Bloch Wave Analysis Research Articles

Bloch wave analysis of the Eshelby twist contrast around end-on screw dislocations in bcc Mo

In high-resolution electron microscopy (HREM), dislocation core structures are examined by tilting the dislocation end-on along the appropriate zone axis. For end-on screw dislocations diffraction contrast is largely due to surface relaxation in the form of the Eshelby twist. In this paper, simulated, many-beam images of end-on, 1 2 〈1 1 1〉 Mo screw dislocations in a thin TEM foil are presented. The diffraction contrast is found to be rotationally symmetric and of very low value. Bloch waves are used to explain the physical origin of the contrast. Diffraction contrast is, however, significantly enhanced by using an annular aperture which includes only the diffracted beams. Annular aperture contrast is largest for foil thicknesses corresponding to minima in the dark-field pendellösung.

Selection rules for Bloch wave scattering for HREM imaging of imperfect crystals along symmetry axes

A Bloch wave analysis is used to investigate high-resolution electron microscope (HREM) imaging of crystals containing atomic displacements due to strain. In the absence of interband scattering, the shifts of peaks and troughs in the image will correspond to the displacements of the atoms in the exit surface. Interband scattering will shift the image peaks away from the actual atom positions and modify the apparent magnitude of the displacement identified by the observed image peak positions. By considering the case of seven-beam imaging of a cubic crystal aligned along a ⟨111⟩ axis, it is shown that the symmetry of the Bloch waves leads to selection rules for the interband scattering, similar to those seen for dipole electron excitations in atoms. It is also shown that, to first order, no intraband scattering can occur.

Multilayer Josephson Junctions as a Multiple Quantum Well Structure

The macroscopic quantum model in conjunction with the transfer matrix method and Bloch wave analysis have been used to compute the energy dispersion equation for Cooper pairs in finite superconducting quantum well (SQW) structures. The energy dispersion equation has been numerically solved by the Cauchy Integration method to obtain the energy levels of quantum states in SQW. The possibility of energy levels and subbands formation for Cooper pairs within the bulk energy gap has been shown. Energy gap squeezing has been derived when the Bragg condition for Bloch wave number is satisfied.

A Bloch wave analysis of optical sectioning in aberration-corrected STEM

The reduction in the focal depth of field that occurs through the use of larger apertures in aberration-corrected STEM allows three-dimensional information to be retrieved by optical depth sectioning. This paper explores depth sectioning in zone-axis crystals using Bloch wave calculations. By decomposing the calculation into the contribution from individual states and from individual partial plane waves in the convergent cone of illumination, we explain the form of the electron intensity in the crystal as a function of depth. Two separate effects are found that can cause the intensity maximum to deviate from that of the expected defocus value. Firstly it is found that the unbound, high angle excited states give rise to a behaviour similar to that of the probe focusing in the vacuum, but with a prefocusing effect due to the lensing effect of the potential of the atomic column. Superimposed upon this prefocused peak is an oscillation due to interference between the channelling 1s state and the rest of the wavefunction. This oscillation can actually prevent an intensity maximum being formed at certain depths in the crystal, and will complicate the interpretation of optical sectioning data.

Justification of the nonlinear Schrödinger equation in spatially periodic media

The dynamics of the envelopes of spatially and temporarily oscillating wave packets advancing in spatially periodic media can approximately be described by solutions of a Nonlinear Schrodinger equation. Here we prove estimates for the error made by this formal approximation using Bloch wave analysis, normal form transformations, and Gronwall’s inequality.

Failure Surfaces for Finitely Strained Two-Phase Periodic Solids Under General In-Plane Loading

For ductile solids with periodic microstructures (e.g., honeycombs, fiber-reinforced composites, cellular solids) which are loaded primarily in compression, their ultimate failure is related to the onset of a buckling mode. Consequently, for periodic solids of infinite extent, one can define as the onset of failure the first occurrence of a bifurcation in the fundamental solution, for which all cells deform identically. By following all possible loading paths in strain or stress space, one can construct onset-of-failure surfaces for finitely strained, rate-independent solids with arbitrary microstructures. The calculations required are based on a Bloch wave analysis on the deformed unit cell. The presentation of the general theory is followed by the description of a numerical algorithm which reduces the size of stability matrices by an order of magnitude, thus improving the computational efficiency for the case of continuum unit cells. The theory is subsequently applied to porous and particle-reinforced hyperelastic solids with circular inclusions of variable stiffness. The corresponding failure surfaces in strain-space, the wavelength of the instabilities, and their dependence on micro-geometry and macroscopic loading conditions are presented and discussed.

HAADF-STEM imaging with sub-angstrom probes: a full Bloch wave analysis.

A full coherent Bloch wave calculation is presented to investigate high-angle annular dark-field image formation for sub-angstrom probes in scanning transmission electron microscopy (STEM). With increasing illumination angle, the contribution of the 1s bound state increases to a maximum at an optimum probe angle, after which we find increasing contributions from high-angle plane wave states around the periphery of the objective aperture. Examination of image contributions from different depths within a crystal shows an oscillatory behavior due to the beating between 1s and non-1s states. The oscillation period reduces with decreasing probe size, while the relative contribution from a specific depth increases. This signifies a changeover from a projection mode of imaging to a depth-slicing mode of imaging. This new mode appears capable of resolving three-dimensional atomic structures in future generation aberration-corrected STEM.

Segmented Transmission-Line Electroabsorption Modulators

We present segmented transmission-line (TML) electroabsorption modulators (EAMs) with characteristic impedance close to 50 /spl Omega/. The segmented TML approach allows us to design a traveling-wave EAM with 50 /spl Omega/ impedance and very high bandwidth. The devices show low return loss (<-15 dB) and excellent frequency response up to 50 GHz, and exhibit a maximum model-extrapolated 3 dBe bandwidth (BW) of 90 GHz. An effective modeling tool based on Bloch-wave analysis is derived. Design considerations and TML properties for periodic TML-EAMs are discussed.

Periodically slotted dielectrically filled parallel-plate waveguide as a leaky-wave antenna: E-polarization case

The analysis of a periodically slotted dielectrically filled parallel-plate waveguide as a leaky-wave antenna for infinite and finite periodic structures for the E-polarization case is discussed. For the analysis of the infinite periodic structure, an integral equation in which Floquet's modes and the strip electric current satisfying the edge condition are used and Bloch wave analysis using the single-slot equivalent circuit parameters are employed. An integro-differential equation with moment-method and transmission-line analysis with equivalent circuit representation are used to analyze the finite N-periodic structure. Results of interest such as complex propagation constant and the radiation pattern for the finite slot number case are observed to be in good agreement with those of the infinite geometry case for the appropriately chosen values of slot numbers, slot width, period, and waveguide height.

A Two-Dimensional Bloch-Wave Method for Dynamical RHEED Calculations

A new two-dimensional Bloch-wave method is developed for dynamical reflection high-energy electron diffraction (RHEED) calculations. In this two-dimensional Bloch-wave analysis, the traverse energy and the wave field for each Bloch wave are calculated by a step-by-step R-matrix method. The accuracy of the method is critically examined in comparison with Ichimiya's method on the experimental rocking curve of an Si(001)–2 × 1 surface. It is demonstrated that two-dimensional Bloch-wave analysis can be used to elucidate the mechanism causing the peaks in rocking curves.

Improvement of single-mode gain margin in gain-coupled DFB lasers

Using the Bloch-wave analysis, this paper investigates the effect of the gain grating on the single-mode condition in DFB lasers. Various factors affecting the threshold gain of gain-coupled DFB lasers are analyzed in some detail. It is shown for the first time that unequal section lengths in the gain grating can have a significant effect on the single-mode gain margin of gain-coupled DFB lasers, especially when the linewidth enhancement factor /spl alpha//sub M/ is large, because the long and shortwavelength Bloch waves are in phase and in antiphase with the index grating of DFB lasers, respectively.

Generalized Bloch wave analysis for fiber and waveguide gratings

We have developed a generalized Bloch wave approach for the analysis of aperiodic gratings. This method yields both a macroscopic (i.e., reflection or transmission coefficient) as well as a microscopic (i.e., dispersion diagram and microstructure of the propagating internal field) characterization of fiber and waveguide aperiodic gratings.

Form birefringence dispersion in periodic layered media

We analyze the form birefringence dispersion of periodic dielectric layered media. Using Bloch wave analysis, we investigate the shape of the normal surface and its dependence on wavelength. Our higher-order uniaxial approximation gives an explicit expression for the additional dispersion of the form birefringence that is due to the finite layer thickness.

Inversion of electron scattering intensity for crystal structure analysis

The validity domain of a recently proposed iterative scheme for the inversion of the wavefunction of fast electrons after dynamical diffraction in a crystal, which is based on reversal of the multislice algorithm, is investigated by Bloch wave analysis. It is shown that the scheme fails when the thickness of the crystal is such that the scattering is sufficiently strong for dominant Bloch wave structure to appear in the wavefunction. The scheme could still be useful, however, in extending the range of interpretable resolution and specimen thickness for which electron scattering can be applied to crystal structure analysis.

Bloch-wave analysis of periodic surface contrast in zone-axis images

The origins of a particular type of dark-field contrast, found to give black-to-white distinction between etch-pits from different (top or bottom) crystal surfaces, are examined using multi-slice image simulations for MgO at the [001] zone-axis setting. The phenomenon is found to be thickness and voltage dependent, and to be associated with the periodically occuring enhancement of dark-field images near thickness of n + 1 2 extinction distances. In addition, the visibility of surface-epitaxed layers is examined. As a result we find that, whereas single-atomic-layers can be imaged using symmetrical dark-field optics, at least five times this thickness of light, (MgO) atoms is required to obtain visibility under standard electron (whole-field) imaging conditions.

Electron diffraction measurements on chromium oxide

Abstract Bloch-wave analysis of partial waves formed by 300 keV electrons in chromium oxide illustrates how an absorptive potential affects convergent-beam contrast in higher-order Laue zones (HOLZs). Mean square displacements 〈u 2〉 of chromium and oxygen atoms are measured from intensities in the series of HOLZ lines containing information on excitation and absorption of each partial wave. Data obtained at 110 and 300K implies a strain contribution to 〈u 2〉 for chromium. Correlation between experiment and theory for HOLZ, zeroth-order Laue zone convergent-beam electron diffraction and large-angle convergent-beam electron diffraction measurements from crystals 74-230 nm thick are reported.

Bloch Wave Analysis of Dispersion and Pulse Propagation in Pure Distributed Feedback Structures

Abstract The excitation and behaviour of monochromatic and pulsed optical Bloch waves in pure distributed feed-back structures are discussed and analysed. The Bloch wave approach, based on a detailed knowledge of the natural optical modes of the periodic structure, is complementary to the more commonly used coupled-wave approach. The inter-relationship between dispersion, field micro-structure and group velocity is discussed, and the effects of group-velocity and higher-order dispersion on pulse propagation treated. Questions about the usefulness of DFB structures for dispersion correction and soliton formation are addressed.

On the uncoupling of surface superlattice reflections in TED analysis of reconstructed surfaces

A Bloch-wave analysis is made of the problem of uncoupling surface superlattice reflections from fundamental reflections in transmission electron diffraction (TED) analysis of reconstructed surfaces. This uncoupling problem is proved to be of crucial importance in determining the structure of reconstructed surfaces, for example the Si(111) 7 × 7 surface [Takayanagi, Tanishiro, Takahashi & Takahashi (1985). Vac. Sci. Technol. A3, 1502–1506; (1985). Surf Sci. 164, 367–392]. It is found that a complete uncoupling, weak coupling and sometimes strong coupling between the bulk scattering and surface superlattice scattering are all possible depending on the diffraction conditions. For a kinematical analysis of reconstructed surfaces to be valid, a weak coupling or a complete uncoupling condition must be realised. General rules for choosing the appropriate diffraction conditions are given.

Kinematic Approximations in TED and RHEED Analysis of Reconstructed Surfaces

In recent years there has been a trend in the structure determination of reconstructed surfaces to use high energy electron diffraction techniques, and to employ a kinematic approximation in analyzing the intensities of surface superlattice reflections. Experimentally this is motivated by the great success of the determination of the dimer adatom stacking fault (DAS) structure of the Si(111) 7 × 7 reconstructed surface.While in the case of transmission electron diffraction (TED) the validity of the kinematic approximation has been examined by using multislice calculations for Si and certain incident beam directions, far less has been done in the reflection high energy electron diffraction (RHEED) case. In this paper we aim to provide a thorough Bloch wave analysis of the various diffraction processes involved, and to set criteria on the validity for the kinematic analysis of the intensities of the surface superlattice reflections.The validity of the kinematic analysis, being common to both the TED and RHEED case, relies primarily on two underlying observations, namely (l)the surface superlattice scattering in the selvedge is kinematically dominating, and (2)the superlattice diffracted beams are uncoupled from the fundamental diffracted beams within the bulk.

High-resolution incoherent imaging of crystals.

A Bloch-wave analysis shows how a signal dependent on the electron intensity at the atom sites can be used to form an incoherent image of a crystal structure. To a good approximation the image is given by a convolution of a compositionally sensitive object function with an appropriate resolution function, and as such can be predicted and interpreted intuitively. Information on a scale below the resolution limit can be interpreted by deconvolution.