Coulomb and quenching effects in small nanoparticle-based spasers

We study numerically the effect of mode mixing and direct dipole-dipole interactions between gain molecules on spasing in a small composite nanoparticles with a metallic core and a dye-doped dielectric shell. By combining Maxwell-Bloch equations with Green's function formalism, we calculate lasing frequency and threshold population inversion for various gain densities in the shell. We find that gain coupling to nonresonant plasmon modes has a negligible effect on spasing threshold. In contrast, the direct dipole-dipole coupling, by causing random shifts of gain molecules' excitation frequencies, hinders reaching the spasing threshold in small systems. We identify a region of parameter space in which spasing can occur considering these effects.Comment: 7 pages, 6 figure

Super-Resolution Imaging by Arrays of High-Index Spheres Embedded in Transparent Matrices

We fabricated thin-films made from polydimethylsiloxane (PDMS) with embedded high-index (n~1.9-2.2) microspheres for super-resolution imaging applications. To control the position of microspheres, such films can be translated along the surface of the nanoplasmonic structure to be imaged. Microsphere-assisted imaging, through these matrices, provided lateral resolution of ~{\lambda}/7 in nanoplasmonic dimer arrays with an illuminating wavelength {\lambda}=405 nm. Such thin films can be used as contact optical components to boost the resolution capability of conventional microscopes.Comment: 3 pages, 3 figures, IEEE proceeding of NAECON June 24-27 (2014

Two problems related to prescribed curvature measures

Existence of convex body with prescribed generalized curvature measures is discussed, this result is obtained by making use of Guan-Li-Li's innovative techniques. In surprise, that methods has also brought us to promote Ivochkina's $C^2$ estimates for prescribed curvature equation in \cite{I1, I}.Comment: 12 pages, Corrected typo

Photonic band structure of highly deformable, self-assembling systems

We calculate the photonic band structure at normal incidence of highly deformable, self-assembling systems - cholesteric elastomers subjected to external stress. Cholesterics display brilliant reflection and lasing owing to gaps in their photonic band structure. The band structure of cholesteric elastomers varies sensitively with strain, showing new gaps opening up and shifting in frequency. A novel prediction of a total band gap is made, and is expected to occur in the vicinity of the previously observed de Vries bandgap, which is only for one polarisation

Fundamental limits of super-resolution microscopy by dielectric microspheres and microfibers

In recent years, optical super-resolution by microspheres and microfibers emerged as a new paradigm in nanoscale label-free and fluorescence imaging. However, the mechanisms of such imaging are still not completely understood and the resolution values are debated. In this work, the fundamental limits of super-resolution imaging by high-index barium-titanate microspheres and silica microfibers are studied using nanoplasmonic arrays made from Au and Al. A rigorous resolution analysis is developed based on the object's convolution with the point-spread function that has width well below the conventional (∼λ/2) diffraction limit, where λ is the illumination wavelength. A resolution of ∼λ/6-λ/7 is demonstrated for imaging nanoplasmonic arrays by microspheres. Similar resolution was demonstrated for microfibers in the direction perpendicular to the fiber axis with hundreds of times larger field-of-view in comparison to microspheres. Using numerical solution of Maxwell's equations, it is shown that extraordinary close point objects can be resolved in the far field, if they oscillate out of phase. Possible super-resolution using resonant excitation of whispering gallery modes is also studied. Keywords: Optical super-resolution; near-field microscopy; confocal microscop

Self-Assembly of Supramolecular Triblock Copolymer Complexes

Four different poly(tert-butoxystyrene)-b-polystyrene-b-poly(4-vinylpyridine) (PtBOS-b-PS-b-P4VP) linear triblock copolymers, with the P4VP weight fraction varying from 0.08 to 0.39, were synthesized via sequential anionic polymerization. The values of the unknown interaction parameters between styrene and tert-butoxystyrene and between tert-butoxystyrene and 4-vinylpyridine were determined from random copolymer blend miscibility studies and found to satisfy 0.031<χS,tBOS<0.034 and 0.39<χ4VP,tBOS<0.43, the latter being slightly larger than the known 0.30<χS,4VP≤0.35 value range. All triblock copolymers synthesized adopted a P4VP/PS core/shell cylindrical self-assembled morphology. From these four triblock copolymers supramolecular complexes were prepared by hydrogen bonding a stoichiometric amount of pentadecylphenol (PDP) to the P4VP blocks. Three of these complexes formed a triple lamellar ordered state with additional short length scale ordering inside the P4VP(PDP) layers. The self-assembled state of the supramolecular complex based on the triblock copolymer with the largest fraction of P4VP consisted of alternating layers of PtBOS and P4VP(PDP) layers with PS cylinders inside the latter layers. The difference in morphology between the triblock copolymers and the supramolecular complexes is due to two effects: (i) a change in effective composition and, (ii) a reduction in interfacial tension between the PS and P4VP containing domains. The small angle X-ray scattering patterns of the supramolecules systems are very temperature sensitive. A striking feature is the disappearance of the first order scattering peak of the triple lamellar state in certain temperature intervals, while the higher order peaks (including the third order) remain. This is argued to be due to the thermal sensitivity of the hydrogen bonding and thus directly related to the very nature of these systems.

A glimpse into the differential topology and geometry of optimal transport

This note exposes the differential topology and geometry underlying some of the basic phenomena of optimal transportation. It surveys basic questions concerning Monge maps and Kantorovich measures: existence and regularity of the former, uniqueness of the latter, and estimates for the dimension of its support, as well as the associated linear programming duality. It shows the answers to these questions concern the differential geometry and topology of the chosen transportation cost. It also establishes new connections --- some heuristic and others rigorous --- based on the properties of the cross-difference of this cost, and its Taylor expansion at the diagonal.Comment: 27 page

Stability of flows associated to gradient vector fields and convergence of iterated transport maps

In this paper we address the problem of stability of flows associated to a sequence of vector fields under minimal regularity requirements on the limit vector field, that is supposed to be a gradient. We apply this stability result to show the convergence of iterated compositions of optimal transport maps arising in the implicit time discretization (with respect to the Wasserstein distance) of nonlinear evolution equations of a diffusion type. Finally, we use these convergence results to study the gradient flow of a particular class of polyconvex functionals recently considered by Gangbo, Evans ans Savin. We solve some open problems raised in their paper and obtain existence and uniqueness of solutions under weaker regularity requirements and with no upper bound on the jacobian determinant of the initial datum