Bloch Wavenumber Research Articles

Band structures of coupled electromagnetic slow waves

In this paper, a modified transfer matrix method is applied for the analysis of electromagnetic slow wave propagation in Kronig–Penny photonic crystals. Analytic expressions for Bloch wavenumbers in terms of normalized frequency are derived and the asymptotic behaviour of electromagnetic slow wave propagation in these structures is investigated at different regimes. The tight binding method is also used for the analysis of electromagnetic slow wave propagation in Kronig–Penny photonic crystals and analytic expressions are given. It has been shown that forbidden wavenumbers rather than forbidden frequencies can be observed under specific circumstances.

Photonic crystal heterostructure waveguides

Photonic crystals with slightly different band-gaps can be stacked to produce heterostructures. Wave-guiding of electromagnetic radiation in such photonic crystal heterostructures is proposed, and analyzed in terms of an equivalent wave admittance, η, and the lateral Bloch wavenumber, kB, for each layer. A transfer matrix formalism is used in order to derive a generalized eigenvalue equation. Two dimensional photonic crystal heterostructure waveguides are expected to exhibit reduced out-of-plane and reflection losses and further increase the coupling of spontaneous emission into the guided mode as compared to conventional photonic crystal waveguides. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Magneto-coupling effect on surface electron states in a semi-infinite superlattice

A theoretical study, within the effective-mass approximation, on the coupling effects of applied magnetic fields along the growth direction of the superlattice on the surface electron states in a semi-infinite semiconductor superlattice is presented by using an effective-barrier height method. The coupling effect is necessarily considered when the difference of the electron effective masses between different materials cannot be neglected. Our numerical results show that the magneto-coupling effect brings about not only the splitting of the surface electron levels but also the definite dependence of the surface levels and its localization degree on magnetic fields and Landau indices. Our results also indicate that the imaginary component of Bloch wave number can serve as a measure of the localization degree of the surface electron states.

Superfluidity of Bose–Einstein condensate in an optical lattice: Landau–Zener tunnellingand dynamical instability

Superflow of a Bose–Einstein condensate in an optical lattice is represented by a Blochwave, a plane wave with periodic modulation of the amplitude. We review the theoreticalresults of the interaction effects in the energy dispersion of the Bloch waves and in thelinear stability of such waves. For sufficiently strong repulsion between the atoms, thelowest Bloch band develops a loop at the edge of the Brillouin zone, with thedramatic consequence of a finite probability of Landau–Zener tunnelling even in thelimit of a vanishing external force. Superfluidity can exist in the central regionof the Brillouin zone in the presence of a repulsive interaction, beyond whichLandau instability takes place where the system can lower its energy by making atransition into states with smaller Bloch wavenumbers. In the outer part of theregion of Landau instability, the Bloch waves are also dynamically unstable inthe sense that a small initial deviation grows exponentially in time. In the innerregion of Landau instability, a Bloch wave is dynamically stable in the absence ofpersistent external perturbations. Experimental implications of our findings will bediscussed.

COUPLED WAVES ON A PERIODICALLY SUPPORTED TIMOSHENKO BEAM

A mathematical model is presented for the propagation of structural waves on an infinitely long, periodically supported Timoshenko beam. The wave types that can exist on the beam are bending waves with displacements in the horizontal and vertical directions, compressional waves and torsional waves. These waves are affected by the periodic supports in two ways: their dispersion relation spectra show passing and stopping bands, and coupling of the different wave types tends to occur. The model in this paper could represent a railway track where the beam represents the rail and an appropriately chosen support type represents the pad/sleeper/ballast system of a railway track. Hamilton's principle is used to calculate the Green function matrix of the free Timoshenko beam without supports. The supports are incorporated into the model by combining the Green function matrix with the superposition principle. Bloch's theorem is applied to describe the periodicity of the supports. This leads to polynomials with several solutions for the Bloch wave number. These solutions are obtained numerically for different combinations of wave types. Two support types are examined in detail: mass supports and spring supports. More complex support types, such as mass/spring systems, can be incorporated easily into the model.

Nonlinear Density Wave Propagation in Layered Superconducting Plasmas

In the present work the propagation of nonlinear density waves in a superconducting layered structure has been investigated. It is seen that the nonlinear Schrodinger (NLS) equation governs the propagation of density waves in a superconducting plasma. It is shown that in general the NLS equation in this case is modulationally unstable. By using the Kronig-Penney model to depict the layered structure we derive a nonlinear dispersion relation relating the nonlinear analog of the Bloch wave number to the propagation frequency. This nonlinear dispersion relation is numerically investigated.

Helicon solitons in a layered semiconductor plasma via Zakharov equations

In the present work we investigate the propagation of helicon envelope solitons in a layered semiconductor plasma. The nonlinear evolution equations governing the propagation of these envelope solitons is the set of Zakharov equations (which are a more generalized form of the nonlinear Schrödinger equation). The set of equations which have a known envelope soliton solution are derived and the relationship between various parameters entering the system is established. In order to investigate the propagation of helicon envelope solitons in a layered medium we use the standard Kronig - Penney model along with its relevant boundary conditions. These boundary conditions are used for the envelope soliton solution thereby connecting the envelope soliton fields across the layers. This in turn leads to a nonlinear dispersion relation which relates the nonlinear analogue of the Bloch wave number with different parameters. We have numerically investigated the dependence of the nonlinear Bloch wave number on the propagation frequency and have established a propagation band and gap structure for the helicon envelope soliton in a layered semiconductor plasma.

Wave Mixing Effects on a Periodically Ribbed Cylindrical Shell

An examination of the response of a periodically ribbed cylindrical shell shows several effects due to the mixing of fast and slow waves on the shell. In wavenumber-frequency space, this mixing manifests itself in the formation of hybrid modes, for example, mixtures of part shear and part flexural vibration. The mixing is strongest when the phase speeds of the different wave types coincide and so reflect the dispersion curves of the flexural Bloch wavenumber. The dispersion curves may be significantly altered by mixing effects, and even for weak wave mixing the polarization of the fast waves can be changed significantly. This effect greatly alters the coupling of the fast waves to an external fluid.

Time harmonic acoustic Bloch wave propagation in periodic waveguides. Part II. Experiment

The propagation of time harmonic acoustic Bloch waves in periodic waveguides is investigated experimentally and the findings compared with those of a theoretical investigation. The waveguides under study are air-filled rectangular aluminum ducts that are loaded with a periodic array of rigidly terminated rectangular side branches. The two parameters that characterize the Bloch waves, the Bloch wave number and the relative component wave amplitude, are measured and found to compare well with the theory. The band structure that is a well-known characteristic of Bloch wave dispersion is clearly evident in both parameters, and the features of the band structure are as predicted. The similarities and differences that, according to the theory, are allowed to exist between the forward and the backward traveling Bloch waves are likewise reflected in the measurements.

Next-nearest neighbour forces in diatomic superlattices

Results are presented for phonons in a 1D diatomic model superlattice. Each frequency omega corresponds to two Bloch wavenumbers Q1 and Q2. For alpha <0.25, where alpha is the ratio of next-nearest to nearest force constants, the results are essentially a perturbation of those found with nearest-neighbour forces only. For alpha >0.25, however, qualitatively new results are found, and in particular both Q1 and Q2 can be real over some frequency range. Illustrative dispersion curves are presented.

Next-nearest neighbour forces in monatomic superlattices

Results are presented for phonons in a ID lattice model superlattice. Each frequency omega corresponds to two Bloch wavenumbers Q1 and Q2. For alpha >0.25, where alpha is the ratio of next-nearest to nearest forces, both wavenumbers can be real over some frequency range. Dispersion curves are given and discussed for various values of alpha and of the mass ratio m2/m1 between the components of the superlattice.