arXiv subjects

Fu Liu

Publications and source records attributed to Fu Liu.

At least 55 records · Page 3Linked to original sources

arXiv2019

Electromagnetic Aspects of Practical Approaches to Realization of Intelligent Metasurfaces

We thoroughly investigate the electromagnetic response of intelligent functional metasurfaces. We study two distinct designs operating at different frequency regimes, namely, a switch-fabric-based design for GHz frequencies and a graphene-based approach for THz band, and discuss the respective practical design considerations. The performance for tunable perfect absorption applications is assessed in both cases.

physics.app-ph↗

arXiv2019

Tunable Perfect Anomalous Reflection in Metasurfaces with Capacitive Lumped Elements

We demonstrate tunable perfect anomalous reflection with metasurfaces incorporating lumped elements. The tunable capacitance of each element provides continuous control over the local surface reactance, allowing for controlling the evanescent field distribution and efficiently tilting the reflected wavefront away from the specular direction. The performance of the metasurface is evaluated for both TE and TM polarization and for reflection to the first and second diffraction order.

physics.app-ph↗

arXiv2019

Software-Defined Metasurface Paradigm: Concept, Challenges, Prospects

HyperSurfaces (HSFs) are devices whose electromagnetic (EM) behavior is software-driven, i.e., it can be defined programmatically. The key components of this emerging technology are the metasurfaces, artificial layered materials whose EM properties depend on their internal subwavelength structuring. HSFs merge metasurfaces with a network of miniaturized custom electronic controllers, the nanonetwork, in an integrated scalable hardware platform. The nanonetwork receives external programmatic commands expressing the desired end-functionality and appropriately alters the metasurface configuration thus yielding the respective EM behavior for the HSF. In this work, we will present all the components of the HSF paradigm, as well as highlight the underlying challenges and future prospects.

physics.app-ph↗

arXiv2018

The strong maximal rank conjecture and moduli spaces of curves

Building on recent work of the authors, we use degenerations to chains of elliptic curves to prove two cases of the Aprodu-Farkas strong maximal rank conjecture, in genus $22$ and $23$. This constitutes a major step forward in Farkas' program to prove that the moduli spaces of curves of genus $22$ and $23$ are of general type. Our techniques involve a combination of the Eisenbud-Harris theory of limit linear series, and the notion of linked linear series developed by the second author.

math.AG↗

arXiv2018

On positivity of Ehrhart polynomials

Ehrhart discovered that the function that counts the number of lattice points in dilations of an integral polytope is a polynomial. We call the coefficients of this polynomial Ehrhart coefficients, and say a polytope is Ehrhart positive if all Ehrhart coefficients are positive (which is not true for all integral polytopes). The main purpose of this article is to survey interesting families of polytopes that are known to be Ehrhart positive and discuss the reasons from which their Ehrhart positivity follows. We also include examples of polytopes that have negative Ehrhart coefficients and polytopes that are conjectured to be Ehrhart positive, as well as pose a few relevant questions.

math.CO↗

arXiv2018

$h^*$-Polynomials With Roots on the Unit Circle

For an $n$-dimensional lattice simplex $Δ_{(1,\mathbf{q})}$ with vertices given by the standard basis vectors and $-\mathbf{q}$ where $\mathbf{q}$ has positive entries, we investigate when the Ehrhart $h^*$-polynomial for $Δ_{(1,\mathbf{q})}$ factors as a product of geometric series in powers of $z$. Our motivation is a theorem of Rodriguez-Villegas implying that when the $h^*$-polynomial of a lattice polytope $P$ has all roots on the unit circle, then the Ehrhart polynomial of $P$ has positive coefficients. We focus on those $Δ_{(1,\mathbf{q})}$ for which $\mathbf{q}$ has only two or three distinct entries, providing both theoretical results and conjectures/questions motivated by experimental evidence.

math.CO↗

arXiv2018

Stanley's non-Ehrhart-positive order polytopes

We say a polytope is Ehrhart positive if all the coefficients in its Ehrhart polynomial are positive. Answering an Ehrhart positivity question posed on Mathoverflow, Stanley provided an example of a non-Ehrhart-positive order polytope of dimension $21$. Stanley's example comes from a certain family of order polytopes. In this paper, we study the Ehrhart positivity question on this family of polytopes. By giving explicit formulas for the coefficients of the Ehrhart polynomials of these polytopes in terms of Bernolli numbers, we determine the sign of each Ehrhart coefficient of each polytope in the family. As a consequence of our result, we conclude that for any positive integer $d \ge 21,$ there exists an order polytope of dimension $d$ that is not Ehrhart positive, and for any positive integer $\ell$, there exists an order polytope whose Ehrhart polynomial has precisely $\ell$ negative coefficients, which answers a question posed by Hibi. We finish this article by discussing the existence of lower-dimensional order polytopes whose Ehrhart polynomials have a negative coefficient.

math.CO↗

arXiv2018

Smooth polytopes with negative Ehrhart coefficients

We present examples of smooth lattice polytopes in dimensions 3 and higher where each coefficient of their Ehrhart polynomials that can potentially be negative is indeed negative. This answers a question by Bruns. We also discuss Berline-Vergne valuations as a useful tool in proving Ehrhart positivity results.

math.CO↗

arXiv2018

On the Relationship Between Ehrhart Unimodality and Ehrhart Positivity

For a given lattice polytope, two fundamental problems within the field of Ehrhart theory are to (1) determine if its (Ehrhart) $h^\ast$-polynomial is unimodal and (2) to determine if its Ehrhart polynomial has only positive coefficients. The former property of a lattice polytope is known as Ehrhart unimodality and the latter property is known as Ehrhart positivity. These two properties are often simultaneously conjectured to hold for interesting families of lattice polytopes, yet they are typically studied in parallel. As to answer a question posed at the 2017 Introductory Workshop to the MSRI Semester on Geometric and Topological Combinatorics, the purpose of this note is to show that there is no general implication between these two properties in any dimension greater than two. To do so, we investigate these two properties for families of well-studied lattice polytopes, assessing one property where previously only the other had been considered. Consequently, new examples of each phenomena are developed, some of which provide an answer to an open problem in the literature. The well-studied families of lattice polytopes considered include zonotopes, matroid polytopes, simplices of weighted projective spaces, empty lattice simplices, smooth polytopes, and $s$-lecture hall simplices.

math.CO↗

arXiv2018

Intercell Wireless Communication in Software-defined Metasurfaces

Tunable metasurfaces are ultra-thin, artificial electromagnetic components that provide engineered and externally adjustable functionalities. The programmable metasurface, the HyperSurFace, concept consists in integrating controllers within the metasurface that interact locally and communicate globally to obtain a given electromagnetic behaviour. Here, we address the design constraints introduced by both functions accommodated by the programmable metasurface, i.e., the desired metasurface operation and the unit cells wireless communication enabling such programmable functionality. The design process for meeting both sets of specifications is thoroughly discussed. Two scenarios for wireless intercell communication are proposed. The first exploits the metasurface layer itself, while the second employs a dedicated communication layer beneath the metasurface backplane. Complexity and performance trade-offs are highlighted.

physics.app-ph↗

arXiv2018

Realizing spin-dependent gauge field with biaxial metamaterials

Artificial magnetic field in electromagnetism is becoming an emerging way as a robust control of light based on its geometric and topological nature. Other than demonstrating topological photonics properties in the diffractive regime using photonic crystals or arrays of waveguides, it will be of great interest if similar manipulations can be done simply in the long wavelength limit, in which only a few optical parameters can be used to describe the system, making the future optical component design much easier. Here, by designing and fabricating a metamaterial with split dispersion surface, we provide a straight-forward experimental realization of spin-dependent gauge field in the real space using a biaxial material. A "magnetic force bending" for light of desired pseudospins is visualized experimentally by such a gauge field as a manifestation of optical spin Hall effect. Such a demonstration is potentially useful to develop pseudospin optics, topological components and spin-enabled transformation optical devices.

physics.optics↗

arXiv2018

Programmable Metasurfaces: State of the Art and Prospects

Metasurfaces, ultrathin and planar electromagnetic devices with sub-wavelength unit cells, have recently attracted enormous attention for their powerful control over electromagnetic waves, from microwave to visible range. With tunability added to the unit cells, the programmable metasurfaces enable us to benefit from multiple unique functionalities controlled by external stimuli. In this review paper, we will discuss the recent progress in the field of programmable metasurfaces and elaborate on different approaches to realize them, with the tunability from global aspects, to local aspects, and to software-defined metasurfaces.

physics.app-ph↗

arXiv2017

Berline-Vergne valuation and generalized permutohedra

Generalizing a conjecture by De Loera et al., we conjecture that integral generalized permutohedra all have positive Ehrhart coefficients. Berline and Vergne construct a valuation that assigns values to faces of polytopes, which provides a way to write Ehrhart coefficients of a polytope as positive sums of these values. Based on available results, we pose a stronger conjecture: Berline-Vergne's valuation is always positive on permutohedra, which implies our first conjecture. This article proves that our strong conjecture on Berline-Vergne's valuation is true for dimension up to 6, and is true if we restrict to faces of codimension up to 3. In addition to investigating the positivity conjectures, we study the Berline-Vergne's valuation, and show that it is the unique construction for McMullen's formula used to describe number of lattice points in permutohedra under certain symmetry constraints. We also give an equivalent statement to the strong conjecture in terms of mixed valuations.

math.CO↗

arXiv2016

Source illusion devices for flexural Lamb waves using elastic metasurfaces

Metamaterials with the transformation method has greatly promoted the development in achieving invisibility and illusion for various classical waves. However, the requirement of tailor-made bulk materials and extreme constitutive parameters associated to illusion designs hampers its further progress. Inspired by recent demonstrations of metasurfaces in achieving reduced versions of electromagnetic cloaks, we propose and experimentally demonstrate source illusion devices to manipulate flexural waves using metasurfaces. The approach is particularly useful for elastic waves due to the lack of form-invariance in usual transformation methods. We demonstrate metasurfaces for shifting, transforming and splitting a point source with "space-coiling" structures. The effects are found to be broadband and robust against a change of source position, with agreement from numerical simulations and Huygens-Fresnel theory. The proposed approach provides an avenue to generically manipulate guided elastic waves in solids, and is potentially useful for applications such as non-destructive testing, enhanced sensing and imaging.

physics.class-ph↗

arXiv2015

Parity-time symmetry from stacking purely dielectric and magnetic slabs

We show that Parity-time symmetry in matching electric permittivity to magnetic permeability can be established by considering an effective Parity operator involving both mirror symmetry and coupling between electric and magnetic fields. This approach extends the discussion of Parity-time symmetry to the situation with more than one material potential. We show that the band structure of a one-dimensional photonic crystal with alternating purely dielectric and purely magnetic slabs can undergo a phase transition between propagation modes and evanescent modes when the balanced gain/loss parameter is varied. The cross-matching between different material potentials also allows exceptional points of the constitutive matrix to appear in the long wavelength limit where they can be used to construct ultrathin metamaterials with unidirectional reflection.

physics.optics↗

arXiv2014

Topological Photonic Phase in Chiral Hyperbolic Metamaterials

Recently the possibility of achieving one-way backscatter immune transportation of light by mimicking the topological order present within certain solid state systems, such as topological insulators, has received much attention. Thus far however, demonstrations of non-trivial topology in photonics have relied on photonic crystals with precisely engineered lattice structures, periodic on the scale of the operational wavelength and composed of finely tuned, complex materials. Here we propose a novel effective medium approach towards achieving topologically protected photonic surface states robust against disorder on all length scales and for a wide range of material parameters. Remarkably, the non-trivial topology of our metamaterial design results from the Berry curvature arising from the transversality of electromagnetic waves in a homogeneous medium. Our investigation therefore acts to bridge the gap between the advancing field of topological band theory and classical optical phenomena such as the Spin Hall effect of light. The effective medium route to topological phases will pave the way for highly compact one-way transportation of electromagnetic waves in integrated photonic circuits.

physics.optics↗

arXiv2014

Gauge Field Optics with Anisotropic Media

By considering gauge transformations on the macroscopic Maxwell's equations, a two dimensional gauge field, with its pseudo magnetic field in the real space, is identified as tilted anisotropy in the constitutive parameters. We show that optical spin Hall effect and one-way edge states become possible simply by using anisotropic media with broadband response. The proposed gauge field also allows us to design an optical isolator based on the Aharonov-Bohm effect. Our approach will be useful in spoof magneto-optics with arbitrary magnetic fields mimicked by metamaterials with subwavelength unit cells. It also serves as a generic way to design polarization-dependent devices.

physics.optics↗

arXiv2014

A Distributive Lattice Connected with Arithmetic Progressions of Length Three

Let $\mathcal{T}$ be a collection of 3-element subsets $S$ of $\{1, \ldots,n\}$ with the property that if $i<j<k$ and $a<b<c$ are two 3-element subsets in $S$, then there exists an integer sequence $x_1 < x_2 < \cdots < x_n$ such that $x_i, x_j, x_k$ and $x_a, x_b, x_c$ are arithmetic progressions. We determine the number of such collections $\mathcal{T}$ and the number of them of maximum size. These results confirm two conjectures of Noam Elkies.

math.CO↗