Diffraction of a mode close to its cut-off by a transversal screen in a planar waveguide

The problem of diffraction of a waveguide mode by a thin Neumann screen is considered. The incident mode is assumed to have frequency close to the cut-off. The problem is reduced to a propagation problem on a branched surface and then is considered in the parabolic approximation. Using the embedding formula approach, the reflection and transmission coefficients are expressed through the directivities of the edge Green's function of the propagation problem. The asymptotics of the directivities of the edge Green's functions are constructed for the case of small gaps between the screen and the walls of the waveguide. As the result, the reflection and transmission coefficients are found. The validity of known asymptotics of these coefficients is studied.Comment: 42 pages, 10 figure

Solid Controllability in Fluid Dynamics

We survey results of recent activity towards studying controllability and accessibility issues for equations of dynamics of incompressible fluids controlled by low-dimensional or, degenerate, forcing. New results concerning controllability of Navier-Stokes/Euler systems on two-dimensional sphere and on a generic two-dimensional domain are represented.Comment: 28 page

Impact of interparticle dipole-dipole interactions on optical nonlinearity of nanocomposites

In this paper, effect of dipole-dipole interactions on nonlinear optical properties of the system of randomly located semiconductor nanoparticles embedded in bulk dielectric matrix is investigated. This effect results from the nonzero variance of the net dipole field in an ensemble. The analytical expressions describing the contribution of the dipole-dipole coupling to nonlinear dielectric susceptibility are obtained. The derived relationships are applicable over the full range of nanoparticle volume fractions. The factors entering into the contribution and depending on configuration of the dipoles are calculated for several cases. It is shown that for the different arrangements of dipole alignments the relative change of this contribution does not exceed 1/3.Comment: 5 pages, 1 figur

Ward identities for strongly coupled Eliashberg theories

We discuss Ward identities for strongly interacting fermion systems described by Eliashberg-type theories. We show that Ward identities are not in conflict with Migdal theorem. We derive diagrammatically Ward identity for a charge vertex in a Fermi liquid, and when a Fermi liquid is destroyed at quantum criticality. We argue that Ward identity for a spin vertex cannot be obtained within Eliashberg theory.Comment: 7 pages, 4 figure

Optical Kerr nonlinearity of disordered all-dielectric resonant high index metasurfaces with negative refraction

The optical Kerr effect of material with negative refraction is estimated for the first time. This is done via three-dimensional finite-difference time-domain (FDTD) simulations of disordered bidisperse metasurfaces consisting of high index (GaP) spheres at the wavelength of 532~nm. The metasurfaces comprise spherical particles randomly arranged on plane having two sizes close to the magnetic and electric dipole Mie resonances. The real part of the effective nonlinear refractive index of the metasurfaces is computed in the vicinity of the Mie resonances where the metasurface possesses the negative index of refraction. The optical Kerr nonlinearity has a peak under the condition for the negative refraction. Intensity-dependent refractive index of the bidisperse metasurfaces is studied through concentration transition to the negative refraction state. It is shown that the nonlinear Kerr coefficient of the monolayer metasurface has maximum when the effective linear refractive index is close to zero.Comment: 5 pages, 4 figure

Giant enhancement and sign inversion of optical Kerr nonlinearity in random high index nanocomposites near Mie resonances

High index dielectric nanoantennas excited at Mie-type resonances have exhibited enormous enhancement of optical nonlinearity. Such nanostructures have been actively studied by researchers in the last years. The present work provides the first numerical analysis study of the optical Kerr effect of nanocomposites consisting of high refractive index (GaP) spheres at the wavelength of 532~nm. This is done by means of three-dimensional finite-difference time-domain (FDTD) simulations. The effective nonlinear refractive index of $0.8$~$\mu$m thick nanocomposites and metasurfaces is evaluated. It is shown that the optical Kerr nonlinearity of the nanocomposites rises by orders in proximity to Mie resonances and may exceed the second-order refractive index of the bulk material. It is revealed that the sign of the effective optical Kerr coefficient is inverted near the Mie resonances. This effect may be of interest in developing nonlinear optical metadevices.Comment: 8 pages, 7 figures, 2 table

Spectral Design for Matrix Hamiltonians: Different Methods of Constructing of a Matrix Intertwining Operator

We study intertwining relations for $n\times n$ matrix non-Hermitian, in general, one-dimensional Hamiltonians by $n\times n$ matrix linear differential operators with nondegenerate coefficients at $d/dx$ in the highest degree. Some methods of constructing of $n\times n$ matrix intertwining operator of the first order of general form are proposed and their interrelation is examined. As example we construct $2\times2$ matrix Hamiltonian of general form intertwined by operator of the first order with the Hamiltonian with zero matrix potential. It is shown that one can add for the final $2\times2$ matrix Hamiltonian with respect to the initial matrix Hamiltonian with the help of intertwining operator of the first order either up to two bound states for different energy values or up to two bound states described by vector-eigenfunctions for the same energy value or up to two bound states described by vector-eigenfunction and associated vector-function for the same energy value.Comment: 39 page

An ODE--based approach to some Riemann--Hilbert problems motivated by wave diffraction

A novel approach to Riemann--Hilbert problems of particular class is introduced. The approach is applicable to problems in which the multiplicative jump is set on a half-line. Such problems are linked to some Wiener--Hopf problems motivated by diffraction theory. The new approach is based on ordinary differential equations: the Riemann--Hilbert problem is reduced to finding a coefficient of an ordinary differential equation and solving this equation. The new method leads to an efficient numerical algorithm and opens a road to new asymptotical and analytical advances.Comment: 17 pages, 7 figure

Asymptotic behaviour of a network of oscillators coupled to thermostats of finite energy

We study the asymptotic behaviour of a finite network of oscillators (harmonic or anharmonic) coupled to a number of deterministic Lagrangian thermostats of finite energy. In particular, we consider a chain of oscillators interacting with two thermostats situated at the boundary of the chain. Under appropriate assumptions we prove that the vector $(p,q)$ of moments and coordinates of the oscillators in the network satisfies $(p,q)(t)\to (0,q_c)$ when $t\to\infty$, where $q_c$ is a critical point of some effective potential, so that the oscillators just stop. Moreover, we argue that the energy transport in the system stops as well without reaching the thermal equilibrium. This result is in contrast to the situation when the energies of the thermostats are infinite, studied for a similar system in [14] and subsequent works, where the convergence to a non-trivial limiting regime was established. The proof is based on a method developed in [22], where it was observed that the thermostats produce some effective dissipation despite the Lagrangian nature of the system.Comment: 22 pages. In comparison with the previous version where a chain of oscillators was considered, the result is generalized to the case when the oscillators form arbitrary networ

Possibility of negative refraction for visible light in disordered all-dielectric resonant high index metasurfaces

Effective refractive index of disordered all-dielectric metasurfaces consisting of gallium phosphide (GaP) spheres is studied by means of three-dimensional finite-difference time-domain (FDTD) simulations at the wavelength of 532~nm. It is shown that a mixture of the high index nanoparticles with sizes close to the first magnetic and electric resonances randomly dispersed on metasurface may possess negative refraction. The dependence of the metasurface effective refractive index on the nanoparticle concentration and size is constructed. The feasibility of negative refraction at large concentrations of the high index resonant spheres is demonstrated. The negative effective refractive index is exhibited only for a monolayer of GaP spheres.Comment: 7 pages, 6 figures, 1 tabl