Equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings

Gähler, Franz; Rhyner, Jakob

doi:10.1088/0305-4470/19/2/020

Cited by 149 publications

(83 citation statements)

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“…To do so we use a generalized version of de Bruijn's dual grid method [19,20,21,22] used for the construction of tiling models of quasicrystals. We begin by selecting a set of D 2-dimensional real-space vectors a (j) [j = 1 .…”

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confidence: 99%

Photonic Quasicrystals for Nonlinear Optical Frequency Conversion

2005

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We present a general method for the design of 2-dimensional nonlinear photonic quasicrystals that can be utilized for the simultaneous phase-matching of arbitrary optical frequency-conversion processes. The proposed scheme-based on the generalized dual-grid method that is used for constructing tiling models of quasicrystals-gives complete design flexibility, removing any constraints imposed by previous approaches. As an example we demonstrate the design of a color fan-a nonlinear photonic quasicrystal whose input is a single wave at frequency ω and whose output consists of the second, third, and fourth harmonics of ω, each in a different spatial direction.PACS numbers: 42.65. Ky, 42.70.Mp, 61.44.Br, 42.79.Nv The problem of phase matching in the interaction of light waves in nonlinear dielectrics became immediately evident as the first theories describing such interaction were developed [1]. Put simply, nonlinear interaction is severely constrained in dispersive materials because the interacting photons must conserve their total energy and momentum. Even the slightest wave-vector mismatch appears as an oscillating phase that averages out the outgoing waves, hence the term "phase mismatch". One approach for treating the problem uses the birefringent properties of specific materials and by playing with the polarizations of the interacting waves achieves phase matching [2,3]. A second approach, suggested over 4 decades ago [1,4] and known today as "quasi-phasematching", is to modulate the sign of the relevant component(s) of the nonlinear dielectric tensor at the period of the oscillating mismatched phase thereby undoing the averaging. Quasi-phase-matching has been generalized from simple 1-dimensional periodic modulation [5] to 2-dimensional periodic modulation [6,7,8,9, 10] as well as 1-dimensional quasiperiodic modulation [11,12,13,14], allowing greater flexibility in phase-matching multiple frequency-conversion processes within the same photonic crystal. Here we present the full generalization of the method that enables the design of nonlinear photonic crystals that can simultaneously phase-match any arbitrary set of frequency-conversion processes in any spatial direction. This design flexibility is ideal for the realization of elaborate multi-step cascading effects [15,16], as demonstrated by the color fan example (Fig. 1) at the end of this article.To understand how the method works it is convenient to adopt the view taken in condensed matter systems. Recall that momentum conservation is a direct consequence of having continuous translation symmetry. In crystals, whether periodic or not, continuous translation symmetry is broken, and momentum conservation is re- placed by the less-restrictive conservation law of crystalmomentum. The total momentum of any set of interacting particles in a crystal-whether they are electrons, phonons, or photons-need only be conserved to within a wave vector from the reciprocal lattice of the crystal, giving rise to so-called umklapp processes. Thus, all one needs to do is to ...

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confidence: 99%

Photonic Quasicrystals for Nonlinear Optical Frequency Conversion

2005

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“…A tiling T of R n can be constructed by taking the dual of the multigrid, meaning a point where m hyperplanes intersect in the multigrid space will correspond to a 2m-sided polytope in the tiling space, with opposite sides being parallel [10].…”

Section: The Multigrid Methodsmentioning

confidence: 99%

Methods for Calculating Empires in Quasicrystals

2017

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This paper reviews the empire problem for quasiperiodic tilings and the existing methods for generating the empires of the vertex configurations in quasicrystals, while introducing a new and more efficient method based on the cut-and-project technique. Using Penrose tiling as an example, this method finds the forced tiles with the restrictions in the high dimensional lattice (the mother lattice) that can be cut-and-projected into the lower dimensional quasicrystal. We compare our method to the two existing methods, namely one method that uses the algorithm of the Fibonacci chain to force the Ammann bars in order to find the forced tiles of an empire and the method that follows the work of N.G. de Bruijn on constructing a Penrose tiling as the dual to a pentagrid. This new method is not only conceptually simple and clear, but it also allows us to calculate the empires of the vertex configurations in a defected quasicrystal by reversing the configuration of the quasicrystal to its higher dimensional lattice, where we then apply the restrictions. These advantages may provide a key guiding principle for phason dynamics and an important tool for self error-correction in quasicrystal growth.

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“…It is well-known that a Penrose tiling can be obtained by the projection of a particularly 'cut' slice of the 5-d euclidian lattice onto a 2-d plane D, 4,11,12 and that its diffraction pattern, 13,14,15 therefore, has five-or tenfold symmetry. It is also known that not all lattice points k in Z 5 can be mapped onto vertices of a Penrose tiling; only those points in a particular 'cut' slice whose projections into the 3-dimensional orthogonal space W are inside the window of acceptance, 11,16 contribute.…”

Section: Grids and The 'Cut And Projection Method'mentioning

confidence: 99%

Quasicrystals: Projections of 5-d Lattice into 2 and 3 Dimensions

Au-Yang¹,

Perk²

2006

Differential Geometry and Physics

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We show that generalized Penrose tilings can be obtained by the projection of a cut plane of a 5-dimensional lattice into two dimensions, while 3-d quasiperiodic lattices with overlapping unit cells are its projections into 3d. The frequencies of all possible vertex types in the generalized Penrose tilings, and the frequencies of all possible types of overlapping 3-d unit cells are also given here. The generalized Penrose tilings are found to be nonconvertable to kite and dart patterns, nor can they be described by the overlapping decagons of Gummelt.

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Equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings

Cited by 149 publications

References 10 publications

Photonic Quasicrystals for Nonlinear Optical Frequency Conversion

Photonic Quasicrystals for Nonlinear Optical Frequency Conversion

Methods for Calculating Empires in Quasicrystals

Quasicrystals: Projections of 5-d Lattice into 2 and 3 Dimensions

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