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Christian Huck

Publications and source records attributed to Christian Huck.

At least 19 recordsLinked to original sources
arXiv2020

Self-Organized Nanorod Arrays for Large-Area Surface-Enhanced Infrared Absorption

Capabilities of highly sensitive surface-enhanced infrared absorption (SEIRA) spectroscopy are demonstrated by exploiting large-area templates ($cm^2$) based on self-organized (SO) nanorod antennas. We engineered highly dense arrays of gold nanorod antennas featuring polarization-sensitive localized plasmon resonances, tunable over a broadband near- and mid-infrared (IR) spectrum, in overlap with the so-called 'functional group' window. We demonstrate polarization-sensitive SEIRA activity, homogeneous over macroscopic areas and stable in time, by exploiting prototype self-assembled monolayers of IR-active octadecanthiol (ODT) molecules. The strong coupling between the plasmonic excitation and molecular stretching modes gives rise to characteristic Fano resonances in SEIRA. The SO engineering of the active hotspots in the arrays allows us to achieve signal amplitude improved up to 5.7%. This figure is competitive to the response of lithographic nanoantennas and is stable when the optical excitation spot varies from the micro- to macroscale, thus enabling highly sensitive SEIRA spectroscopy with cost-effective nanosensor devices.

physics.app-ph↗
arXiv2019

Number-theoretic positive entropy shifts with small centraliser and large normaliser

Higher-dimensional binary shifts of number-theoretic origin with positive topological entropy are considered. We are particularly interested in analysing their symmetries and extended symmetries. They form groups, known as the topological centraliser and normaliser of the shift dynamical system, which are natural topological invariants. Here, our focus is on shift spaces with trivial centralisers, but large normalisers. In particular, we discuss several systems where the normaliser is an infinite extension of the centraliser, including the visible lattice points and the $k$-free integers in some real quadratic number fields.

math.DS↗
arXiv2017

On the logarithmic probability that a random integral ideal is $\mathscr A$-free

This extends a theorem of Davenport and Erdös on sequences of rational integers to sequences of integral ideals in arbitrary number fields $K$. More precisely, we introduce a logarithmic density for sets of integral ideals in $K$ and provide a formula for the logarithmic density of the set of so-called $\mathscr A$-free ideals, i.e. integral ideals that are not multiples of any ideal from a fixed set $\mathscr A$.

math.NT↗
arXiv2015

On weak model sets of extremal density

The theory of regular model sets is highly developed, but does not cover examples such as the visible lattice points, the k-th power-free integers, or related systems. They belong to the class of weak model sets, where the window may have a boundary of positive measure, or even consists of boundary only. The latter phenomena are related to the topological entropy of the corresponding dynamical system and to various other unusual properties. Under a rather natural extremality assumption on the density of the weak model set we establish its pure point diffraction nature. We derive an explicit formula that can be seen as the generalisation of the case of regular model sets. Furthermore, the corresponding natural patch frequency measure is shown to be ergodic. Since weak model sets of extremal density are generic for this measure, one obtains that the dynamical spectrum of the hull is pure point as well.

math.DS↗
arXiv2015

On pattern entropy of weak model sets

We study point sets arising from cut-and-project constructions. An important class is weak model sets, which include squarefree numbers and visible lattice points. For such model sets, we give a non-trivial upper bound on their pattern entropy in terms of the volume of the window boundary in internal space. This proves a conjecture by R.V. Moody.

math.CO↗
arXiv2015

Ergodic properties of visible lattice points

Recently, the dynamical and spectral properties of square-free integers, visible lattice points and various generalisations have received increased attention. One reason is the connection of one-dimensional examples such as $\mathscr B$-free numbers with Sarnak's conjecture on the `randomness' of the Möbius function, another the explicit computability of correlation functions as well as eigenfunctions for these systems together with intrinsic ergodicity properties. Here, we summarise some of the results, with focus on spectral and dynamical aspects, and expand a little on the implications for mathematical diffraction theory.

math.DS↗
arXiv2014

Radial spacing distributions from planar points sets

In this paper, we explore the radial projection method for locally finite point sets and provide numerical examples for different types of order. The main question is whether the method is suitable to analyse order in a quantitive way. Our findings indicate that the answer is affermative. In this context, we also study local visibility conditions for certain types of aperiodic point sets.

math.DS↗
arXiv2014

Discrete tomography: Magic numbers for $N$-fold symmetry

We consider the problem of distinguishing convex subsets of $n$-cyclotomic model sets $\varLambda$ by (discrete parallel) X-rays in prescribed $\varLambda$-directions. In this context, a `magic number' $m_{\varLambda}$ has the property that any two convex subsets of $\varLambda$ can be distinguished by their X-rays in any set of $m_{\varLambda}$ prescribed $\varLambda$-directions. Recent calculations suggest that (with one exception in the case $n=4$) the least possible magic number for $n$-cyclotomic model sets might just be $N+1$, where $N=\operatorname{lcm}(n,2)$.

math.MG↗
arXiv2014

Dynamical properties of $k$-free lattice points

We revisit the visible points of a lattice in Euclidean $n$-space together with their generalisations, the $k$th-power-free points of a lattice, and study the corresponding dynamical system that arises via the closure of the lattice translation orbit. Our analysis extends previous results obtained by Sarnak and by Cellarosi and Sinai for the special case of square-free integers and sheds new light on previous joint work with Peter Pleasants.

math.DS↗
arXiv2013

Experimental Verification of the Spectral Shift between Near- and Far-Field Peak Intensities of Plasmonic Nanoantennas

Theory predicts a distinct spectral shift between the near- and far-field optical responses of plasmonic antennas. Here we combine near-field optical microscopy and far-field spectroscopy of individual infrared-resonant nanoantennas to verify experimentally this spectral shift. Numerical calculations corroborate our experimental results. We furthermore discuss the implications of this effect in surface-enhanced infrared spectroscopy (SEIRS).

physics.optics↗
arXiv2012

Magic numbers in the discrete tomography of cyclotomic model sets

We report recent progress in the problem of distinguishing convex subsets of cyclotomic model sets $\varLambda$ by (discrete parallel) X-rays in prescribed $\varLambda$-directions. It turns out that for any of these model sets $\varLambda$ there exists a `magic number' $m_{\varLambda}$ such that any two convex subsets of $\varLambda$ can be distinguished by their X-rays in any set of $m_{\varLambda}$ prescribed $\varLambda$-directions. In particular, for pentagonal, octagonal, decagonal and dodecagonal model sets, the least possible numbers are in that very order 11, 9, 11 and 13.

math-ph↗
arXiv2011

Solution of a uniqueness problem in the discrete tomography of algebraic Delone sets

We consider algebraic Delone sets $\varLambda$ in the Euclidean plane and address the problem of distinguishing convex subsets of $\varLambda$ by X-rays in prescribed $\varLambda$-directions, i.e., directions parallel to nonzero interpoint vectors of $\varLambda$. Here, an X-ray in direction $u$ of a finite set gives the number of points in the set on each line parallel to $u$. It is shown that for any algebraic Delone set $\varLambda$ there are four prescribed $\varLambda$-directions such that any two convex subsets of $\varLambda$ can be distinguished by the corresponding X-rays. We further prove the existence of a natural number $c_{\varLambda}$ such that any two convex subsets of $\varLambda$ can be distinguished by their X-rays in any set of $c_{\varLambda}$ prescribed $\varLambda$-directions. In particular, this extends a well-known result of Gardner and Gritzmann on the corresponding problem for planar lattices to nonperiodic cases that are relevant in quasicrystallography.

math.MG↗
arXiv2008

On the Existence of $U$-Polygons of Class $c\geq 4$ in Planar Point Sets

For a finite set $U$ of directions in the Euclidean plane, a convex non-degenerate polygon $P$ is called a $U$-polygon if every line parallel to a direction of $U$ that meets a vertex of $P$ also meets another vertex of $P$. We characterize the numbers of edges of $U$-polygons of class $c\geq4$ with all their vertices in certain subsets of the plane and derive explicit results in the case of cyclotomic model sets.

math.MG↗
arXiv2008

A Note on Coincidence Isometries of Modules in Euclidean Space

It is shown that the coincidence isometries of certain modules in Euclidean $n$-space can be decomposed into a product of at most $n$ coincidence reflections defined by their non-zero elements. This generalizes previous results obtained for lattices to situations that are relevant in quasicrystallography.

math.MG↗
arXiv2008

Uniqueness in Discrete Tomography of Delone Sets with Long-Range Order

We address the problem of determining finite subsets of Delone sets $\varLambda\subset\R^d$ with long-range order by $X$-rays in prescribed $\varLambda$-directions, i.e., directions parallel to non-zero interpoint vectors of $\varLambda$. Here, an $X$-ray in direction $u$ of a finite set gives the number of points in the set on each line parallel to $u$. For our main result, we introduce the notion of algebraic Delone sets $\varLambda\subset\R^2$ and derive a sufficient condition for the determination of the convex subsets of these sets by $X$-rays in four prescribed $\varLambda$-directions.

math.MG↗
arXiv2008

Discrete Tomography of F-Type Icosahedral Model Sets

We address the problem of uniquely reconstructing F-type icosahedral quasicrystals from few images produced by quantitative high resolution transmission electron microscopy and explain recent results in the discrete tomography of these sets.

math-ph↗
arXiv2008

Discrete Tomography of Icosahedral Model Sets

The discrete tomography of B-type and F-type icosahedral model sets is investigated, with an emphasis on reconstruction and uniqueness problems. These are motivated by the request of materials science for the unique reconstruction of quasicrystalline structures from a small number of images produced by quantitative high resolution transmission electron microscopy.

math.MG↗