Band Structure Of Photonic Crystals Research Articles

CONVERGENT ADAPTIVE FINITE ELEMENT METHODS FOR PHOTONIC CRYSTAL APPLICATIONS

We prove the convergence of an adaptive finite element method for computing the band structure of two-dimensional periodic photonic crystals with or without compact defects in both the TM and TE polarization cases. These eigenvalue problems involve non-coercive elliptic operators with discontinuous coefficients. The error analysis extends the theory of convergence of adaptive methods for elliptic eigenvalue problems to photonic crystal problems, and in particular deals with various complications which arise essentially from the lack of coercivity of the elliptic operator with discontinuous coefficients. We prove the convergence of the adaptive method in an oscillation-free way and with no extra assumptions on the initial mesh, beside the conformity and shape regularity. Also we present and prove the convergence of an adaptive method to compute efficiently an entire band in the spectrum. This method is guaranteed to converge to the correct global maximum and minimum of the band, which is a very useful piece of information in practice. Our numerical results cover both the cases of periodic structures with and without compact defects.

Band structure of photonic crystal with dispersive and lossy materials using Dirichlet-to-Neumann wave vector eigen equation method

We present the Dirichlet-to-Neumann wave vector eigen equation (DtN-WVEE) method to calculate the band structure of photonic crystal (PhC) with dispersive and lossy materials, and then use finite-difference time-domain method to verify the validity of the results of the DtN-WVEE method. Since the material is lossy with a non-zero imaginary part of epsilon, the wave vector of PhC band structure also has a non-zero imaginary part which results in the reduction of the intensity of the mode. We further present the band structure which gives the relationship among the normalized frequency, the real part of the wave vector, and the imaginary part of the wave vector. To our knowledge, the band structure of lossy PhC has not been well studied yet, and our results can be useful in designing the devices using surface plasmon polariton effect.

Controlling surface plasmon polaritons by a static and/or time-dependent external magnetic field

We have demonstrated numerically by using of Fourier modal method (FMM) that the interface between a metal and a uniformly magnetized two-dimensional photonic crystal fabricated from a transparent dielectric magneto-optical (MO) material possesses a one-way frequency range in which a surface plasmon polariton (SPP) is allowed to propagate only in one direction. The time-reversal symmetry breaking is implied by the MO properties of the photonic crystal material, namely, bismuth iron garnet (BIG), which may be magnetically saturated by fields of the order of tens of milli tesla. The results obtained by FMM have been validated by a theoretical model and a standard plane-wave method that yield separately a nonreciprocal dispersion relation for the SPP and the band structure of the two-dimensional magneto-optical photonic crystal (2D MOPhC), respectively. These spectra represent the key characteristics assuring the functionality of the one-way waveguide associated with the both underlying mechanisms, namely, time-reversal symmetry breaking and a suppression of disorder-induced backscattering. By using a generalized finite-difference time-domain (FDTD) method, which allows studying the propagation of electromagnetic (EM) waves through media with a tensor MO permittivity, we studied transport properties of the one-way waveguide. We examined the influence of specific types of boundary conditions on one-way functionality in the presence of a static external magnetic field and have shown that the SPP can be dynamically controlled by applying a time-dependent magnetic field. By evaluating the Fourier transform of the energy density, we have analyzed the behavior of the field patterns observed in the waveguide in the case of ac magnetic field, and have interpreted new and interesting features associated with the redistribution of the EM field that may offer new mechanisms for dynamical control of SPP flow.

Photonic band structures of one-dimensional photonic crystals doped with plasma

The photonic band structures (PBSs) of oblique incidence propagation in one-dimensional plasma-doped photonic crystals (PCs) are investigated carefully. When the lattice constant of plasma-doped PCs is less than the incident wavelength, the PC becomes anisotropic. Therefore, the dielectric constant of PC is converted into a complex tensor dielectric constant. This determines the PBSs of PCs. In the present paper, one-dimensional PCs are taken as an example to study both normal and absorption PBSs. Using both the effective medium approximation and the transfer matrix method, we can derive the dispersion relation for PCs. The dependence of the plasma filling factor on the effective dielectric constant and PBSs is calculated and discussed.

Epitaxially Regrown GaAs-Based Photonic Crystal Surface-Emitting Laser

A GaAs-based epitaxially regrown photonic crystal surface-emitting laser is proposed and demonstrated at room temperature. The photonic crystal band-structure is mapped through the angular dependence of subthreshold electroluminescence, allowing the photonic crystal coupling coefficients to be determined.

Design of Two-Dimensional Low-Dielectric Photonic Crystal and Its Terahertz Waveguide Application

We have systematically investigated the band structure of a two-dimensional photonic crystal (PC) made of a low permittivity dielectric. We present design of PCs showing photonic band gaps. Subsequently, we propose a terahertz (THz) low-dielectric PC slab waveguide sandwiched between two parallel metal plates. The THz wave propagation on the device was simulated by a finite-difference time-domain method. Taking into account the THz absorption of dielectric and the finite electrical conductivity of the metal, we estimated the propagation loss of a guided mode on the PC slab waveguide. We obtained lower propagation loss than those of typical THz-transmission-lines.

Photonic band structures of two-dimensional magnetized plasma photonic crystals

By using modified plane wave method, photonic band structures of the transverse electric polarization for two types of two-dimensional magnetized plasma photonic crystals are obtained, and influences of the external magnetic field, plasma density, and dielectric materials on the dispersion curves are studied, respectively. Results show that two areas of flat bands appear in the dispersion curves due to the role of external magnetic field, and the higher frequencies of the up and down flat bands are corresponding to the right-circled and left-circled cutoff frequencies, respectively. Adjusting external magnetic field and plasma density can not only control positions of the flat bands, but also can control the location and width of the local gap; increasing relative dielectric constant of the dielectric materials makes omni-direction gaps appear.

Improved fake mode free plane-wave expansion and three planar-slab waveguides method

We compare the extra modes introduced by plane-wave expansion with the two-dimensional (2D) and the three-dimensional (3D) supercell approximation method, and analyze the differences between those two kinds of extra modes. In the band structure of 2D photonic crystal (PhC) using the 2D supercell approximation, the extra modes are created through band folding, and those extra modes can really exist in the 2D PhC. However, in the band structure of a 2D PhC slab using the 3D supercell approximation, the extra modes are mainly generated by the coupling between supercells, and cannot exist in the real 2D PhC slab, thus these extra modes are defined as fake modes in our paper. Through a detailed analysis of the origin of the fake modes and of the energy distribution of the fake modes and of the real modes, we propose an improved fake mode free plane-wave expansion and three planar slab waveguides method which removes the fake modes. This is the first time such a fake mode free band structure is presented. This result can be definitely helpful in designing 3D devices.

An a posteriori error estimator for [formula omitted]-adaptive discontinuous Galerkin methods for computing band gaps in photonic crystals

In this paper we propose and analyze an hp-adaptive discontinuous finite element method for computing the band structure of 2D periodic photonic crystals. The spectrum of a 2D photonic crystal is approximated by computing the discrete spectrum of members of a family of periodic Hermitian eigenvalue problems on the primitive cell, parametrized by a two-dimensional parameter — the quasimomentum. We propose a residual-based error estimator and show that it is reliable and efficient for all eigenvalue problems in the family. In particular we prove that if the error estimator converges to zero, then the distance of the computed eigenfunction from the true eigenspace also converges to zero, and so the computed eigenvalue converges to a true eigenvalue. The results hold for eigenvalues of any multiplicity. We illustrate the benefits of the resulting hp-adaptive method numerically, both for fully periodic crystals and also for crystals with defects.

Nonadiabatic switching of a photonic band structure: Ultrastrong light-matter coupling and slow-down of light

We map out the band structure of a one-dimensional photonic crystal while a 12-fs control pulse activates ultrastrong interaction with quantized electronic transitions in semiconductor quantum wells. Phase-locked multi-terahertz transients trace the buildup of a large vacuum Rabi splitting and an unexpected asymmetric formation of the upper and lower polariton bands. The pronounced flattening of the photonic bands causes a slow-down of the group velocity by one order of magnitude on the time scale of the oscillation period of light.

Rhombohedral photonic crystals by triple-exposure interference lithography: Complete photonic band gap

The band structure of 3D photonic crystal that could be synthesized by means of interference lithography with triple-exposure two-beam interference technique has been investigated. As a result of the geometry optimization the optimal conditions for maximal band gaps with different refractive index contrast have been obtained. The refractive index threshold for gap opening equaled to 2.14 has been predicted for this method of photonic crystals synthesis. This value is close to the refractive index thresholds of the best known structures. The continuous transition between simple cubic, face centered cubic and bulk centered cubic symmetries has been considered.

Graphene-based one-dimensional photonic crystal

A novel type of one-dimensional (1D) photonic crystal formed by an array of periodically located stacks of alternating graphene and dielectric stripes embedded into a background dielectric medium is proposed. The wave equation for the electromagnetic wave propagating in such a structure is solved in the framework of the Kronig–Penney model. The frequency band structure of the 1D graphene-based photonic crystal is obtained analytically as a function of the filling factor and the thickness of the dielectric between the graphene stripes. The photonic frequency corresponding to the electromagnetic wave localized by a defect of the photonic crystal formed by an extra dielectric placed in the position of one stack of alternating graphene and dielectric stripes is obtained.

Pressure, temperature and plasma frequency effects on the band structure of a 1D semiconductor photonic crystal

In this work using the transfer-matrix formalism we study pressure, temperature and plasma frequency effects on the band structure of a 1D semiconductor photonic crystal made of alternating layers of air and GaAs. We have found that the temperature dependence of the photonic band structure is negligible, however, its noticeable changes are due mainly to the variations of the width and the dielectric constant of the layers of GaAs, caused by the applied hydrostatic pressure. On the other hand, by using the Drude's model, we have studied the effects of the hydrostatic pressure by means of the variation of the effective mass and density of the carriers in n-doped GaAs, finding firstly that increasing the amount of n-dopants in GaAs, namely, increasing the plasma frequency, the photonic band structure is shifted to regions of higher frequencies, and secondly the appearance of two regimes of the photonic band structure: one above the plasma frequency with the presence of usual Bragg gaps, and the other, below this frequency, where there are no gaps regularly distributed, with their width diminishing with the increasing of the plasma frequency as well as with the appearance of more bands, but leaving a wide frequency range in the lowest part of the spectrum without accessible photon states. Also, we have found characteristic frequencies in which the dielectric constant equals for different applied pressures, and from which to higher or lower values the photonic band structure inverts its behavior, depending on the value of the applied hydrostatic pressure. We hope this work may be taken into account for the development of new perspectives in the design of new optical devices.

Adaptive finite element methods for computing band gaps in photonic crystals

In this paper we propose and analyse adaptive finite element methods for computing the band structure of 2D periodic photonic crystals. The problem can be reduced to the computation of the discrete spectra of each member of a family of periodic Hermitian eigenvalue problems on a unit cell, parametrised by a two-dimensional parameter - the quasimomentum. These eigenvalue problems involve non-coercive elliptic operators with generally discontinuous coefficients and are solved by adaptive finite elements. We propose an error estimator of residual type and show it is reliable and efficient for each eigenvalue problem in the family. In particular we prove that if the error estimator converges to zero then the distance of the computed eigenfunction from the true eigenspace also converges to zero and the computed eigenvalue converges to a true eigenvalue with double the rate. We also prove that if the distance of a computed sequence of approximate eigenfunctions from the true eigenspace approaches zero, then so must the error estimator. The results hold for eigenvalues of any multiplicity. We illustrate the benefits of the resulting adaptive method in practice, both for fully periodic structures and also for the computation of eigenvalues in the band gap of structures with defect, using the supercell method.

The plane wave expansion method applied to thin plates

The plane wave expansion method refers to a computational technique in electromagnetics to solve Maxwell's equations. This method is popular in the photonic crystal community, because it is possible to obtain the band structure (dispersion-relation) of photonic crystals with specific forms. Since photonic crystals are periodic systems, the method expands all parameters crystal in Fourier series (plane waves). These expansions transform the wave equation in an eigenvalue equation. In this work, we will follow the line of previous investigations where the plane wave method was applied to the classical theory of thin plates. In particular the plane wave expansion method will be applied to the Mindlin equation. This equation describes the out-of-plane vibrations of thin plates in a better way—the classical theory of thin plates.

Dyadic Green's function study of band structures of dispersive photonic crystals

We present here in terms of a dyadic Green's function (DGF) a general description of optical phenomena in photonic crystal (PC) structures, described particularly by frequency-dependent components, assuming that PC structures are decomposed into their relatively simple constituent parts via conductivity tensors. We demonstrate this approach by explicitly calculating the DGFs for electromagnetic waves propagating in the one- and two-dimensional dispersive PCs consisting of a periodic array of identical metallic quantum wells and a periodic square array of identical metallic quantum wires, each embedded in a three-dimensional dispersive medium. By means of the explicit analytic dispersion relations, which result from the frequency poles of the corresponding DGFs, we also calculate the band structures of these dispersive PCs by simple numerical means. Our analysis shows that the band structures calculated from our DGF approach conform well with those calculated from the traditional computational methods.

Improved fake mode free plane wave expansion method

We analyze the origin of the fake modes introduced by the plane wave expansion method with three-dimension (3D) supercell approximation. Through the detailed analysis of the energy distribution of fake modes and real modes, we propose the plane wave expansion-three planar-slab waveguides method to remove the fake modes and obtain the fake mode free band structure of a two-dimensional air hole photonic crystal slab. To the best of our knowledge, this is the first time that such a fake mode free photonic crystal band structure is presented. Our method is also definitely useful in designing other 3D devices.

Band gaps of two-dimensional antiferromagnetic photonic crystal

In an external magnetic field, the band structure of a two-dimensional photonic crystal (PC) composed of parallel antiferromagnetic cylinders in a background dielectric is investigated with a Green’s function method. The cylinders with two resonant frequencies form a square lattice and are characterized by a magnetic permeability tensor. In our numerical calculation, we find that this method allows fast convergence and is available in both the resonant and non-resonant frequency ranges. In the non-resonant range, the PC is similar in band structure to an ordinary dielectric PC. Two electromagnetic band gaps, however, appear in the resonant frequency region, and their frequency positions and widths are governed by the external field. The dependence of the electromagnetic gaps on the cylinder radius also is discussed.

Study on electro-optic properties of two-dimensional PLZT photonic crystal band structure

The band characteristics of two-dimensional (2D) lead lanthanum zirconate titanate (PLZT) photonic crystals are analyzed by finite element method. The electro-optic effect of PLZT can cause the refractive index change when it is imposed by the applied electric field, and the band structure of 2D photonic crystals based on PLZT varies accordingly. The effect of the applied electric field on the structural characteristics of the first and second band gaps in 2D PLZT photonic crystals is analyzed in detail. And the results show that for each band gap, the variations of start wavelength, cut-off wavelength and bandwidth are proportional to quadratic of the electric field.

Band structure of two-dimensional square lattice photonic crystals of circular dispersive metamaterial rods

By virtue of the efficiency of the Dirichlet-to-Neumann map method, the details of the band structure of a two-dimensional square lattice photonic crystal composed of dispersive metamaterial circular rods in air background has been studied. We show that there are two flat bands at the band structure of the system for both H-polarization and E-polarization. These flat bands are created around the magnetic resonance frequency, surface plasmon frequency and magnetic surface plasmon frequency. We realized that the modes with frequencies lying above the resonance frequency behave like resonant cavity modes created in a single metallic cylindrical waveguide. While, due to the relatively large and imaginary refractive index of the metamaterial rods at the frequencies lying below the resonance frequency, the modes are localized modes with negligible penetration into the rods. Moreover, the modes are localized at the interface of the cylindrical metamaterial rods and the air background for the frequencies around the surface plasmon frequency and the magnetic surface plasmon frequency.