonic crystal slabs consisting of multiple layers of spherical scatterers. Our technique can be used for photonic band structure calculations. It can also be combined with existing FDTD methods for the analysis of periodic structures at an oblique incidence, as well as for modeling point sources in a periodic environment.

Spectral approximation of quadratic operator polynomials arising in photonic band structure calculations

Galerkin spectral approximation theory for non-self-adjoint quadratic operator polynomials with periodic coefficients is considered. The main applications are complex band structure calculations in metallic photonic crystals, periodic waveguides, and metamaterials. We show that the spectrum of the considered operator polynomials consists of isolated eigenvalues of finite multiplicity with a nonzero imaginary part. The spectral problem is equivalent to a non-compact block operator matrix and norm convergence is shown for a block operator matrix having the same generalized eigenvectors as the original operator. Convergence rates of finite element discretizations are considered and numerical experiments with the $$p$$ p -version and the $$h$$ h -version of the finite element method confirm the theoretical convergence rates.

Macroscopic optical response and photonic bands

We develop a formalism for the calculation of the macroscopic dielectric response of composite systems made of particles of one material embedded periodically within a matrix of another material, each of which is characterized by a well-defined dielectric function. The nature of these dielectric functions is arbitrary, and could correspond to dielectric or conducting, transparent or opaque, absorptive and dispersive materials. The geometry of the particles and the Bravais lattice of the composite are also arbitrary. Our formalism goes beyond the long-wavelength approximation as it fully incorporates retardation effects. We test our formalism through the study of the propagation of electromagnetic waves in two-dimensional photonic crystals made of periodic arrays of cylindrical holes in a dispersionless dielectric host. Our macroscopic theory yields a spatially dispersive macroscopic response which allows the calculation of the full photonic band structure of the system, as well as the characterization of its normal modes, upon substitution into the macroscopic field equations. We can also account approximately for the spatial dispersion through a local magnetic permeability and analyze the resulting dispersion relation, obtaining a region of left handedness.

The finite element method applied to the study of two-dimensional photonic crystals and resonant cavities

A critical assessment of the finite element (FE) method for studying two-dimensional dielectric photonic crystals is made. Photonic band structures, transmission coefficients, and quality factors of various two-dimensional, periodic and aperiodic, dielectric photonic crystals are calculated by using the FE (real-space) method and the plane wave expansion or the finite difference time domain (FDTD) methods and a comparison is established between those results. It is found that, contrarily to popular belief, the FE method (FEM) not only reproduces extremely well the results obtained with the standard plane wave method with regards to the eigenvalue analysis (photonic band structure and density of states calculations) but it also allows to study very easily the time-harmonic propagation of electromagnetic fields in finite clusters of arbitrary complexity and, thus, to calculate their transmission coefficients in a simple way. Moreover, the advantages of using this real space method in the context of point defect cluster quality factor calculations are also stressed by comparing the results obtained with this method with those obtained with the FDTD one. As a result of this study, FEM comes out as an stable, robust, rigorous, and reliable tool to study light propagation and confinement in both periodic and aperiodic dielectric photonic crystals and clusters.

Photonic polycrystal in the greenish-white scales of the African longhorn beetleProsopocera lactator(Cerambycidae)

Three-dimensional photonic-crystal grains were found in the scales of the longhorn beetle Prosopocera lactator (Cerambycidae). The local geometric structure can be described as a face-centered-cubic array of spheres, connected by short rods, reminiscent of the "ball-and-stick" models used by solid-state chemists to visualize atomic structures. Based on scanning electron microscopy, x-ray nanotomography, optical measurements, photonic band-structure calculations, and computer simulations of the reflectance, the desaturated greenish coloration is shown to arise from the observed photonic polycrystalline structure. X-ray nanotomography is revealed as a very promising tool for photonic-crystal morphology studies.

Coexistence of guided mode resonance and extraordinary optical transmission in metal/dielectric/metal photonic crystal slab

This study examines the optical properties of metal-dielectric-metal (MDM) photonic crystal slabs (PCS) which consist of a 100-nm freestanding silicon nitride membrane that is sandwiched on either side by a 20-nm-thick gold film with an array of hexagonal tapered holes. It shows the extraordinary optical transmission phenomenon that is generated from the coupling of internal surface plasmon polaritons that are excited by dual gold layers. In addition, the presence of transmission dips can be attributed to the guided mode resonance of silicon nitride based PCS. Numerical calculation of photonic band structure confirms this attribution. This coexistence of extraordinary optical transmission and the guided mode resonance in MDM PCS may contribute to the development of multifunctional optical devices.

Spiral-staircase photonic structures of metallic nanorods

In this paper, we report a thorough investigation of the optical properties of a 3D spiral-staircase structure of metallic nanorods by means of rigorous full-electrodynamic calculations.Weanalyzethenatureofthephotoniceigenmodesofthis crystal in the light of group theory and explain their physical origin. Moreover, we study the change in the polarization state of a wave transmitted through a finite slab of the crystal and reveal the occurrence of strong optical activity effects. Finally, we provide a consistent interpretation of negative refraction,whichcanoccurwithincertainregionsoffrequency and angles of incidence on a specific surface of the crystal, by reference to relevant isofrequency contours obtained from detailed photonic band-structure calculations. It is worth noting that the structure under consideration does not support purely left- and right-circularly polarized (LCP and RCP, respectively) eigenmodes, in contrast to mostly studied chiral metamaterials with, e.g., C4 symmetry. 17‐23,34‐36,38‐40,42,43,45,46

Photoelectrochemical study of TiO2 inverse opals

The optical and photoelectrochemical behaviour of TiO2 inverse opals and TiO2 disordered macroporous structures has been investigated by means of UV-vis spectroscopy, cyclic voltammetry, chronopotentiometry and electrochemical impedance spectroscopy. These techniques, together with photonic band structure calculations, confirmed the better absorption of light of TiO2 inverse opals, due to the slow light phenomenon, and indeed the status of metamaterials for such structures. TiO2 inverse opals and TiO2 disordered macroporous structures are shown to have the same recombination rate of photogenerated charge carriers, but TiO2 inverse opals are characterized by a better reactivity towards O2 which can also contribute to the improvement of the photocatalytic activity observed by several groups.

Photonic Band Structure of Dispersive Metamaterials Formulated as a Hermitian Eigenvalue Problem

We formulate the photonic band structure calculation of any lossless dispersive photonic crystal and optical metamaterial as a Hermitian eigenvalue problem. We further show that the eigenmodes of such lossless systems provide an orthonormal basis, which can be used to rigorously describe the behavior of lossy dispersive systems in general.

2차원 광결정 군속도의 특징

A plane wave expansion method (PWEM) was applied for photonic band structure calculation. We examined zero group velocity modes in photonic crystal. The zero group velocity modes were obtained at the second band along F-K direction. We expanded higher order Brillouin zone (BZ) to find the locations of zero group velocity modes and to investicate their properties. We found twelve locations, inside the first Brillouin zone, where the group velocities became zero. Also, we calculated band structure and group velocity in off-plane configuration.