Two-dimensional Periodic Structures Research Articles

Fabrication and sensing application of a silver nanoshell array structure by a micro-flow injection method.

A two-dimensional periodic metallic spherical shell array structure with controllable geometric parameters was fabricated on the target substrate by microsphere templating and magnetron sputtering. The micro-flow injection method was used to prepare a two-dimensional colloidal microsphere template, and reactive ion etching (RIE) was used to change the spherical spacing. The geometric parameters and spectral characteristics of the spherical shell array structure were analyzed with the simulation software FDTD solutions. The nanostructural morphology and optical properties of the samples were characterized by scanning electron microscopy (SEM) and optical spectral analysis (OSA). The refractive index sensing application based on the principle of the local surface plasmon resonance (LSPR) and plasmonic interference was realized. The results show that the spherical shell arrays structure is sensitive to the surrounding environment, the refractive index sensitivity of spherical shell array structure is 527.07 nm per RIU and 922.25 nm per RIU, and the quality factor FOM is 16.5 and 15.3, respectively. The techniques demonstrated can produce large-area periodic nanostructure arrays with ultra-large production in cost-competitive ways. In addition, these properties make them applicable to multiple applications, such as surface plasmon sensors and various optical device.

Efficient method for the solution of Maxwell’s equations for nanostructured materials

The calculation of the electromagnetic field in nanostructured materials and nano-optoelectronic devices, when the wavelength of the incident radiation is comparable with the size of the structural elements, requires the exact solution of Maxwell's equations. In this case, a very promising numerical approach is the spectral element method, which combines the geometric flexibility of finite elements with high precision of spectral methods. In this paper the implementation of the spectral element method based on the Dirichlet-to-Neumann map for solving Maxwell’s equations is discussed. The application of the method for two-dimensional periodic structures, such as diffraction gratings and a metal nanowire array in a dielectric matrix, is demonstrated.

Theoretical investigation of subwavelength structure fabrication based on multi-exposure surface plasmon interference lithography

Abstract We propose a method of fabricating subwavelength structures based on multi-exposure surface plasmon interference lithography. This new nanolithography technique fabricates various subwavelength structures breaking the diffraction limit, which differs from the common method of writing various micro structures by using a two-laser-beam interference with sample rotation. Analysis of the optical field distributions obtained using the proposed nanolithography technique, which was performed using the theory of surface plasmon interference lithography in combination with coordinate matrix transformation, is also provided. Various special subwavelength structures are demonstrated by presenting simulated optical field distributions corresponding to different sample rotations, including two-dimensional lattice structures and periodic quasi-hexagonal structures. These structures are potentially useful in micro- and nano-optics applications.

Polarization-Independent Narrowband Near-Perfect Absorption Based on One-Dimension Embedded Aluminum Grating

In the past, the polarization-insensitive absorption is realized mainly employing the two-dimensional periodic structure of fourfold rotational symmetry, which greatly increases the manufacturing complexity and cost. Here, we present the numerical design of a polarization-independent near-perfect absorber incorporating one-dimensional embedded aluminum grating. The absorption peaks near 99% at 520-nm wavelength for different polarization angles are achieved. The underlying mechanism associated with the resonance is attributed to the magnetic polariton resonance and the cavity-mode resonance for TM and TE polarization, respectively, and further explained by the inductor-capacitor circuit model and the eigen equation of cavity mode. Furthermore, the effects of geometrical parameters of the nano-cavity on the absorption performance are discussed. The proposed structure may provide a new way to achieve polarization-independent absorption with one-dimensional meta-surface, which has broad applications in plasmonic sensors, photo-detectors, and so on.

Miniature Spectroscopes with Two-Dimensional Guided-Mode Resonant Metal Grating Filters Integrated on a Photodiode Array.

A small spectroscope with 25 color sensors was fabricated by combining metamaterial color filters and Si photodiodes. The metamaterial color filters consisted of guided-mode resonant metal gratings with subwavelength two-dimensional periodic structures. Transmittance characteristics of the color filters were designed to obtain peak wavelengths proportional to grating periods. For each color sensor, a peak wavelength of the spectral sensitivity could be tuned in the range of visible wavelengths by adjusting each grating period. By performing spectrum reconstruction using Tikhonov regularization, the spectrum of an incident light was obtained from the signal of photodiodes. Several monochromatic lights were made incident on the fabricated device and the spectral characteristics of the incident light were reconstructed from the output signals obtained from the respective color sensors. The peak wavelengths of the reconstructed spectra were in good agreement with the center wavelengths of the monochromatic lights.

Band gaps for wave propagation in 2-D periodic composite structures incorporating microstructure effects

A new model for determining band gaps for wave propagation in two-dimensional (2-D) periodic composite structures is developed using a modified couple stress theory. The general equation of motion and boundary conditions in the elasto-dynamics of the modified couple stress theory are first derived by a variational formulation based on Hamilton’s principle. The in-plane and anti-plane wave equations incorporating microstructure effects are then obtained explicitly from the general equation of motion. The plane wave expansion method and the Bloch theorem for periodic media are used to solve the in-plane and anti-plane wave equations, which are reduced to an eigenvalue problem in each case. The band gaps are determined from solving the characteristic equation and plotting the resulting eigen-frequencies. The new model recovers the classical elasticity-based model when microstructure effects are not considered. To quantitatively illustrate the newly developed model, a parametric study is conducted for 2-D periodic composite structures containing circular and square inclusions. The numerical results reveal that the microstructure effects on the band gaps are significant only when the unit cell size is small for both the composite structures. In addition, it is found that the volume fraction has a significant effect on the band gap size, and the inclusion shape has a large influence on the band gaps.

Topological protection of photonic mid-gap defect modes

Defect modes in two-dimensional periodic photonic structures have found use in a highly diverse set of optical devices. For example, photonic crystal cavities confine optical modes to subwavelength volumes and can be used for Purcell enhancement of nonlinearity, lasing, and cavity quantum electrodynamics. Photonic crystal fiber defect cores allow for supercontinuum generation and endlessly-single-mode fibers with large cores. However, these modes are notoriously fragile: small changes in the structure can lead to significant detuning of resonance frequency and mode volume. Here, we show that a photonic topological crystalline insulator structure can be used to topologically protect the resonance frequency to be in the middle of the band gap, and therefore minimize the mode volume of a two-dimensional photonic defect mode. We experimentally demonstrate this in a femtosecond-laser-written waveguide array, a geometry akin to a photonic crystal fiber. The topological defect modes are determined by a topological invariant that protects zero-dimensional states (defect modes) embedded in a two-dimensional environment; a novel form of topological protection that has not been previously demonstrated.

Semi-phenomenological effective permittivity approach to metallic periodic structures.

A detailed review of the theory of effective permittivity for one- and two-dimensional periodic structures shows its limited validity for metal-dielectric structures in the visible and near infra-red if the feature dimensions are comparable with the metal skin depth. We propose a phenomenological correction to the static formulae using a realistic assumption for the electric field behavior inside the metal features. This approach allows us to obtain analytical expressions for the effective permittivity in the case when the electric field is not sufficiently homogeneous within the unit cell of the gratings. A comparison with the numerical results of the Fourier modal method demonstrates the validity of the analytical formulae. Additional study is made on the impedance approximation at the outer boundaries of the periodical structure in order to propose analytical formulae for the reflection coefficient that permits better correspondence with the numerical results. The link between the values of effective permittivity and permeability defined as the ratios between the averaged fields, and the metamaterial permittivity and permeability is discussed.

Microwave Photonic Crystal from Spirals

Features of microwave propagation through a two-dimensional periodic structure of cylindrical spiral coils were studied in the frequency range from 8.5 to 12 GHz. The frequency dependence of the effective refractive index nef of the structure is measured. The features of measurements are noted and the structure reflectance is estimated in various regions of the frequency range under study. In the frequency range where nef ~ 0, a surface wave propagating around the structure is detected.

An efficient method for the dynamic responses of periodic structures based on the physical features of the structure and group theory

This paper proposes an efficient numerical method for solving the dynamic responses of one-dimensional and two-dimensional periodic structures. Efficiently solving a system of linear equations is a crucial issue for computation of the dynamic responses of large-scale periodic structures. By using condensation technology and based on the periodic properties of a structure, the scale of the system of linear equations that corresponds to the entire structure is reduced. Based on the properties of the periodic structure, it is strictly proved in mathematics that, within a time step, the external force applied on a local unit cell can cause only the dynamic responses of a limited number of adjacent unit cells. By using this conclusion, the dynamic responses of periodic structures can be converted into the response analysis of some small-scale structures. Then, the dynamic responses of the small-scale structures are converted into the response analysis of the cyclic periodic structures, which can be efficiently solved by using the group theory. Outstanding efficiency and low computational resources are achieved because the dynamic responses for the entire structure are converted into that of some small-scale structures. Numerical examples are presented to illustrate the high efficiency and low computational resources of the proposed method.

Bound states in the continuum on periodic structures surrounded by strong resonances

Bound states in the continuum (BICs) are trapped or guided modes with their frequencies in the frequency intervals of the radiation modes. On periodic structures, a BIC is surrounded by a family of resonant modes with their quality factors approaching infinity. Typically the quality factors are proportional to $1/|\beta - \beta_*|^2$, where $\beta$ and $\beta_*$ are the Bloch wavevectors of the resonant modes and the BIC, respectively. But for some special BICs, the quality factors are proportional to $1/|\beta -\beta_*|^4$. In this paper, a general condition is derived for such special BICs on two-dimensional periodic structures. As a numerical example, we use the general condition to calculate special BICs, which are antisymmetric standing waves, on a periodic array of circular cylinders, and show their dependence on parameters. The special BICs are important for practical applications, because they produce resonances with large quality factors for a very large range of $\beta$.

3D printed two-dimensional periodic structures with tailored in-plane dynamic responses and fracture behaviors

High-performance biological structural materials such as bone and spider silk have evolved soft and stiff components to optimize the load transfer path. This inspired many researchers to achieve engineering composites with significantly improved properties. In this paper, we designed a class of 2D elastomer filled composites with periodic units. The designs of these composites were realized by 3D printing of a soft elastomer and a stiff plastic. The plastic constructed a honeycomb-like mesh, in which the soft elastomer formed isolated inclusions. Variation of the mesh geometries, the elastomer content, and the constituent material properties could tune the in-plane dynamic responses and fracture behaviors. Dynamic mechanical analysis measurements showed two tanδ peaks, corresponding to the elastomer and the plastic. By changing the geometry, a very wide range of storage modulus values (about three orders of magnitudes) could be achieved at room temperature. The fracture strain of the composites was found to increase with the elastomer content, and an obvious brittle-ductile transition was observed. Tough samples showed several stress plateaus and a trumpet-shaped crack profile. Increasing the vertex angles of the rhombus filler geometry was found to result in a brittle-ductile transition. In both experiments and simulations, a forward-backward crack propagation mode was observed, indicating that soft elastomer inclusions stabilize the crack propagation. Compared to staggered composites with stiff inclusions, the current design with soft inclusions showed higher modulus, tensile strength and toughness.

Efficient and accurate method for 2D periodic structures based on the physical features of the transient heat conduction

An efficient and accurate method is developed to solve the transient heat conduction problem in two-dimensional (2D) periodic structures. For a 2D periodic structure, according to the physical features of the transient heat conduction, the periodic property of the structure, and the physical meaning of the matrix exponential, it is demonstrated that the matrix exponential for a reasonable time step is a sparse matrix containing many identical elements. Next, based on the superposition principle of linear systems and the algebraic structure of the matrix exponential, computation of the response of original 2D periodic structure is transformed into computation of the responses of small-scale models with several unit cells. Finally, the precise integration method (PIM) is used to compute the temperature responses of the small-scale models. The proposed method not only inherits the accuracy and stability of the PIM but also achieves significantly improved computational time and storage requirements. A series of numerical examples demonstrate that the proposed method is more efficient than the Crank-Nicholson method and provides highly precise solutions even with a larger time step.

Powerful surface-wave oscillators with one-dimensional and two-dimensional periodic planar structures

Powerful surface-wave oscillators with one-dimensional and two-dimensional periodic planar structures

Prediction of transmission, reflection and absorption coefficients of periodic structures using a hybrid Wave Based – Finite Element unit cell method

This paper presents a hybrid Wave Based Method – Finite Element unit cell method to predict the absorption, reflection and transmission properties of arbitrary, two-dimensional periodic structures. The planar periodic structure, represented by its unit cell combined with Bloch–Floquet periodicity boundary conditions, is modelled within the Finite Element Method, allowing to represent complex geometries and to include any type of physics. The planar periodic structure is coupled to semi-infinite acoustic domains above and/or below, in which the dynamic pressure field is modelled with the Wave Based Method, applying a wave function set that fulfills the Helmholtz equation and satisfies the Sommerfeld radiation condition and the Bloch–Floquet periodicity conditions inherently. The dynamic fields described within both frameworks are coupled using a direct coupling strategy, accounting for the mutual dynamic interactions via a weighted residual formulation. The method explicitly accounts for the interaction between the unit cell and the surrounding acoustic domain, also accounting for higher order periodic waves. The convergence of the method is analysed and its applicability is shown for a variety of problems, proving it to be a useful tool combining the strengths of two methods.

Bound states in the continuum on periodic structures: perturbation theory and robustness.

On periodic structures, a bound state in the continuum (BIC) is a standing or propagating Bloch wave with a frequency in the radiation continuum. Some BICs (e.g., antisymmetric standing waves) are symmetry protected, since they have incompatible symmetry with outgoing waves in the radiation channels. The propagating BICs do not have this symmetry mismatch, but they still crucially depend on the symmetry of the structure. In this Letter, a perturbation theory is developed for propagating BICs on two-dimensional periodic structures. The Letter shows that these BICs are robust against structural perturbations that preserve the symmetry, indicating that these BICs, in fact, are implicitly protected by symmetry.

Domain Decomposition CN-FDTD Method for Analyzing Dispersive Metallic Gratings

In this paper, an efficient domain decomposition (DD) technique is originally introduced into the unconditionally stable Crank–Nicolson finite-difference time-domain (CN-FDTD) method for analyzing the extraordinary optical transmission (EOT) phenomenon of periodic metallic gratings. Being caused by the evanescent waves propagating along the metal-dielectric interface in the visible and near infrared regions, the dispersion of the considered metal in this case is expressed by the Drude model and solved with a generalized auxiliary differential equation (ADE) technique. The periodic boundary condition is applied to the two-dimensional periodic metallic grating structures due to their periodicity. Then, the standard unsplit-field perfectly matched layer is derived as the absorbing boundary condition for CN-FDTD based on the ADE concept. In DD structures, the whole computational domain is divided into several subdomains to reduce the matrix size and save calculating time. Furthermore, the reverse Cuthill–Mckee scheme for the lower–upper decomposition is applied to the subdomain matrices individually to improve the efficiency of DD-CN-FDTD. Finally, two numerical examples are calculated and the physical mechanism of the EOT phenomenon is investigated. The results from the proposed method demonstrate its accuracy and efficiency for the nanostructures.

Generation of high-power Cherenkov superradiance pulses using 2D periodic slow-wave structures based on oversized cylindrical waveguides

We propose to stabilize the azimuthal structure of superradiance pulses using two-dimensional (2D) periodic Bragg structures that simultaneously play the role of a slow-wave system. Analysis performed in the framework of a quasi-optical approach is confirmed by the results of 3D particle-in-cell modeling that demonstrates the possibility of using the proposed system for generating multi-GW nanosecond radiation pulses in the millimeter waveband.

Analytical normalization of resonant states in photonic crystal slabs and periodic arrays of nanoantennas at oblique incidence

We present an analytical formulation for the normalization of resonant states at oblique incidence in one- and two-dimensional periodic structures with top and bottom boundaries to homogeneous space, such as photonic crystal slabs and arrays of nanoantennas. The normalization is validated by comparing the resonant state expansion using one and two resonant states with numerically exact results. The predicted changes of resonance frequency and linewidth due to perturbations of refractive index or geometry can be used to study resonantly enhanced refractive index sensing as well as the influence of disorder. In addition, the normalization is essential for the calculation of the Purcell factor.

Monolayer-graphene-based perfect absorption structures in the near infrared.

Subwavelength perfect optical absorption structures based on monolayer-graphene are analyzed and demonstrated experimentally. The perfect absorption mechanism is a result of critical coupling relating to a guided mode resonance of a low index two-dimensional periodic structure. Peak absorption over 99% at wavelength of 1526.5 nm with full-width at half maximum (FWHM) about 18 nm is demonstrated from a fabricated structure with period of 1230 nm, and the measured results agree well with the simulation results. In addition, the influence of geometrical parameters of the structure and the angular response for oblique incidence are analyzed in detail in the simulation. The demonstrated absorption structure in the presented work has great potential in the design of advanced photo-detectors and modulators.