Weak Limit Research Articles

Tackling the inverse problem in ellipsometry: analytic expressions for supported coatings with nonuniform refractive index profiles in the thin film and weak contrast limits

Ellipsometry is a powerful tool for the evaluation of the refractive index profile of a film or coating supported on a solid substrate. A well-acknowledged problem, however, is the inverse problem: Given a set of data, under what conditions can a refractive index profile be determined unambiguously? To this end, a series expansion of the Abeles matrix method has been applied to an arbitrary refractive index profile to determine analytic expressions of the ellipsometric ratio ρ. Two types of expressions are found: The thin film limit in which the film thickness L is much less than the wavelength of the incident light beam ( L≪λ ) and the weak contrast limit in which the refractive index of the coating is very near the refractive index of the supporting substrate. In the thin film limit, the first two terms in the series expansion are relatively straight-forward, and they depend on two types of integrals involving the difference between the dielectric profile of the coating and the dielectric constant of the substrate. While higher order terms are possible, they are quite convoluted and do not assist in the inverse problem. In the weak contrast limit, however, the series expansion of ρ depends on the moments of the difference between the dielectric profile of the coating and the dielectric constant of the substrate, allowing an analytic expression that applies to coatings that are even much larger than the incident light beam. The expressions associated with both limits are verified through comparison to the numerical evaluation of ρ with the Abeles matrix method. The results demonstrate that through judicious selection of the substrate refractive index and incident wavelength, conditions can be created that permit critical insights into the inverse problem for either thin coatings or for coatings that are very near the refractive index of the substrate.

Rotation number and eigenvalues of two-component modified Camassa–Holm equations

In this paper, we study the spectral problem y1y2x=−1212λm−12λn12y1y2for the two-component modified Camassa–Holm equation, where m,n are two potentials. By introducing the rotation number ρ(λ) and studying its properties, we prove that for any integer k, the periodic or anti-periodic eigenvalues are the endpoints of the interval λ∈R:ρ(λ)=−k2. Moreover, we prove that as nonlinear functionals of potentials, such eigenvalues are continuous in potentials with respect to the weak topologies in the Lebesgue space L1[0,T]. Finally, we apply the trace formula to give some estimates of the periodic eigenvalues when the L1 norms of potentials are given.

Dephasingless plasma wakefield photon acceleration.

Sandberg and Thomas [Phys. Rev. Lett. 130, 085001 (2023)0031-900710.1103/PhysRevLett.130.085001] proposed a scheme to generate ultrashort, high-energy pulses of XUV photons though dephasingless photon acceleration in a beam-driven plasma wakefield. An ultrashort laser pulse is placed in the plasma wake behind a relativistic electron bunch such that it experiences a comoving negative density gradient and therefore shifts up in frequency. Using a tapered density profile provides phase-matching between driver and witness pulses. In this paper, we give the details of the wakefield solutions and phase-matching conditions used to generate the phase-matching density profile. The short, high-density, and weak driver limits are considered. We show, explicitly, the numerical algorithm used to calculate the density profiles.

Three-dimensional nonlinear ion acoustic waves near critical density in magnetized negative ion plasmas

The nonlinear dynamics of three-dimensional ion acoustic waves near critical density in plasmas consisting of electrons, positive and negative ions in the presence of a uniform external magnetic field are investigated. In the weak nonlinear and dispersive limit, the nonlinear wave is shown to be governed by a Gardner–Zakharov–Kuznetsov (GZK) equation, which is also shown to be integrable in the sense of Painlevé only on the traveling wave frame. The invariant Painlevé analysis provides the analytical solutions in the closed form of solitary, cnoidal, double layer and bipolar structures. The Lyapunov stability analysis shows that the stationary solitary structure is always unstable due to the three-dimensional effect. Finally, the stability of one-dimensional solitary wave has been checked in the transverse direction and the regions of stability (and instability) have been separated in terms of parameters. The results are in qualitative agreement with the observations in magnetized plasmas with negative ions.

Observation of quantum oscillations in the extreme weak anharmonic limit

Observation of quantum oscillations in the extreme weak anharmonic limit

On the Hole Argument and the Physical Interpretation of General Relativity

Einstein presented the Hole Argument against General Covariance, understood as invariance with respect to a change in coordinates, as a consequence of his initial failure to obtain covariant equations that, in the weak static limit, contain Newton’s law. Fortunately, about two years later, Einstein returned to General Covariance, and found these famous equations of gravity. However, the rejection of his Hole Argument carries a totally different vision of space-time. Its substantivalism notion, which is an essential ingredient in Newtonian theory and also in his special theory of relativity, has to be replaced, following Descartes and Leibniz’s relationalism, by a set of “point-coincidences”.

Representative Functions, Variational Convergence and Almost Convexity

We develop a new epi-convergence based on the use of bounded convergent nets on the product topology of the strong topology on the primal space and weak star topology on the dual space of a general real Banach space. We study the propagation of the associated variational convergences through conjugation of convex functions defined on this product space. These results are then applied to the problem of construction of a bigger-conjugate representative function for the recession operator associated with a maximal monotone operator on this real Banach space. This is then used to study the relationship between the recession operator of a maximal monotone operator and the normal–cone operator associated with the closed, convex hull of the domain of that monotone operator. This allows us to show that the strong closure of the domain of any maximal monotone operator is convex in a general real Banach space.

Weak Sensitive Compactness for Linear Operators

Weak Sensitive Compactness for Linear Operators

Strong Solutions and Mild Solutions for Sturm-Liouville Differential Inclusions

Existence results for a Cauchy problem driven by a semilinear differential Sturm-Liouville inclusion are achived by proving, in a preliminary way, an existence theorem for a suitable integral inclusion. In order to obtain this proposition we use a recent fixed point theorem that allows us to work with the weak topology and the De Blasi measure of weak noncompactness. So we avoid requests of compactness on the multivalued terms. Then, by requiring different properties on the map p involved in the Sturm-Liouville inclusion, we are able to establish the existence of both mild solutions and strong ones for the problem examinated. Moreover we focus our attention on the study of controllability for a Cauchy problem governed by a suitable Sturm-Liouville equation. Finally we precise that our results are able to study problems involving a more general version of a semilinear differential Sturm-Liouville inclusion.

Limits of conformal images and conformal images of limits for planar random curves

We address the scaling limits of random curves arising from, e.g., planar lattice models, especially in rough domains. The well-known precompactness conditions of Kemppainen and Smirnov (2017) show that certain crossing probability estimates guarantee the subsequential weak convergence of the random curves in the topology of unparametrized curves, as well as in a topology inherited from the unit disc via conformal maps. We complement this result by proving that proceeding to weak limit commutes with changing topology, i.e., limits of conformal images are conformal images of limits, with minimal boundary regularity assumptions on the domains where the random curves lie. Such rough boundaries are especially interesting if, in the context of multiple random curves, a limit candidate is defined in terms of iterated SLE-type processes with \kappa \in (4, 8) , and one hence needs to study (boundary-touching) curves in domains slit by other random curves.

GLADE: Gravitational Light-Bending Astrometry Dual-Satellite Experiment

Light bending is one of the classical tests of general relativity and is a crucial aspect to be taken into account for accurate assessments of photon propagation. In particular, high-precision astrometry can constrain theoretical models of gravitation in the weak field limit applicable to the Sun neighborhood. We propose a concept for experimental determination of the light deflection close to the Sun in the 10−7 to 10−8 range, in a modern rendition of the 1919 experiment by Dyson, Eddington and Davidson, using formation flying to generate an artificial long-lasting eclipse. The technology is going to be demonstrated by the forthcoming ESA mission PROBA3. The experimental setup includes two units separated by 150 m and aligned to the mm level: an occulter and a small telescope (0.3 m diameter) with an annular field of view covering a region 0∘.7 from the Sun. The design is compatible with a space weather payload, merging several instruments for observation of the solar corona and environment. We discuss the measurement conditions and the expected performance.

Local weak limit of preferential attachment random trees with additive fitness

AbstractWe consider linear preferential attachment trees with additive fitness, where fitness is the random initial vertex attractiveness. We show that when the fitnesses are independent and identically distributed and have positive bounded support, the local weak limit can be constructed using a sequence of mixed Poisson point processes. We also provide a rate of convergence for the total variation distance between the r-neighbourhoods of a uniformly chosen vertex in the preferential attachment tree and the root vertex of the local weak limit. The proof uses a Pólya urn representation of the model, for which we give new estimates for the beta and product beta variables in its construction. As applications, we obtain limiting results and convergence rates for the degrees of the uniformly chosen vertex and its ancestors, where the latter are the vertices that are on the path between the uniformly chosen vertex and the initial vertex.

From continuous to discrete: weak limit of normalized Askey–Wilson measure

From continuous to discrete: weak limit of normalized Askey–Wilson measure

Tailoring performance of perovskite-based tunneling photodetector for portable monitoring of ultraviolet radiation risk

Portable sensing systems have a broad application prospect in the field of personal healthcare via continuous monitoring of surroundings. Ultraviolet (UV) light, deemed to be a main inducement of skin-related cancers, calls for the development of UV detectors with properties of high sensitivity and portability. Herein, we constructed a CsPbCl3 film-based tunneling UV sensor (CTUS) based on the photo-induced tunneling mechanism. By optimizing the thickness of the tunneling layer inserted through an atomic layer deposition process, dark current of the CTUS was suppressed by three orders of magnitude, thus obtaining a salient UV detecting performance (ultralow dark current of 100 pA, high responsivity of 285 mA/W, broad linear dynamic range of 95 dB, fast response time of 2.1 μs, and weak light detection limit of 150 nW), which is comparable to the commercial Si-based PIN-type UV detectors. A UV wristband was further built by merging the CTUS with a self-designed flexible printed circuit board. Co-assisted by a self-developed APP, real-time and continuous monitoring of sun radiation to prevent diseases due to excessive sunburn was demonstrated. The proof-of-concept system will pave the way for portable device development, thus bridging the gap between technology innovation and real life.

On bounded sets in [formula omitted

In this note we study some topological properties of the bounded sets of the locally convex space Ck(X) of all real-valued continuous functions defined on a Tychonoff space X equipped with the compact-open topology as well as of the bounded sets of a (DF)-space. We show that, assuming X is a Lindelöf Čech-complete space, if X is scattered then each bounded set in Ck(X) is a Fréchet-Urysohn space under the weak topology. In the opposite direction, if X is pseudocompact and each bounded set in Ck(X) is Fréchet-Urysohn under the weak topology, then X is k-scattered. We also derive a new condition, related to the closed bounded sets, for a (DF)-space to be quasibarrelled, which we apply to the bounded sets of Ck(X). Finally, we get a sort of Krein-like theorem for Ck(X) when X is a Lindelöf Σ-space.

Second Harmonic Generation Control in 2D Layered Materials: Status and Outlook

AbstractSecond harmonic generation (SHG) as an essential nonlinear optical effect, has gradually shifted its research trend toward the integration and miniaturization of photonic and optoelectronic on‐chip devices in recent years. 2D layered materials (2DLMs) open up a new research paradigm of nonlinear optics due to their large second‐order susceptibility, atomically thin structure, and perfect phase‐matching. However, 2DLMs are facing a bottleneck of weak SHG conversion efficiency limit caused by short light–matter interaction lengths at a nanoscale. Moreover, advances in integrated on‐chip SHG devices based on 2DLMs rely on the continuing development of novel strategies with tunable and efficient SHG responses. Here, this review provides a comprehensive overview of recent progress in exploring highly efficient and tunable SHG responses in 2DLMs. Various modulation and enhancement strategies for the SHG response of 2DLMs are extensively studied and systematically discussed, which can be classified into two categories: symmetry breaking and light‐matter interaction enhancement. Moreover, remaining challenges and outlooks toward further extending and realizing the practical applications of 2DLMs in nonlinear on‐chip integrated devices with SHG modulation and enhancement characteristics are discussed.

Microwave electrometry with Rydberg atoms in a vapor cell using microwave amplitude modulation

We have theoretically and experimentally studied the dispersive signal of the Rydberg atomic electromagnetically-induced transparency (EIT) Autler–Townes (AT) splitting spectra obtained using amplitude modulation of the microwave (MW) electric field. In addition to the two zero-crossing points interval Δf zeros, the dispersion signal has two positive maxima with an interval defined as the shoulder interval Δf sho, which is theoretically expected to be used to measure a much weaker MW electric field. The relationship of the MW field strength E MW and Δf sho is experimentally studied at the MW frequencies of 31.6 GHz and 9.2 GHz respectively. The results show that Δf sho can be used to characterize the much weaker E MW than that of Δf zeros and the traditional EIT–AT splitting interval Δf m; the minimum E MW measured by Δf sho is about 30 times smaller than that by Δf m. As an example, the minimum E MW at 9.2 GHz that can be characterized by Δf sho is 0.056 mV/cm, which is the minimum value characterized by the frequency interval using a vapor cell without adding any auxiliary fields. The proposed method can improve the weak limit and sensitivity of E MW measured by the spectral frequency interval, which is important in the direct measurement of weak E MW.

Weak gravitational lensing and fundamental frequencies of Einstein–Euler–Heisenberg black hole

This paper is suggested to analyze the gravitational weak lensing and fundamental frequencies in the framework of the Einstein–Euler–Heisenberg (EEH) black hole. We compute the deflection angle of light by the EEH black hole in weak field limits. Which represents that the bending of light is a global and topological effect. For this purpose, we deduce the Gaussian curvature and apply the Gauss–Bonnet theorem (GBT). Furthermore, we determine the deflection angle at which light is deflected by a plasma medium. We also look into how an EEH black hole behaves graphically in vacuum and plasma medium. Moreover, we study the fundamental frequencies with three different models of EEH black hole.

On second order differential inclusion driven by quasi-variational–hemivariational inequalities

In this paper we investigate an evolution problem (SDQHVI) which constitutes of the second order differential inclusion driven by a quasi-variational–hemivariational inequality (QHVI) with perturbation operator in Banach spaces. Unlike the existing literature on differential inclusions, on the one hand, the second order differential operator is not assumed to be compact; on the other hand, the solvability of the inequality is proved under non-coercive condition. More precisely, first, it is shown that the solution set of the QHVI with perturbation operator is nonempty, bounded, closed and convex under the KKM theorem and the Minty formulation. Then, an associated multivalued map with the solution set of the QHVI with perturbation operator is introduced, and we prove that it is upper semicontinuous and measurable. Finally, by utilizing Kakutani–Fan–Glicksberg fixed point theorem, a weak topology technique and properties of strong continuous cosine operators, we prove the existence of mild solutions for SDQHVI.

Comparison Theorems for Weak Topologies (3)

Weak topology on a nonempty set X is defined as the smallest or weakest topology on X with respect to which a given (fixed) family of functions on X is continuous. In an effort to construct and analyze strictly stronger weak topologies, only four weak topologies (namely, the Arens topology, the Mackey topology, the weakened topology, and the strong topology) have been compared before now. Also, all these weak topologies were constructed with an eye on only the polars of linear spaces (i.e. with a focus on only linear functionals on linear spaces). To that extent, the study of strictly stronger weak topologies before now can be described as a study of linear topological spaces. Here in Part 3 of our Comparison Theorems for Weak Topologies: We established and clarified the place of our comparison theorems in the context of the weakened topology, the Arens topology, the Mackey topology, and the strong topology that have up to now been compared. We showed that there are many other weak topologies between the four already compared weak topologies—constructible even by the use of polars. In particular, we showed that we can we find a weak topology stronger than the one hitherto known and referred to as the strong topology. Then we showed that if two weak topologies, generated by one family of functions on a set, are strictly comparable, then there exist in a range space two strictly comparable topologies which induce the weak topologies. The rather simplistic view that” The weak topology is Hausdorff” has been held by many authors for very long time. We changed that narrative here, as we stated and proved (with examples) that:” Some weak topologies are Hausdorff while others are not.”