Semi-infinite Crystal Research Articles

Influence of the Atomic Structure on the Raman Spectra of Graphite Edges

A study of step edges in graphite with different atomic structures combining Raman spectroscopy and scanning probe microscopy is presented. The orientation of the carbon hexagons with respect to the edge axis, in the so-called armchair or zigzag arrangements, is distinguished spectroscopically by the intensity of a disorder-induced Raman peak. This effect is explained by applying the double resonance theory to a semi-infinite graphite crystal and by considering the one-dimensional character of the defect.

Manifestation of surface phonons in far infrared reflectivity of diamond‐type semiconductors

The coupling of surface phonons with light at (001) surfaces of diamond-structure crystals and its manifestation in far-infrared anisotropy spectra are theoretically studied. We apply the adiabatic bond charge model to describe short-range mechanical interactions together with long-range Coulomb forces and radiation fields, and we solve the corresponding system of coupled equations for the electromagnetic field and the lattice vibrations. We calculate far-infrared normal reflectance spectra of (001) surfaces of semi-infinite diamond-type crystals. In particular, we analyse reflectance spectra for the Si(001) (2 × 1) surface, which exhibit a resonance structure associated with the excitation of surface phonon modes. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Applying a modified plane-wave expansion method to the calculations of transmittivity and reflectivity of a semi-infinite photonic crystal.

We propose a modified plane-wave expansion method to calculate transmittivity and reflectivity of a semi-infinite photonic crystal (PC) with interface. This method is based on an expanded completeness basis, including both the propagation and evanescence modes. We use this approach to deal with two kinds of problems: one is to determine the normal direction of the largest attenuation strength for a semi-infinite PC in the gap frequencies; the other is to calculate the transmittivity and reflectivity of a PC slab. To demonstrate the extensive utilization of our approach, we revisit the same system as studied by Phys. Rev. B 52, 8992 (1995)] and find that our results are in good agreement with ones obtained by Sakoda's paper.

Light propagation in semi-infinite photonic crystals and related waveguide structures

A transfer-matrix method (TMM) that employs the plane-wave expansion of electromagnetic (EM) fields has been developed to handle EM wave propagation in semi-infinite photonic crystal and related waveguide structures. The major aim is to account for wave scattering only at the concerned boundary and to completely remove multiple reflections in the presence of other structural boundaries. It turns out that the scattering problem is closely connected to the eigenmodes of the transfer matrix for the unit cell of the crystal. A natural boundary condition is imposed to describe the asymptotic propagation behavior of scattered EM waves in a region far away from the interface. Theories for a variety of important structures have been systematically set up. These include wave propagation in a semi-infinite photonic crystal, a coated semi-infinite photonic crystal, a heterostructure formed by two different semi-infinite photonic crystals face to face, and a complex sandwiched structure formed by two semi-infinite photonic crystals separated by a general grating slab. In combination with a supercell technique, the developed formulations can also be used to handle photonic crystal waveguide structures. We have applied the developed TMM to two-dimensional photonic crystal and related waveguide structures. The first is the coupling of an external wave into a photonic crystal (photonic crystal waveguide) and the related inverse problems of coupling of a Bloch's wave (guided wave) out of the photonic crystal (photonic crystal waveguide). The second is scattering of a guided wave by a cavity introduced into a photonic crystal waveguide. The developed TMM can help to understand optical properties and design optimal structures of individual functional elements in an optical integrated circuit built in a photonic crystal environment.

All-frequency effective medium theory of a photonic crystal

We consider light propagation in a finite photonic crystal. The transmission and reflection from a one-dimensional system are described in an effective medium theory, which reproduces exactly the results of transfer matrix calculations. We derive simple formulas for the reflection from a semi-infinite crystal, the local density of states in absorbing crystals, and discuss defect modes and negative refraction.

Electromagnetic analysis of ultra-refraction and negative refraction

From the notion of dispersion diagram of a photonic crystal, an analytic study of ultra-refraction and negative refraction phenomena is achieved. In these phenomena, a photonic crystal simulates a material having an optical index that is positive and less than unity (ultra-refraction) or negative (negative refraction). The study is made in both harmonic and time-domain regimes in conditions close to experimental ones: a beam or a light pulse illuminates a semi-infinite photonic crystal. The form of the transmitted light as well as its phase and group velocities are described analytically. Special attention is paid to frequencies very close to the edge of a gap, where an effective index independent of the incidence angle can be defined. The present analysis extends to the case of the so-called ‘left-handed materials’ recently reported in the literature.

Anomalous refractive properties of a two-dimensional photonic band-gap prism

An analysis of the optical response of a triangular-shaped photonic band-gap prism is presented. Numerical simulations have been performed in the framework of multiple-scattering theory, which is applied considering spot illumination to avoid diffraction effects. First of all, refractive properties in the frequency range below the first TM band gap are analyzed and compared with the available experimental data. It validates the approach employed and supports the predictions obtained in the frequency range above the gap. At these high frequencies we found an unusual superprism effect characterized by an angle sensitivity and a frequency sensitivity of the intensity of outgoing beams. We report several representative examples that could be used in device applications. The results are interpreted in terms of the corresponding semi-infinite photonic crystal, through the analysis of the coupling between external radiation and bulk eigenmodes, using the two-dimensional-layer-Korringa-Kohn-Rostoker method. The procedure presented here constitutes a simple but functional alternative to the methods used until now with the same purpose.

Interfaces of semi-infinite smectic liquid crystals and equations of state of infinite smectic stacks of semiflexible manifolds.

In this paper, we first elucidate the classical problem of the elastic free energy of a semi-infinite smectic-A liquid crystals, that fills the semispace above an interface (a boundary smectic layer) of a given shape. For the free energy of this interface, we obtain an effective interface Hamiltonian that takes into account the system discreteness introduced by the layered character of smectic-A liquid crystals. It is thus applicable to both short and long wavelength fluctuations of the interface shape. Next, we use our interface Hamiltonian to develop an efficient approach to the statistical mechanics of stacks of N semiflexible manifolds, such as two-dimensional smectic phases of long semiflexible polymers and three-dimensional lamellar fluid membrane phases. Within our approach, doing the practically interesting thermodynamic limit N--> infinity is reduced to considering a small stack, with just a few interacting manifolds, representing a subsystem of an infinite smectic. This dramatic reduction in the number of degrees of freedom is achieved by treating the first (the last) manifold of the small stack as an interface with the semi-infinite smectic medium below (above) the small stack. We illustrate our approach by considering in detail two-dimensional sterically stabilized smectic liquid crystals of long semiflexible polymers with hard-core repulsion. Smectic bulk (N= infinity ) equation of state and the universal constant characterizing entropic repulsion in these phases are obtained with a high accuracy from numerical simulations of small subsystems with just a few semiflexible polymers.

Systematic design of antireflection coating for semi-infinite one-dimensional photonic crystals using Bloch wave expansion

We present a systematic method for designing a perfect antireflection coating (ARC) for a semi-infinite one-dimensional (1D) photonic crystal (PC) with an arbitrary unit cell. We use Bloch wave expansion and time reversal symmetry, which leads exactly to analytic formulas of structural parameters for the ARC and renormalized Fresnel coefficients of the PC. Surface immittance (admittance and impedance) matching plays an essential role in designing the ARCs of 1D PCs, which is shown together with a practical example.

Field-induced anomalies of the polariton spectrum of a bounded crystal with spatial dispersion: I. A semi-infinite crystal

Conditions are determined under which an external magnetic field leads to a new mechanism of formation of surface TM polaritons in crystals possessing dynamic magnetoelectric interaction and spatial dispersion simultaneously.

Three-beam X-ray diffraction in a semi-infinite perfect crystal.

The approach to multibeam X-ray diffraction by Shen [Acta Cryst. (1986). A42, 525-533] has been carefully re-examined for three-beam Laue-Laue diffraction in a perfect non-absorbing semi-infinite crystal. A modified version of Shen's theory is shown to be equivalent to the Takagi-Taupin formalism. The perturbation of the two-beam integrated power when a third beam is being excited is thus expressed by non-diverging terms. Both Aufhellung and Umweganregung terms are recovered.

Bragg Diffraction of X-Rays by Single Crystals with Large Microdefects

Dynamical wave fields formed in imperfect single crystals by diffusely scattered waves have been considered in the two-beam case of diffraction for the homogeneous distribution of microdefects with large sizes. Wave vectors of the constituent plane waves include the complex dispersion corrections accounting for multiple diffuse scattering processes. The corresponding dynamical diffuse scattering amplitudes and cross-sections in vacuum have been derived for both reflection and transmission directions. The diffuse component of crystal reflectivity has been calculated in the approximation of semiinfinite crystal and has been integrated over exit angles for two microdefect types: spherical clusters and prismatic dislocation loops. The obtained formula for the diffuse reflectivity has been analyzed and compared with the known kinematical one.

Bragg Diffraction of X-Rays by Single Crystals with Large Microdefects

Dynamical wave fields formed in imperfect single crystals by diffusely scattered waves have been considered in the two-beam case of diffraction for the homogeneous distribution of microdefects with large sizes. Wave vectors of the constituent plane waves include the complex dispersion corrections accounting for multiple diffuse scattering processes. The corresponding dynamical diffuse scattering amplitudes and cross-sections in vacuum have been derived for both reflection and transmission directions. The diffuse component of crystal reflectivity has been calculated in the approximation of semiinfinite crystal and has been integrated over exit angles for two microdefect types: spherical clusters and prismatic dislocation loops. The obtained formula for the diffuse reflectivity has been analyzed and compared with the known kinematical one.

Scattering of light on the surfaces of photonic crystals

We present a new method to calculate the scattering of light at the surface of a photonic crystal. The problem is solved in terms of virtual surface-current distributions and the calculation takes full advantage of the existing infinite-space plane-wave expansion method for obtaining the photonic band structure. Working with surface currents makes the calculations less time-consuming by means of reduction of the dimensionality in the problem. The method is tested and illustrated for semi-infinite two-dimensional photonic crystals of small and large dielectric contrast.

Scattering of Light by Photonic Crystals

A new method is developed to calculate the scattering of light at the surface of a photonic crystal. The problem is solved in terms of virtual surface-current distributions and the calculation takes full advantage of the infinite-space plane-wave expansion method for obtaining the photonic band structure. Working with surface currents makes the calculations less time-consuming by means of reduction of the dimensionality in the problem. The method is applied to a semi-infinite dielectric continuum and for semi-infinite two-dimensional photonic crystals of small and large dielectric contrast.

Bragg Diffraction of X-Rays by Single Crystals with Large Microdefects I. Generalized Dynamical Theory

Basic equations of the dynamical scattering theory in momentum space have been considered in the two-beam approximation for single crystals containing randomly distributed microdefects commensurable with extinction length. The amplitudes of coherent and diffusely scattered waves inside the crystal have been found by using the perturbation theory with average and fluctuating parts of crystal polarizability as small parameters. In the complex dispersion corrections to the wave vectors of coherent and diffuse waves, the imaginary parts of which describe the attenuation of these waves due to diffuse scattering, the dynamical effects in diffuse scattering and their dependences on the incidence angle have been taken into account. These corrections also take account of the influence of any multiple diffuse scattering processes. The coherent component of crystal reflectivity has been calculated in the approximation of semiinfinite crystal for an arbitrary diffraction geometry.

Approximate solution of the Takagi-Taupin equations for a semi-infinite crystal in the three-beam Laue-Laue case.

An analytical approximate solution of the Takagi-Taupin equations for a symmetrical three-beam Laue-Laue case in a perfect non-absorbing semi-infinite crystal slab has been obtained. The expression, a second-order expansion, is valid for situations where the effective crystal thickness does not exceed half the actual extinction length and it is shown to be in perfect agreement with the full numerical solution of the fundamental equations.

Surface shear horizontal waves associated with a periodic array of wires deposited on a substrate

The vibrational and electronic spectra of a semi-infinite crystal with a planar surface are modified by the presence of surface inhomogeneities or roughness such as ridges or grooves, quantum wires or tips. We develop a Green's function formalism to investigate the localized and resonant acoustic modes of shear horizontal polarization associated with the surface of a substrate supporting a single and a periodic array of wires. Each material is assumed to be an isotropic elastic medium. The calculation can be applied to an arbitrary choice of the shape and elastic parameters of the wires. The surface modes are obtained as well-defined peaks of the densities of states (DOS). In this paper, we calculate the variation of the density of states associated with the adsorption of a single wire, and the dispersion curves of the surface modes for a periodic array of wires on the flat surface of a substrate. We discuss their behaviors as a function of the elastic parameters and the relationship between resonant modes of the single wire and dispersion curves of the surface modes for a periodic structure.

The features of light reflection from the interface of a semi-infinite nonlinear crystal upon two-photon excitation of biexcitons

The features of light reflection from the interface of a semi-infinite semiconductor are studied upon two-pulse, two-photon excitation of biexcitons from the crystal ground state. It is shown that the reflectivity is a complex multistable function of the amplitudes of the fields of incident pulses. The appearence of solitary closed domains or horn-shaped structures is predicted.

Interface states in photonic crystal heterostructures

We study the interface states formed in a heterostructure composed of two semi-infinite two-dimensional photonic crystals with different filling fractions. Photonic band structure calculations by means of a supercell technique show that localized interface states are absent in this simple heterostructure; they can be created either by lateral lattice slipping or by increasing the interface separation.