Search

In this paper we report a few numerical tests by using a slight extension of the Matlab code fhbvm in [8], implementing Fractional HBVMs, a recently introduced class of numerical methods for solving Initial Value Problems of Fractional Differential Equations (FDE-IVPs). The reported experiments are aimed to give evidence of its effectiveness., Comment: 10 pages, 4 figures, minor typos fixed

In this paper, a class of high-order methods to numerically solve Functional Differential Equations with Piecewise Continuous Arguments (FDEPCAs) is discussed. The framework stems from the expansion of the vector field associated with the reference differential equation along the shifted and scaled Legendre polynomial orthonormal basis, working on a suitable extension of Hamiltonian Boundary Value Methods. Within the design of the methods, a proper generalization of the perturbation results coming from the field of ordinary differential equations is considered, with the aim of handling the case of FDEPCAs. The error analysis of the devised family of methods is performed, while a few numerical tests on Hamiltonian FDEPCAs are provided, to give evidence to the theoretical findings and show the effectiveness of the obtained resolution strategy., Comment: 29 pages, 23 figures

In this paper we describe the efficient numerical implementation of Fractional HBVMs, a class of methods recently introduced for solving systems of fractional differential equations. The reported arguments are implemented in the Matlab code fhbvm, which is made available on the web. An extensive experimentation of the code is reported, to give evidence of its effectiveness., Comment: 26 pages, 6 figures

This study investigates the impact of monochromatic lighting on visual acuity (VA) and contrast sensitivity (CS). Traditional assessments of VA and CS are typically conducted under the illumination of ``white'' light, but variations in color temperature can influence outcomes. Utilizing data from an exhibition by Olafur Eliasson, where a room was illuminated with low-pressure sodium lamps, creating an almost monochromatic yellow light, we compared visual assessments in the yellow room with conventional lighting in a white room. For VA, the results show no significant differences between the two lighting conditions, while for CS, a more nuanced situation is observed. The bias in CS measurements is clinically relevant, and the p-value suggests that further investigation with a larger, more diverse sample may be worthwhile. Despite limitations, such as higher illumination conditions than standard protocols, the unique ``laboratory'' offered by the exhibition facilitated measurements not easily achievable in a traditional setting., Comment: Submitted to Nuovo Cimento, special number for "SIF simposio Optometria": this version is before referee comments

In this paper we consider the numerical solution of fractional terminal value problems (FDE-TVPs). In particular, the proposed procedure uses a Newton-type iteration which is particularly efficient when coupled with a recently-introduced step-by-step procedure for solving fractional initial value problems (FDE-IVPs), able to produce spectrally accurate solutions of FDE problems. Some numerical tests are reported to make evidence of its effectiveness., Comment: 23 pages, 4 figures, 7 tables, one typo fixed

Hyperuniform disordered photonic materials have recently been shown to display large, complete photonic band gaps and isotropic optical properties, and are emerging as strong candidates for a plethora of optoelectronic applications, making them competitive with many of their periodic and quasiperiodic counterparts. In this work, high quality factor optical cavities in hyperuniform disordered architectures are fabricated through semiconductor slabs and experimentally addressed by scanning near-field optical microscopy. The wide range of confined cavity modes that we detect arise from carefully designed local modifications of the dielectric structure. Previous works on hyperuniform disordered photonic systems have previously identified several Anderson localized states spectrally located at the PBG edges with relatively high quality factors. In this work, by engineering the structural parameters of the cavity, we achieve an experimental quality factor of order 6000 (higher than the one of the Anderson states) and we demonstrate that three types of localized modes of different nature coexist within a small area and in a relatively narrow spectral window of the disordered correlated system. Their compatibility with general boundary constraints, in contrast with ordered architectures that suffer strict layout constraints imposed by photonic crystals' axes orientation, makes optical cavities in disordered hyperuniform patterns a flexible optical insulator platform for planar optical circuits., Comment: 9 pages, 5 figures

A trust-region algorithm is presented for finding approximate minimizers of smooth unconstrained functions whose values and derivatives are subject to random noise. It is shown that, under suitable probabilistic assumptions, the new method finds (in expectation) an $\epsilon$-approximate minimizer of arbitrary order $ q \geq 1$ in at most $\mathcal{O}(\epsilon^{-(q+1)})$ inexact evaluations of the function and its derivatives, providing the first such result for general optimality orders. The impact of intrinsic noise limiting the validity of the assumptions is also discussed and it is shown that difficulties are unlikely to occur in the first-order version of the algorithm for sufficiently large gradients. Conversely, should these assumptions fail for specific realizations, then "degraded" optimality guarantees are shown to hold when failure occurs. These conclusions are then discussed and illustrated in the context of subsampling methods for finite-sum optimization.

All-dielectric, sub-micrometric particles have been successfully exploited for light management in a plethora of applications at visible and near-infrared frequency. However, the investigation of the intricacies of the Mie resonances at the sub-wavelength scale has been hampered by the limitation of conventional near-field methods. Here we address spatial and spectral mapping of multi-polar modes of a Si island by hyper-spectral imaging. The simultaneous detection of several resonant modes allows to clarify the role of substrate and incidence angle of the impinging light, highlighting spectral splitting of the quadrupolar mode and resulting in different spatial features of the field intensity. We explore theoretically and experimentally such spatial features. Details as small as 200 nm can be detected and are in agreement with simulations based on a Finite Difference Time Domain method. Our results are relevant to near-field imaging of dielectric structures, to the comprehension of the photophysics of resonant Mie structures, to beam steering and to the resonant coupling with light emitters. Our analysis paves the way for a novel approach to control the spatial overlap of a single emitter with localized electric field maxima.

We analyze the change in the spontaneous decay rate, or Purcell effect, of an extended quantum emitter in a structured photonic environment. Based on a simple theory, we show that the cross-density of states is the central quantity driving interferences in the emission process. Using numerical simulations in realistic photonic cavity geometries, we demonstrate that a structured cross-density of states can induce subradiance or superradiance, and change subtantially the emission spectrum. Interestingly, the spectral lineshape of the Purcell effect of an extended source cannot be predicted from the sole knowledge of the spectral dependence of the local density of states.

Magnetoplasmonics is highly promising to devise active optical elements: modulating the plasmon resonance condition with magnetic field can boost the performance of refractometric sensors and nanophotonic optical devices. Nevertheless, real life applications are hampered by the magnetoplasmonic trilemma: 1) a good plasmonic metal has sharp optical resonances but low magneto-optical response; 2) a magnetic metal has strong magneto-optical response but a very broad plasmonic resonance; 3) mixing the two components degrades the quality of both features. To overcome the trilemma, we use a different class of materials, transparent conductive oxide nanocrystals (NCs) with plasmonic response in the near infrared. Although non-magnetic, they combine a large cyclotron frequency (due to small electron effective mass) with sharp plasmonic resonances. We benchmark the concept with F- and In- doped CdO (FICO) and Sn-doped In2O3 (ITO) NCs to boost the magneto-optical Faraday rotation and ellipticity, reaching the highest magneto-optical response for a non-magnetic plasmonic material, and exceeding the performance of state-of-the-art ferromagnetic nanostructures. The magnetoplasmonic response of these NCs was rationalized with analytical model based on the excitation of circular magnetoplasmonic modes. Finally, proof of concept experiments demonstrated the superior performance of FICO NCs with respect to current state of the art in magnetoplasmonic refractometric sensing, approaching the sensitivity of leading localized plasmon refractometric methods with the advantage of not requiring complex curve fitting., Comment: Main text: 9 pages, 3 figure. Supporting information: 8 pages, 6 figures

Intrinsic noise in objective function and derivatives evaluations may cause premature termination of optimization algorithms. Evaluation complexity bounds taking this situation into account are presented in the framework of a deterministic trust-region method. The results show that the presence of intrinsic noise may dominate these bounds, in contrast with what is known for methods in which the inexactness in function and derivatives' evaluations is fully controllable. Moreover, the new analysis provides estimates of the optimality level achievable, should noise cause early termination. It finally sheds some light on the impact of inexact computer arithmetic on evaluation complexity.

This paper focuses on regularisation methods using models up to the third order to search for up to second-order critical points of a finite-sum minimisation problem. The variant presented belongs to the framework of [3]: it employs random models with accuracy guaranteed with a sufficiently large prefixed probability and deterministic inexact function evaluations within a prescribed level of accuracy. Without assuming unbiased estimators, the expected number of iterations is $\mathcal{O}\bigl(\epsilon_1^{-2}\bigr)$ or $\mathcal{O}\bigl(\epsilon_1^{-{3/2}}\bigr)$ when searching for a first-order critical point using a second or third order model, respectively, and of $\mathcal{O}\bigl(\max[\epsilon_1^{-{3/2}},\epsilon_2^{-3}]\bigr)$ when seeking for second-order critical points with a third order model, in which $\epsilon_j$, $j\in\{1,2\}$, is the $j$th-order tolerance. These results match the worst-case optimal complexity for the deterministic counterpart of the method. Preliminary numerical tests for first-order optimality in the context of nonconvex binary classification in imaging, with and without Artifical Neural Networks (ANNs), are presented and discussed., Comment: 9 pages

In the framework of photonics with all-dielectric nanoantennas, sub-micro-metric spheres can be exploited for a plethora of applications including vanishing back-scattering, enhanced directivity of a light emitter, beam steering, and large Purcell factors. Here, the potential of a high-throughput fabrication method based on aerosol-spray is shown to form quasi-perfect sub-micrometric spheres of polycrystalline TiO 2 . Spectroscopic investigation of light scattering from individual particles reveals sharp resonances in agreement with Mie theory, neat structural colors, and a high directivity. Owing to the high permittivity and lossless material in use, this method opens the way toward the implementation of isotropic meta-materials and forward-directional sources with magnetic responses at visible and near-UV frequencies, not accessible with conventional Si- and Ge-based Mie resonators., Comment: 8 pages, 6 figures

The long dreamed quantum internet would consist of a network of quantum nodes (solid-state or atomic systems) linked by flying qubits, naturally based on photons, travelling over long distances at the speed of light, with negligible decoherence. A key component is a light source, able to provide single or entangled photon pairs. Among the different platforms, semiconductor quantum dots are very attractive, as they can be integrated with other photonic and electronic components in miniaturized chips. In the early 1990s two approaches were developed to synthetize self-assembled epitaxial semiconductor quantum dots (QDs), or artificial atoms, namely the Stranski-Krastanov (SK) and the droplet epitaxy (DE) method. Because of its robustness and simplicity, the SK method became the workhorse to achieve several breakthroughs in both fundamental and technological areas. The need for specific emission wavelengths or structural and optical properties has nevertheless motivated further research on the DE method and its more recent development, the local-droplet-etching (LDE), as complementary routes to obtain high quality semiconductor nanostructures. The recent reports on the generation of highly entangled photon pairs, combined with good photon indistinguishability, suggest that DE and LDE QDs may complement (and sometime even outperform) conventional SK InGaAs QDs as quantum emitters. We present here a critical survey of the state of the art of DE and LDE, highlighting the advantages and weaknesses, the obtained achievements and the still open challenges, in view of applications in quantum communication and technology., Comment: 26 pages, 5 figures

We present self-assembly of InAs/InAlAs quantum dots by droplet epitaxy technique on vicinal GaAs(111)A substrates. The small miscut angle, while maintaining the symmetries imposed to the quantum dot from the surface, allows fast growth rate thanks to the presence of preferential nucleation sites at the step edges. A 100 nm InAlAs metamorphic layer with In content > 50% is already almost fully relaxed with a very flat surface. The quantum dots emit at the 1.3 {\mu}m telecom O-band with the fine structure splitting as low as 16 {\mu}eV, thus making them suitable as photon sources in quantum communication networks using entangled photons., Comment: 13 pages, 4 figures

Hydrogenation of nitrogen (N) doped GaAs allows for reversible tuning of the bandgap and the creation of site controlled quantum dots through the manipulation of N-nH complexes, N-nH complexes, wherein a nitrogen atom is surrounded by n hydrogen (H) atoms. Here we employ cross-sectional scanning tunneling microscopy (X-STM) to study these complexes in the GaAs (110) surface at the atomic scale. In addition to that we performed density functional theory (DFT) calculations to determine the atomic properties of the N-nH complexes. We argue that at or near the (110) GaAs surface two H atoms from N-nH complexes dissociate as an H$_2$ molecule. We observe multiple features related to the hydrogenation process, of which a subset is classified as N-1H complexes. These N-1H related features show an apparent reduction of the local density of states (LDOS), characteristic to N atoms in the GaAs (110) surface with an additional apparent localized enhancement of the LDOS located in one of three crystal directions. N-nH features can be manipulated with the STM tip. Showing in one case a switching behavior between two mirror-symmetric states and in another case a removal of the localized enhancement of the LDOS. The disappearance of the bright contrast is most likely a signature of the removal of an H atom from the N-nH complex., Comment: 11 pages, 12 figures

A regularization algorithm allowing random noise in derivatives and inexact function values is proposed for computing approximate local critical points of any order for smooth unconstrained optimization problems. For an objective function with Lipschitz continuous $p$-th derivative and given an arbitrary optimality order $q \leq p$, it is shown that this algorithm will, in expectation, compute such a point in at most $O\left(\left(\min_{j\in\{1,\ldots,q\}}\epsilon_j\right)^{-\frac{p+1}{p-q+1}}\right)$ inexact evaluations of $f$ and its derivatives whenever $q\in\{1,2\}$, where $\epsilon_j$ is the tolerance for $j$th order accuracy. This bound becomes at most $O\left(\left(\min_{j\in\{1,\ldots,q\}}\epsilon_j\right)^{-\frac{q(p+1)}{p}}\right)$ inexact evaluations if $q>2$ and all derivatives are Lipschitz continuous. Moreover these bounds are sharp in the order of the accuracy tolerances. An extension to convexly constrained problems is also outlined., Comment: 22 pages

We here adapt an extended version of the adaptive cubic regularisation method with dynamic inexact Hessian information for nonconvex optimisation in [3] to the stochastic optimisation setting. While exact function evaluations are still considered, this novel variant inherits the innovative use of adaptive accuracy requirements for Hessian approximations introduced in [3] and additionally employs inexact computations of the gradient. Without restrictions on the variance of the errors, we assume that these approximations are available within a sufficiently large, but fixed, probability and we extend, in the spirit of [18], the deterministic analysis of the framework to its stochastic counterpart, showing that the expected number of iterations to reach a first-order stationary point matches the well known worst-case optimal complexity. This is, in fact, still given by O(epsilon^(-3/2)), with respect to the first-order epsilon tolerance. Finally, numerical tests on nonconvex finite-sum minimisation confirm that using inexact first and second-order derivatives can be beneficial in terms of the computational savings., Comment: 33 pages