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Spaceplates have emerged in the context of nonlocal metasurfaces, enabling the compression of optical systems by minimizing the required empty space between their components. In this work, we design and analyze spaceplates that support resonances with opposite symmetries, operating under the so-called Huygens' condition. Using the temporal coupled-mode theory, we demonstrate that the spatial compression provided by Huygens' spaceplates is twice that of conventional single-resonance counterparts. Additionally, they can support broader operational bandwidths and numerical apertures, facilitating the reduction of chromatic aberrations. Moreover, Huygens' spaceplates maintain nearly full transparency over a wide frequency and angular range, allowing their straightforward cascading for multi-frequency broadband operation. Finally, we propose a physical implementation of a Huygens' spaceplate for optical frequencies based on a photonic crystal slab geometry., Comment: 9 pages, 5 figures

The catalyst-free synthesis of graphene on dielectrics prevents the damage induced by the transfer process. Although challenging, to master this synthesis would boost the integration of graphene on consumer electronics since defects hinder its optoelectronic properties. In this work, the influence of the different surface terminations of c-plane sapphire substrates on the synthesis of graphene via plasma-enhanced chemical vapour deposition (PECVD) is studied. The different terminations of the sapphire surface are controlled by a plasma etching process. A design of experiments (DoE) procedure was carried out to evaluate the major effects governing the etching process of four different parameters: i.e. discharge power, time, pressure and gas employed. In the characterization of the substrate, two sapphire surface terminations were identified and characterized by means of contact angle measurements, being a hydrophilic (hydrophobic) surface the fingerprint of an Al- (OH-) terminated surface, respectively. The defects within the synthesized graphene were analysed by Raman spectroscopy. Notably, we found that the ID/IG ratio decreases for graphene grown on OH-terminated surfaces. Furthermore, two different regimes related to the nature of graphene defects were identified and depending on the sapphire terminated surface are bound either to vacancy or boundary like defects. Finally, studying the density of defects and the crystallite area, as well as their relationship with the sapphire surface termination paves the way for increasing the crystallinity of the synthesized graphene., Comment: 27 pages, 6 figures

In this work we study some topological aspects of function spaces arising in Stieltjes differential calculus. Chief among them are compactness results related to the Ascoli-Arzel\`a and Kolmogorov-Riesz theorems, as well as their applications to Stieltjes-Sobolev spaces and decomposable functions.

In this work, we define the notions of Wronskian and simplified Wronskian for Stieltjes derivatives and study some of their properties in a similar manner to the context of time scales or the usual derivative. Later, we use these tools to investigate second order linear differential equations with Stieltjes derivatives to find linearly independent solutions, as well as to derive the variation of parameters method for problems with $g$-continuous coefficients. This theory is later illustrated with some examples such as the study of the one-dimensional linear Helmholtz equation with piecewise-constant coefficients.

This work revolves around the study of differentiability in the Stieltjes sense of a product of functions. A formula for the first order derivative has been obtained in the past, which is similar to the usual one with some extra terms in its expression. The aim of this paper is to take this behavior into account to study under which conditions we can guarantee the existence of higher order derivatives, while obtaining some other interesting results for the Stieltjes derivative along the way. We also investigate the regularity of the product of two functions.

Invisibility cloaks have become one of the most outstanding developments among the wide range of applications in the field of metamaterials. So far, most efforts in invisibility science have been devoted to achieving practically realizable cloak designs and to improving the effectiveness of these devices in reducing their scattering cross-section (SCS), a scalar quantity accounting for the total electromagnetic energy scattered by an object. In contrast, little attention has been paid to the opposite side of the technology: the development of more efficient techniques for the detection of invisibility devices. For instance, the SCS ignores the phase change introduced by the cloak, as well as the angular dependence of the incident and scattered waves. Here, we propose to take advantage of the smarter way in which diffraction tomography processes all this overlooked information to improve the efficiency in unveiling the presence of invisibility devices. We show that this approach not only results in a considerable sensitivity enhancement in the detection of different kinds of cloaks based on both scattering cancellation and transformation optics, but also enables us to even obtain images depicting the approximate shape and size of the cloak. As a side result, we find that scattering cancellation coatings may be used to enhance the contrast of tomographic images of closely-packed particles. Our work could be easily extended to the detection of sound cloaks., Comment: 10 pages, 11 figures

This work deals with the obtaining of solutions of first and second order Stieltjes differential equations. We define the notions of Stieltjes derivative on the whole domain of the functions involved, provide a notion of n-times continuously Stieltjes-differentiable functions and prove existence and uniqueness results of Stieltjes differential equations in those spaces. We also present the Green's functions associated to the different problems and an application to the Stieltjes harmonic oscillator., Comment: 40 pages, 11 figures

The coherent coupling between a quartz electro-mechanical resonator at room temperature and trapped ions in a 7-tesla Penning trap has been demonstrated for the first time. The signals arising from the coupling remain for integration times in the orders of seconds. From the measurements carried out, we demonstrate that the coupling allows detecting the reduced-cyclotron frequency ($\nu_+$) within times below 10~ms and with an improved resolution compared to conventional electronic detection schemes. A resolving power $\nu_+/\Delta \nu_+=2.4\times10^{7}$ has been reached in single measurements. In this publication we present the first results, emphasizing the novel features of the quartz resonator as fast non-destructive ion-trap detector together with different ways to analyze the data and considering aspects like precision, resolution and sensitivity., Comment: 19 pages, 10 figures

This work is devoted to the mathematical analysis of Stieltjes Bochner spaces and their applications to the resolution of a parabolic equation with Stieltjes time derivative. This novel formulation allows us to study parabolic equations that present impulses at certain times or lapses where the system does not evolve at all and presents an elliptic behavior. We prove several theoretical results related to existence of solution, and propose a full algorithm for its computation, illustrated with some realistic numerical examples related to population dynamics., Comment: Preprint

This work is devoted to the obtaining of a new numerical scheme based in quadrature formulas for the Lebesgue-Stieltjes integral for the approximation of Stieltjes ordinary differential equations. This novel method allows us to numerically approximate models based in Stieltjes ordinary differential equations for which no explicit solution is known. We prove several theoretical results related to the consistency, convergence and stability of the numerical method. We also obtain the explicit solution of the Stieltjes linear ordinary differential equation, and we use it to validate the numerical method. Finally, we present some numerical results that we have obtained for a realistic population model based in a Stieltjes differential equation., Comment: Preprint

In this work we introduce a well-posed mathematical model for the processes involved in the artificial circulation of water, in order to avoid eutrophication phenomena, for instance, in a lake. This novel and general formulation is based on the modified Navier-Stokes equations following the Smagorinsky model of turbulence, and presenting a suitable nonhomogeneous Dirichlet boundary condition. For the analytical study of the problem, we prove several theoretical results related to existence, uniqueness and smoothness for the solution of this recirculation model.

In this work, the artificial recirculation of water is presented and analyzed, from the perspective of the optimal control of partial differential equations, as a tool to prevent eutrophication effects in large waterbodies. A novel formulation of the environmental problem, based on the coupling of nonlinear models for hydrodynamics, water temperature and concentrations of the different species involved in the eutrophication processes, is introduced. After a complete and rigorous analysis of the existence of optimal solutions, a full numerical algorithm for their computation is proposed. Finally, some numerical results for a realistic scenario are shown, in order to prove the efficiency of our approach.

This work is devoted to the analysis and resolution of a well-posed mathematical model for several processes involved in the artificial circulation of water in a large waterbody. This novel formulation couples the convective heat transfer equation with the modified Navier-Stokes system following a Smagorinsky turbulence model, completed with a suitable set of mixed, nonhomogeneous boundary conditions of diffusive, convective and radiative type. We prove several theoretical results related to existence of solution, and propose a full algorithm for its computation, illustrated with some realistic numerical examples.