A simple derivation of quasi-crystalline spectra

Zia, R. K. P.; Dallas, William J.

doi:10.1088/0305-4470/18/7/002

Cited by 156 publications

(38 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The result derived algebraically in the previous section can also be derived geometrically (Zia & Dallas, 1985) by means of an irrational projection from two dimensions into one dimension. Referring to Fig.…”

Section: Projection From Two Dimensions To Onementioning

confidence: 88%

Diffraction patterns from tilings with fivefold symmetry

Prince¹

1987

Acta Cryst Sect A

View full text Add to dashboard Cite

A procedure involving projection from sixdimensional to three-dimensional space to describe objects that give sharp diffraction with fivefold symmetry can be reduced to the easier problem of projection from two dimensions to one dimension. This result is used to derive an explicit formula for the quasilattice contribution to the diffracted intensity for an arbitrary size and shape of the selection region. The predictions of this formula are compared with the electron diffraction patterns obtained from rapidly solidified aluminium-manganese alloys, and it is concluded that the edges of the rhombic faces of the three-dimensional objects from which models for these alloy structures may be constructed is larger than that used in previous analyses by a factor of z 3, where r is the golden mean. It is shown that the quasilattice density is proportional to the volume of the selection region in the complementary threedimensional space into which a lattice point in sixdimensional space must project in order for the point to be included in the direct space; this results in important constraints on the possible structures of these alloys.

show abstract

Section: Projection From Two Dimensions To Onementioning

confidence: 88%

Diffraction patterns from tilings with fivefold symmetry

Prince¹

1987

Acta Cryst Sect A

View full text Add to dashboard Cite

show abstract

“…The spectrum for the projection of a square lattice onto a line was given in a work by Zia and Dallas [11]. Here we treat the most general periodic case-that is-projection of any NPC built upon a periodic 2-D lattice.…”

Section: Discussionmentioning

confidence: 99%

“…The values are just the projection of the set of all reciprocal lattice vectors along the propagation direction . If the inclination of the propagating beams with respect to the lattice primitive vectors is irrational the spectrum is characteristic of a quasi-periodic structure [9], [11]. An example for this scenario is given in Section III-C.…”

Section: Projection-based Qpmmentioning

confidence: 99%

“…Given a specific process (defined by a specific ) find which order can give a solution in which angle? To solve this problem let us define again the phase-matching condition (6) in yet another way (11) A geometric representation of a solution is very illustrative as can be seen in Fig. 5: drawing the newly defined vectors and as an orthogonal basis we can see that the solution is equal to the projection of the diagonal of the rectangle whose edges are the basis vectors.…”

Section: B Possibilities For Phase Matchingmentioning

confidence: 99%

See 1 more Smart Citation

Analysis of Colinear Quasi-Phase-Matching in Nonlinear Photonic Crystals

Bahabad

Voloch²,

Arie³

2008

IEEE J. Quantum Electron.

View full text Add to dashboard Cite

Abstract-Standard quasi-phase-matching (QPM) schemes assume exact momentum relations between the interacting beams, thus necessitating the direct use of a reciprocal lattice vector to satisfy momentum balance. Usage of finite width beams for colinear interactions permits to use a projection of the reciprocal lattice vector along the propagation direction of the beams. We analytically derive this result and exam the new options given by this projection-based QPM for the analysis and design of nonlinear optical devices.

show abstract

“…Another direction is to investigate the possibilities of decorating the quasilattices in physical space (Henley, 1986;Kumar, Sahoo & Athithan, 1986) constructed by means of either the projection or the section method. However, surprisingly little attention has been paid to modified window functions (the projection method) or atomic surfaces (the section method) (Zia & Dallas, 1985;Bak, 1986;DiVincenzo, 1986;Elser, 1986). The purpose of this paper is to investigate which kinds of windows (atomic surfaces) correspond to quasilattices obtained by means of 'simple' deflation rules* and what the corresponding effects are on the squared Fourier transforms and diffraction patterns of these quasilattices.…”

Section: Introductionmentioning

confidence: 99%