Diffractive optics with harmonic radiation in 2d nonlinear photonic crystal waveguide

The propagation of modulated light in a 2d nonlinear photonic waveguide is investigated in the framework of diffractive optics. It is shown that the dynamics obeys a nonlinear Schr?dinger equation at leading order. We compute the first and second corrector and show that the latter may describe some dispersive radiation through the structure. We prove the validity of the approximation in the interval of existence of the leading term.     

Conservative finite-difference scheme for the problem of propagation of a femtosecond pulse in a photonic crystal with combined nonlinearity

A conservative finite-difference scheme is constructed for the problem of propagation of a light pulse in a one-dimensional nonlinear photonic crystal with combined nonlinearity. The invariants of the corresponding differential problem and their difference analogues are given. The scheme is compared with those based on the widespread splitting method. For combined cubic and quadratic nonlinearity in photonic crystal layers, it is shown that the classical splitting method is ineffective, since it requires time steps that are smaller by one or more orders of magnitude. The finite-difference scheme proposed conserves the propagation invariants, which cannot be achieved for splitting schemes even on considerably finer grids. Nonreflecting conditions substantially improve the efficiency of conservative finite-difference schemes as applied to the simulation of complex nonlinear effects in photonic crystals, which require much smaller steps in space and time than those used in the case of linear propagation. The simulation is based on the approach proposed by the authors for the given class of problems.     

Corpuscular modeling of the interaction of a laser pulse with plasma

Results from computer modeling according to the particle interaction of ultrastrong and ultrashort laser pulses with a two-layer target are presented. Particle acceleration regimes are demonstrated in a computing experiment.     

Mathematical analysis of plasmonic resonance for 2-D photonic crystal

In this article, we study the plasmonic resonance of infinite photonic crystal mounted by the double negative nanoparticles in two dimensions. The corresponding physical model is described by the Helmholz equation with so called Bloch wave condition in a periodic domain. By using the quasi-periodic layer potential techniques and the spectral theorem of quasi-periodic Neumann–Poincaré operator, the quasi-static expansion of the near field in the presence of nanoparticles is derived. Furthermore, when the magnetic permeability of nanoparticles satisfies the Drude model, we give the conditions under which the plasmonic resonance occurs, and the rate of blow up of near field energy with respect to nanoparticle's bulk electron relaxation rate and filling factor are also obtained. It indicates that one can appropriately control the bulk electron relaxation rate or filling factor of nanoparticle in photonic crystal structure such that the near field energy attains its maximum, and enhancing the efficiency of energy utilization.     

The nonlinear interaction of a laser beam with a plasma pellet

A model is developed which simulates the interaction of a laser beam with a plasma. The model allows for a nonlinear dependence of the density on the amplitude of the electric field. Numerical results show that for strong nonlinearities time-harmonic fields cannot be obtained. Instead pulses of electromagnetic energy (solitary waves) are formed inside the plasma and carry energy away from the plasma.     

Methods for computer simulation of crystal sputtering

A computer program for molecular dynamics simulation of sputtering is developed. Numerical methods for the integration of the equations of motion are described. Algorithms for constructing models of single-and polycrystalline targets are proposed. The applicability of various ion-atom and atom-atom interaction potentials to sputtering computations is analyzed.     

Mathematical analysis of the guided waves in photonic crystal fibers

The propagation of guided waves in photonic crystal fibers (PCFs) is studied. The structure of a PCF can be regarded as a perfect two-dimensional photonic crystal with a line defect along the invariant direction. This problem can be treated as an eigenvalue problem for a family of noncompact self-adjoint operators. We prove that line defects do not change the essential spectrum of the associated “background” medium. This result plays a key role for studying the influence of line defects on the “background” spectrum. A modified Combes-Thomas estimate is also formulated.     

Prime orbit theorems with multi-dimensional constraints for axiom a flows

In this paper we give a new proof of an asymptotic formula for the number of closed orbits of an Axiom A flow subject to certain constraints due to S. P. Lalley. We consider finite group extensions and, for transitive Anosov flows, give an application to homology. We also discuss asymptotics for closed orbits in a fixed homology class, extending a result of Katsuda and Sunada.Supported by S.E.R.C. grant no. 88001623.     

Recursive methods for a multi-dimensional risk process with common shocks

In this paper, a multi-dimensional risk model with common shocks is studied. Using a simple probabilistic approach via observing the risk processes at claim instants, recursive integral formulas are developed for the survival probabilities as well as for a class of Gerber-Shiu expected discounted penalty functions that include the surplus levels at ruin. Under the assumption of exponential or mixed Erlang claims, the recursive integrals can be simplified to give recursive sums which are computationally more tractable. Numerical examples including an optimal capital allocation problem are also given towards the end.     

Visualizing the results of a mathematical information system for identification of materials by interaction with a short-period terahertz pulse

The article describes a software interface for visualizing the results of experimental data processing in a computer information system for analysis and identification of materials, including explosives. The interface provides a wide range of graphic tools that draw distributions of spectral intensities, instantaneous Fourier spectra, spectrograms, and 3D plots viewable with different degrees of detail. The package has been tested with excellent results using physical experimental data.     

Central limit results for jump diffusions with mean field interaction and a common factor

A system of N weakly interacting particles whose dynamics is given in terms of jump-diffusions with a common factor is considered. The common factor is described through another jump-diffusion and the coefficients of the evolution equation for each particle depend, in addition to its own state value, on the empirical measure of the states of the N particles and the common factor. A Central Limit Theorem, as N → ∞, is established. The limit law is described in terms of a certain Gaussian mixture. An application to models in mathematical finance of self-excited correlated defaults is described.     

Particle method simulation of the resonance interaction of a high-frequency field with a highly nonhomogeneous collisionless plasma

Translated from Metody Matematicheskogo Modelirovaniya i Vychislitel'noi Diagnostiki, pp. 260–265, Izd. Moskovskogo Universiteta, Moscow, 1990.     

Quantum equations of motion for multimode laser generation with a spatial dependence of the atom interaction with the field taken into account

We derive equations of motion for the electromagnetic field operators a_q′ ⁺ a_q″ for a three-level multimode laser with a spatial dependence of the interaction of atoms with the field of a standing wave in a cavity taken into account. We calculate and analyze the dynamics of means of photon numbers in the field modes and of the correlation function of field modes. We explore the effect of intermode correlations on the dynamics of establishing stationary laser generation. We find that taking the spatial dependence of the interaction of atoms with the field and the intermode correlation into account in investigating the means of photon numbers leads to revealing new properties of laser generation, such as saturation of the laser radiation intensity in a single-mode regime and generation of short light pulses of side below-threshold modes with the amplitudes depending on the initial state of the field in a cavity.     

Numerical analysis of the effect of small geometrical imperfections on photonic crystal wires

Imperfections in the geometry of photonic crystal wires (PCWs) are always a result of current manufacture techniques. The influence of such random defects on the propagating modes in PCWs is studied computationally by means of a full vectorial finite-element method using a perfectly matched layer (PML) as absorbing boundary condition, and the numerical results show a large sensitivity of the birefringence to the geometrical defects in the transversal section of the fibre. It has also been found that imperfections in PCWs have a greater effect on wave propagation than those in photonic crystal fibres (PCFs).