Nonlinear waves in optical media

In nonlinear optical systems quasi-monochromatic waves satisfy the classical, constant dispersion and dispersion managed nonlocal, nonlinear Schrödinger equations, both of which exhibit localized pulse solutions. Current research has shown that mode-locked lasers are also described by dispersion managed equations. In spatial systems recent developments have attracted considerable interest in 2D photonic lattices. A computational method is introduced to find these and other localized waves in nonlinear optical media with vanishing and non-vanishing boundary conditions.     

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.     

Solitons in liquid crystals: Recent developments

Liquid crystal is a state of matter intermediate between isotropic liquid and anisotropic crystal. The mechanical and optical properties of liquid crystals are highly nonlinear. Consequently, they are naturally soliton-bearing media. After a brief general introduction, five topics in recent developments on solitons in liquid crystals are presented, namely (i) optical solitons, (ii) solitons in nematics under a rotating magnetic field, (iii) solitons in electroconvective nematics, (iv) incommensurate solitons in smectic A, and (v) the soliton model for the chevron structure in ferroelectric smectic C^* and in smectic A.     

Spectrum created by line defects in periodic structures

We study a Helmholtz‐type spectral problem related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a three‐dimensional periodic medium; the defect is infinitely extended in one direction, but compactly supported in the remaining two. This perturbation introduces guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. We will show that even small perturbations lead to additional spectrum in the spectral gaps of the unperturbed operator and investigate some properties of the spectrum that is created.     

Linear and nonlinear electromagnetic waves in modulated honeycomb media

Wave dynamics in topological materials has been widely studied recently. A striking feature is the existence of robust and chiral wave propagations that have potential applications in many fields. A common way to realize such wave patterns is to utilize Dirac points, which carry topological indices and is supported by the symmetries of the media. In this work, we investigate these phenomena in photonic media. Starting with Maxwell's equations with a honeycomb material weight as well as the nonlinear Kerr effect, we first prove the existence of Dirac points in the dispersion surfaces of transverse electric and magnetic Maxwell operators under very general assumptions of the material weight. Our assumptions on the material weight are almost the minimal requirements to ensure the existence of Dirac points in a general hexagonal photonic crystal. We then derive the associated wave packet dynamics in the scenario where the honeycomb structure is weakly modulated. It turns out the reduced envelope equation is generally a two-dimensional nonlinear Dirac equation with a spatially varying mass. By studying the reduced envelope equation with a domain-wall-like mass term, we realize the subtle wave motions, which are chiral and immune to local defects. The underlying mechanism is the existence of topologically protected linear line modes, also referred to as edge states. However, we show that these robust linear modes do not survive with nonlinearity. We demonstrate the existence of nonlinear line modes, which can propagate in the nonlinear media based on high-accuracy numerical computations. Moreover, we also report a new type of nonlinear modes, which are localized in both directions.     

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.     

Spatial Vector Solitons in Nonlinear Photonic Crystal Fibers

We study spatial vector solitons in a photonic crystal fiber (PCF) made of a material with the focusing Kerr nonlinearity. We show that such two-component localized nonlinear waves consist of two mutually trapped components confined by the PCF linear and the self-induced nonlinear refractive indices, and they bifurcate from the corresponding scalar solitons. We demonstrate that, in a sharp contrast with an entirely homogeneous nonlinear Kerr medium where both scalar and vector spatial solitons are unstable and may collapse, the periodic structure of PCF can stabilize the otherwise unstable two-dimensional spatial optical solitons. We apply the matrix criterion for stability of these two-parameter solitons, and verify it by direct numerical simulations.     

Coupled dielectric waveguides with photonic crystal properties

A system of planar waveguides coupled through a periodic set of small windows is considered. It is shown that the weak periodic coupling of the waveguides leads to an additional eigenvalue band separated by a gap from the threshold of a continuous spectrum branch. Thus, the system has photonic crystal properties, which can be used to construct optical fiber devices. This system can also play the role of a SCISSOR device for reducing the group velocity of light, in optical delay lines, etc.     

Nonlinear dynamics in double square-well potentials

The first example of the coexistence of Josephson oscillations with a self-trapping regime is found in the context of the coherent nonlinear dynamics in a double square-well potential. We prove the simultaneous existence of symmetric, antisymmetric, and asymmetric stationary solutions of the associated Gross-Pitaevskii equation, which explains this macroscopic bistability. We illustrate and confirm the effect with numerical simulations. This property allows suggesting experiments with Bose-Einstein condensates in engineered optical lattices or with weakly coupled optical waveguide arrays. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 292–303, August, 2007.     

Conservative finite-difference scheme for the problem of propagation of a femtosecond pulse in a nonlinear photonic crystal with nonreflecting boundary conditions

Conservative finite-difference schemes are constructed for the problems of self-action of a femtosecond laser pulse and of second-harmonic generation in a one-dimensional nonlinear photonic crystal with nonreflecting boundary conditions. The invariants of the governing equations are found taking into account these conditions. Nonreflecting conditions substantially improve the efficiency of conservative finite-difference schemes used in the modeling 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 numerical experiments performed show that the boundary reflects no more than 0.01% of the transmitted energy, which corresponds to the truncation error in the boundary conditions. The amplitude of the reflected pulse is less than that of the pulse transmitted through the boundary by two (and more) orders of magnitude. The simulation is based on the approach proposed by the authors for the given class of problems.     

A method of variation of boundaries for waveguide grating couplers

We describe a method for calculating the solution of the electromagnetic field in a non-rectilinear open waveguide by using a series expansion, starting from the field of a rectilinear waveguide. Our approach is based on a method of variation of boundaries. We prove that the obtained series expansion converges and we provide a radiation condition at infinity in such a way that the problem has a unique solution. Our approach can model several kinds of optical devices which are used in optical integrated circuits. Numerical examples will be shown for the case of finite aperiodic waveguide grating couplers.     

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).     

NUMERICAL METHODS FOR MAXWELL＇S EQUATIONS IN INHOMOGENEOUS MEDIA WITH MATERIAL INTERFACES

In this paper, we will present some recent results on developing numerical methods for solving Maxwell‘s equations in inhomogeneous media with material interfaces. First,we will present a second order upwinding embedded boundary method - a Cartesian grid based finite difference method with special upwinding treatment near the material interfaces. Second, we will present a high order discontinuous spectral element with Dubinar orthogonal polynomials on triangles. Numerical results on electromagnetic scattering and photonic waveguide will be included.     

Analysis of the radiation properties of a planar antenna on a photonic crystal substrate

This paper is concerned with the rigorous investigation of the radiation properties of a planar patch antenna on a photonic crystal substrate. Under the assumptions that the driving frequency of the antenna lies within the band gap of the photonic crystal substrate and that the crystal satisfies a symmetry condition, we prove that the power radiated into the substrate decays exponentially. To do this, we reduce the radiation problem to the study of the well‐posedness of a weakly singular integral equation on the patch antenna, and to the study of the asymptotic behaviour of the corresponding Green's function. We also provide a mathematical justification of the use of a photonic crystal substrate as a perfect mirror at any incidence angle. Copyright © 2001 John Wiley & Sons, Ltd.     

Second kind integral equation formulation for the mode calculation of optical waveguides

We present a second kind integral equation (SKIE) formulation for calculating the electromagnetic modes of optical waveguides, where the unknowns are only on material interfaces. The resulting numerical algorithm can handle optical waveguides with a large number of inclusions of arbitrary irregular cross section. It is capable of finding the bound, leaky, and complex modes for optical fibers and waveguides including photonic crystal fibers (PCF), dielectric fibers and waveguides. Most importantly, the formulation is well conditioned even in the case of nonsmooth geometries. Our method is highly accurate and thus can be used to calculate the propagation loss of the electromagnetic modes accurately, which provides the photonics industry a reliable tool for the design of more compact and efficient photonic devices. We illustrate and validate the performance of our method through extensive numerical studies and by comparison with semi-analytical results and previously published results.     

Optical solitons in a power law media with fourth order dispersion

In this paper, a closed form optical soliton solution is obtained for the nonlinear Schrödinger’s equation with fourth order dispersion in a power law media. The solitary wave ansatze is used to carry out the integration of this equation. Finally, a numerical simulation is given for the closed form soliton solution.     

Laser molecular-beam epitaxy and second-order optical nonlinearity of BaTiO3/SrTiO3 superlattices

A series of c-axis oriented BaTiO3/SrTiO3 superlattices with the atomic-scale precision were epitaxially grown on single-crystal SrTiO3 (100) substrates using laser molecular-beam epitaxy (LMBE). A periodic modulation of the intensity of reflection high-energy electron diffraction (RHEED) in BaTiO3 and SrTiO3 layers was observed and attributed to the lattice-misfit-induced periodic variation of the terrace density in film surface. The relationship between the second-order nonlinear optical susceptibilities and the superlattice structure was systematically studied. The experimental and theoretical fitting results indicate that the second-order nonlinear optical susceptibilities of BaTiO3/SrTiO3 superlattices were greatly enhanced with the maximum value being more than one order of magnitude larger than that of bulk BaTiO3 crystal. The mechanism of the enhancement of the second-order optical nonlinearity was discussed by taking into account the stress-induced lattice distortion and polarization enhancement.     

Partially Coherent Waves in Nonlinear Periodic Lattices

We study the propagation of partially coherent (random-phase) waves in nonlinear periodic lattices. The dynamics in these systems is governed by the threefold interplay between the nonlinearity, the lattice properties, and the statistical (coherence) properties of the waves. Such dynamic interplay is reflected in the characteristic properties of nonlinear wave phenomena (e.g., solitons) in these systems. While the propagation of partially coherent waves in nonlinear periodic systems is a universal problem, we analyze it in the context of nonlinear photonic lattices, where recent experiments have proven their existence.     

Propagation of weakly guided waves in a Kerr nonlinear medium using a perturbation approach

The equations are represented in a simplified format with only a few leading terms needed in the expansion. The set of equations are then solved numerically using vector finite element method. To validate our algorithm, we analyzed a two-dimensional rectangular waveguide consisting of a linear core and nonlinear identical cladding. The exact nonlinear solutions for three different modes of propagations, TE0, TE1, and TE2 modes are generated and compared with the computed solutions. Next, we investigate the effect of a more intense monochromatic field on the propagation of a “weak” optical field in a fully three-dimensional cylindrical waveguide.