Band Structure For Electromagnetic Waves Research Articles

Tunability of photonic band gaps in one- and two-dimensional photonic crystals based on ZnS particles embedded in TiO2 matrix

Using the Maxwell-Garnett theory, the evolution of the refractive index of titanium dioxide (TiO2) doped with zinc sulfide (ZnS) particles is presented. The presence of the nano-objects in the host matrix allows us to obtain a new composite material with tunable optical properties. We find that the filling factor of ZnS nanoparticles greatly alters photonic band gaps (PBGs). We have calculated also the photonic band structure for electromagnetic waves propagating in a structure consisting of ZnS rods covered with the air shell layer in 2D hexagonal and square lattices by the finite difference time domain (FDTD) method. The rods are embedded in the TiO2 background medium with a high dielectric constant. Such photonic lattices present complete photonic band gaps (CPBGs). Our results show that the existence of the air shell layer leads to larger complete photonic gaps. We believe that the present results are significant to increase the possibilities for experimentalists to realize a sizeable and larger CPBG.

Comment on “Photonic bands in two-dimensional microplasma array. I. Theoretical derivation of band structures of electromagnetic waves” [J. Appl. Phys. 101, 073304 (2007)

Recently, theoretical derivation of band structures of electromagnetic waves in two-dimensional microplasma array has been induced by Osamu Sakai et al. [J. Appl. Phys. 101, 073304 (2007)] using a modified plane wave expansion (PWE) method and a frequency-dependent finite difference time–domain (FDTD) method. This report reveals band diagrams with the effects of plasma electron collision frequency, especially focuses on the TE wave by nonmagnetized plasma. Although the band diagrams of TE wave and formulas of calculation look correct at first glance, there are some mistakes in the report which are unfortunately ignored by the authors. The correct formulas of the modified PWE method and FDTD method will be proposed.

The mechanism of the formation of passbands of electromagnetic waves in one-dimensional ordered or disordered structures containing single-negative metamaterials

We propose a mechanism of the formation of band structures of electromagnetic waves in one-dimensional periodic or nonperiodic systems containing single-negative metamaterials. The formation of passbands in these periodic systems is attributed to two factors, i.e., the existence of localized modes and ‘phase’ matching condition. It is completely different from the Bragg scattering. For the wave transmission in disordered systems, the two factors also play an important role. If they are satisfied, there are abnormal transmission behaviors of electromagnetic waves. It implies that the formation of passbands may not depend on periodic structures.

Photonic bands in two-dimensional microplasma arrays. I. Theoretical derivation of band structures of electromagnetic waves

Two theoretical approaches appropriate for two-dimensional plasma photonic crystals reveal dispersions of propagating waves including photonic (electromagnetic) band gaps and multiflatbands. A modified plane-wave expansion method yields dispersions of collisional periodical plasmas, and the complex-value solution of a wave equation by a finite difference method enables us to obtain dispersions with structure effects in an individual microplasma. Periodical plasma arrays form band gaps as well as normal photonic crystals, and multiflatbands are present below the electron plasma frequency in the transverse electric field mode. Electron elastic collisions lower the top frequency of the multiflatbands but have little effect on band gap properties. The spatial gradient of the local dielectric constant resulting from an electron density profile widens the frequency region of the multiflatbands, as demonstrated by the change of surface wave distributions. Propagation properties described in dispersions including band gaps and flatbands agree with experimental observations of microplasma arrays.

Analysis of photonic band gap structure for the design of photonic devices

A detailed analysis, based on Kronig–Penney model and finite-difference time-domain (FDTD) method, is used to explain the air-filling factor effect on the optical properties of defect-free photonic crystals. By the use of the Kronig–Penney model, we calculated the photonic band structure for electromagnetic waves in a structure consisting of a periodic square array of dielectric rods of lattice constant a separated by air holes. Gaps in the resulting band structures are found for waves of both polarisations. We analysed the air-filling factor effect on both polarisations in low and high frequency regions. It is shown that the frequency of the lower TE (transverse-electric) band edge is independent of the air-filling factor in the low frequency region. The opposite behaviour holds for the upper band edge, growing rapidly with the air-filling factor. Using the FDTD we simulated the electric field as the pulse propagates through the structure. The results of both approaches are compared, and the operation characteristics of the measuring air-filling factor device are described. We investigate the optical properties of a single and two defects incorporated in the PC, which can be potentially applied to ultra small surface-emitting-type channel drop filter. It is shown that the frequency and polarisation of the dropped light can be controlled by changing the size and/or shape of the defect. The electric field distribution calculations show that the electric field for a given frequency is located only at the defect, which means that each defect can detect only its corresponding wavelength. To cite this article: F. Ouerghi et al., C. R. Physique 5 (2004).

Fabrication of Two-Dimensional InP Photonic Band-Gap Crystals by Reactive Ion Etching with Inductively Coupled Plasma

We fabricated two-dimensional (2D) InP photonic band-gap crystals by reactive ion etching (RIE) with a SiCl4/Ar inductively coupled plasma (ICP) chemistry, and characterized their reflective characteristics in the optical wavelength region by Fourier-transformed infrared reflection absorption spectrometry (FTIR-RAS). The photonic band-gap crystals consisted of a periodic array of parallel air rods of circular cross section whose intersections with a perpendicular plane form a triangular lattice in InP substrates. Prior to the fabrication of the periodic array of air rods, the photonic band structure for electromagnetic waves was calculated theoretically in the sample structure and was predicted to appear in the optical wavelength region. In RIE with the SiCl4/Ar ICP chemistry, we systematically investigated the InP etch rate and the etch selectivity of InP over SiO2 as functions of various etching parameters, to fabricate deep air rods with a vertical profile. The effect of the N2O addition to the SiCl4/Ar ICP chemistry was investigated and it was revealed that the addition of a small amount of N2O results in an improvement in the vertical profile with a slight increase in the InP etch rate. The InP etch rate and the etch selectivity of InP over SiO2 depended strongly on the SiCl4 flow rate. In FTIR-RAS measurements, characteristic features were observed in the optical wavelength region, depending on the diameter of the air rods. Behaviors of the features were discussed in relation to theoretically calculated densities of states.

Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation

We present an approach that allows calculating photonic band structures of electromagnetic waves propagating in periodic systems containing dispersive and highly absorptive materials characterized by a dielectric function, which is frequency dependent and has a non-negligible imaginary part. This method, which provides a complementary approach to that of the transfer-matrix method, is based on the use of a position-dependent dielectric function and the plane-wave technique. The use of the complex form of the dielectric function transforms Maxwell's equations into a generalized nonlinear eigenvalue problem. At low filling fractions of the dispersive and absorptive component (f\ensuremath{\leqslant}1%), the generalized eigenvalue problem is reduced to a problem of solving sets of simultaneous nonlinear equations which correspond to the diagonal terms of the matrix equation in the plane-wave representation, with the nondiagonal elements taken into account perturbatively. The resulting complex band structure yields, in addition to the dispersion curves, the attenuation of each mode as it propagates through the system. We first consider a model system represented by a one-dimensional (1D) periodic array of alternating layers of vacuum, and a metal characterized by the complex frequency-dependent dielectric function. To calculate the photonic band structure of this system we employ, in addition to the transfer-matrix method and our perturbative plane-wave approach, a standard linearization technique which solves the general nonlinear eigenvalue problem by the diagonalization of an equivalent, enlarged, matrix. We then apply both our perturbative plane-wave approach and the linearization scheme to obtain the photonic band structures of an infinite array of parallel, infinitely long metallic rods whose intersections with a perpendicular plane form a simple square lattice. The interesting features associated with the presence of dissipation displayed by the photonic band structures, such as an asymmetric behavior of the absorption coefficient and the lifetime of each electromagnetic wave for wave vectors near the Brillouin-zone boundaries, the splitting of the lifetimes of degenerate modes, and the different dependences of the real and imaginary parts of the complex photonic band structure on the polarization of the electromagnetic waves in 2D systems, are discussed.

Photonic band structures of two-dimensional systems fabricated from rods of a cubic polar crystal

By the use of the plane-wave method, we calculate the photonic band structure for electromagnetic waves of E and H polarization propagating in a system consisting of an infinite array of identical, infinitely long, parallel cylinders of circular cross section, embedded in vacuum, whose intersections with a perpendicular plane form a square or triangular lattice. The cylinders are fabricated from GaAs, which represents a cubic, polar crystal material, containing two atoms in a primitive unit cell, for which the dielectric function has the form \ensuremath{\epsilon}(\ensuremath{\omega})=${\mathrm{\ensuremath{\epsilon}}}_{\mathrm{\ensuremath{\infty}}}$ (${\mathrm{\ensuremath{\omega}}}_{\mathrm{L}}^{2g}$-${\mathrm{\ensuremath{\omega}}}^{2}$ )/(${\mathrm{\ensuremath{\omega}}}_{\mathrm{T}}^{2g}$-${\mathrm{\ensuremath{\omega}}}^{2}$ ), where ${\mathrm{\ensuremath{\epsilon}}}_{\mathrm{\ensuremath{\infty}}}$ is the optical frequency dielectric constant, while ${\mathrm{\ensuremath{\omega}}}_{\mathrm{L}}$ and ${\mathrm{\ensuremath{\omega}}}_{\mathrm{T}}$ are the frequencies of the longitudinal optical and transverse optical vibration modes of infinite wavelength, respectively. For electromagnetic waves of both polarizations the problem of obtaining the photonic band structure is reduced to the solution of a generalized eigenvalue problem. In comparison with the dispersion curves of electromagnetic waves in vacuum, the photonic band structures are the most significantly affected by the particular form of the dielectric function we have assumed for frequencies in the vicinity of the polariton gap ${\mathrm{\ensuremath{\omega}}}_{\mathrm{T}}$ ${\mathrm{\ensuremath{\omega}}}_{\mathrm{L}}$ . For small filling fractions of the GaAs cylinders the photonic band structure for each polarization, except within the frequency range of the polariton gap, is a slightly perturbed version of the dispersion curves of electromagnetic waves in vacuum. For higher values of the filling fraction the dispersion curves deviate substantially from the dispersion curves of electromagnetic waves in vacuum within a broader frequency range, and in the case of E polarization, the photonic band structures reveal the existence of absolute band gaps. As the most striking feature, the calculated photonic band structures yield additional, nearly dispersionless bands that for waves of both polarizations occur at and below ${\mathrm{\ensuremath{\omega}}}_{\mathrm{T}}$ and in the case of H polarization occur also within the frequency range ${\mathrm{\ensuremath{\omega}}}_{\mathrm{T}}$ ${\mathrm{\ensuremath{\omega}}}_{\mathrm{L}}$ where \ensuremath{\epsilon}(\ensuremath{\omega}) is negative. The results for H polarization and small filling fractions of the GaAs cylinders are reproduced by an approach in which the zeros of a determinant are sought. A possible origin of the flat bands obtained in this polarization by both approaches is discussed.

Photonic band structures of two-dimensional systems containing metallic components.

By use of the plane-wave method, we calculate the photonic band structure for electromagnetic waves of E and H polarization propagating in a system consisting of an infinite array of identical, infinitely long, parallel, metal cylinders of circular cross section, embedded in vacuum, whose intersection with a perpendicular plane forms a simple square or triangular lattice. The dielectric function of the metal from which the cylinders are formed has the simple, free-electron form \ensuremath{\epsilon}(\ensuremath{\omega})=1-(${\mathrm{\ensuremath{\omega}}}_{\mathit{p}}^{2}$/${\mathrm{\ensuremath{\omega}}}^{2}$), where ${\mathrm{\ensuremath{\omega}}}_{\mathit{p}}$ is the plasma frequency of the conduction electrons. For electromagnetic waves of both polarizations, the problem of obtaining the photonic band structure is reduced to the solution of a standard eigenvalue problem despite the frequency dependence of the dielectric constant of the two-dimensional, periodic system. For electromagnetic waves of E polarization the photonic band structure at low filling fractions of the metallic cylinders is a slightly perturbed version of the dispersion curves of electromagnetic waves in vacuum, except for the appearance of a band gap below the lowest frequency band, whose width increases with increasing filling fraction. A band gap between the first and second bands is present in the photonic band structure of the square lattice, but no band gap is found in the band structure of the triangular lattice. In the case of electromagnetic waves of H polarization the photonic band structure at low filling fractions of the metallic cylinders is also, for the most part, a slightly perturbed version of the dispersion curves of electromagnetic waves in vacuum, but possesses additional, nearly dispersionless, bands in the frequency range \ensuremath{\omega}${\mathrm{\ensuremath{\omega}}}_{\mathit{p}}$. A possible origin of these flat bands is described.

Photonic band structure of two-dimensional systems: The triangular lattice

By the use of a position-dependent dielectric constant and the plane-wave method, we have calculated the photonic band structure for electromagnetic waves in a structure consisting of a periodic array of parallel dielectric rods of circular cross section, whose intersections with a perpendicular plane form a triangular lattice. The rods are embedded in a background medium with a different dielectric constant. The electromagnetic waves are assumed to propagate in a plane perpendicular to the rods, and two polarizations of the waves are considered. Absolute gaps in the resulting band structures are found for waves of both polarizations, and the dependence of the widths of these gaps on the ratio of the dielectric constants of the rods and of the background, and on the fraction of the total volume occupied by the rods, is investigated.

Two-dimensional photonic band structures

By the use of a position-dependent dielectric constant we have calculated the photonic band structure for electromagnetic waves in a structure that consists of a periodic array of parallel dielectric rods of circular cross section, whose intersections with a perpendicular plane constitute a square lattice. The rods are embedded in a background medium with a different dielectric constant. Two polarizations of the electromagnetic waves are considered, and they are assumed to propagate in the plane perpendicular to the rods. Absolute gaps in the resulting band structures are found for both polarizations, and the properties of these gaps as functions of the ratio of the dielectric constants of the rods and of the background, and of the fraction of the total volume occupied by the rods, are investigated.