Vasyl V. Yatsyk

Generation and Resonance Scattering of Waves on Cubically Polarisable Layered Structures

In this paper, mathematical models for the analysis of processes of generation and resonance scattering of wave packets on a transversely inhomogeneous, isotropic, cubically polarisable, non-magnetic, linearly polarised (E polarisation) medium with a non-linear, layered dielectric structure, and methods of their numerical simulation are considered. In general, electromagnetic waves in a non-linear medium with a cubic polarisability can be described by an infinite system of non-linear differential equations. In the study of particular non-linear effects it proves to be possible to restrict the examination to a finite number of equations, and also to leave certain terms in the representation of the polarisation coefficients, which characterise the physical problem under investigation. Here we investigate the situation where the incident field consists of a packet of three waves oscillating with single, double and triple frequency. An intense field at the basic frequency leads to the generation of the third harmonic, i.e. of a field at the triple frequency. In this case it is possible to reduce the mathematical model to a system of two equations, where only the non-trivial terms in the expansion of the polarisation coefficients are taken into account (see Angermann & Yatsyk (2010)). The consideration of a weak field at the double frequency or at both the double and triple frequencies allows to analyse its influence on the generation process of the third harmonic. In this situation, the mathematical model consists of three differential equations. The rigorous formulation finally leads to a system of boundary-value problems of Sturm-Liouville type, which can be equivalently transformed into a system of one-dimensional non-linear integral equations (defined along the height of the structure) with respect to the complex Fourier amplitudes of the scattered fields in the non-linear layer at the basic and multiple frequencies. In the paper both the variational approach to the approximate solution of the system of non-linear boundary-value problems of Sturm-Liouville type (based on the application of a finite element method) and an iterative scheme of the solution of the system of non-linear integral equations (based on the application of a quadrature rule to each of the non-linear integral equations) are considered. 8

DIFFRACTION OF ELECTROMAGNETIC WAVES BY A LAYER FILLED WITH A KERR-TYPE NONLINEAR MEDIUM

The diffraction of a plane wave by a transversely inhomogeneous isotropic nonmagnetic linearly polarized dielectric layer filled with a Kerr-type nonlinear medium is considered. The analytical and numerical solution techniques are developed. The diffraction problem is reduced to a singular boundary value problem for a semilinear second-order ordinary differential equation with a cubic nonlinearity and then to a cubic-nonlinear integral equation (IE) of the second kind and to a system of nonlinear operator equations of the second kind solved using iterations. Sufficient conditions of the unique solvability are obtained using the contraction principle.

Resonance Properties of Scattering and Generation of Waves on Cubically Polarisable Dielectric Layers

In this paper we investigate the problem of scattering and generation of waves on an isotropic, non-magnetic, linearly polarised (E-polarisation), non-linear, layered, cubically polarisable, dielectric structure, which is excited by a packet of plane waves, in the range of resonant frequencies. We consider wave packets consisting of both strong electromagnetic fields at the excitation frequency, leading to the generation of waves, and of weak fields at the multiple frequencies, which do not lead to the generation of harmonics but influence on the process of scattering and generation of waves. The analysis of the quasi-homogeneous electromagnetic fields of the non-linear dielectric layered structure made it possible to derive a condition of phase synchronism of waves. If the classical formulation of the mathematical model is supplemented by this condition of phase synchronism, we arrive at a rigorous formulation of a system of boundary-value problems with respect to the components of the scattered and generated fields (see Angermann & Yatsyk (2011)). This system is transformed to equivalent systems of non-linear problems, namely a system of one-dimensional non-linear Fredholm integral equations of the second kind and a system of non-linear boundary-value problems of Sturm-Liouville type. The numerical algorithms of the solution of the non-linear problems are based on iterative procedures which require the solution of a linearised system in each step. In this way the approximate solution of the non-linear problems is described by means of solutions of linearised problems with an induced dielectric permeability. The analytical continuation of these problems into the region of complex values of the frequency parameter allows us to switch to the analysis of spectral problems. The corresponding eigen-frequencies form a discrete, countable set of points, with the only possible accumulation point at infinity, and lie on a complex two-sheeted Riemann surface. In the frequency domain, the resonant scattering and generation properties of non-linear structures are determined by the proximity of the excitation frequencies of the non-linear structures to the complex eigen-frequencies of the corresponding homogeneous linear spectral problems with the induced non-linear dielectric permeability of the medium. 15

Resonance scattering of electromagnetic waves by a Kerr nonlinear dielectric layer

Diffraction of a plane electromagnetic wave by a dielectric layer is considered in the resonance region without allowance for multiple frequencies. The layer is assumed to be transversely inhomogeneous, isotropic, nonmagnetic, linearly polarized, and weakly Kerr nonlinear. A method based on solution of a nonlinear Fredholm integral equation of the second kind is developed. Sufficient conditions for the existence and uniqueness of a solution are obtained.

The Effect of Weak Fields at Multiple Frequencies on the Scattering and Generation of Waves by Nonlinear Layered Media

The solution of the system of integral equations is approximated by the help of numerical algorithms. Those include the application of suitable quadrature rules and iterative procedures to solve the resulting nonlinear algebraic problems. Since in each iteration step the solution of linear algebraic systems is required, the approximate solution of the nonlinear problems is described by means of solutions of linear problems with an induced nonlinear permittivity. In continuation of our results from previous works ([3], [4]), where we only considered one excitation field at the basic frequency, here results of calculations of characteristics of the

The multifunctional process of resonance scattering and generation of oscillations by nonlinear layered structures

Abstract The paper focuses on the development of a mathematical model, an effective algorithm and a self-consistent numerical analysis of the multifunctional properties of resonant scattering and generation of oscillations by nonlinear, cubically polarizable layered structures. The multifunctionality of such layered media is caused by the nonlinear mechanism between interacting oscillations—the incident oscillations (exciting the nonlinear layer from the upper and lower half-spaces) as well as the scattered and generated oscillations at the frequencies of excitation/scattering and generation. The study of the resonance properties of scattering and generation of oscillations by a nonlinear structure with a controllable permittivity in dependence on the variation of the intensities of the components of the exciting wave package is of particular interest. In the present paper, we extend our former results, and furthermore we analyze the realizability of multifunctional properties of nonlinear electromagnetic objects with a controllable permittivity. The results of our investigations (i) demonstrate the possibility to control the scattering and generation properties of the nonlinear structure via the intensity of the incident field, (ii) indicate the possibility of increasing the multifunctionality of electronic devices, of designing frequency multipliers, and other electrodynamic devices containing nonlinear dielectrics with controllable permittivity.

Diffraction by a Kerr-type Nonlinear Dielectric Layer

The difiraction of a plane wave by a transversely inhomogeneous isotropic nonmag- netic linearly polarized dielectric layer fllled with a Kerr-type nonlinear medium is considered. The difiraction problem is reduced to a cubic-nonlinear integral equation (IE) of the second kind and to a system of nonlinear operator equations of the second kind solved using iterations. Su-cient conditions of the IE unique solvability are obtained using the contraction principle.

Mathematical Models of Electrodynamical Processes of Wave Scattering and Generation on Cubically Polarisable Layers

Results of a self-consistent computational analysis based on a mathematical model of resonance scattering and generation of waves on an isotropic nonmagnetic nonlinear layered dielectric structure excited by a packet of plane waves are presented, where the analysis is performed in the domain of resonance frequencies. Physically interesting properties of the nonlinear permittivities of the layers as well as their scattering and generation characteristics are obtained, for instance the characteristic dynamical behaviour of the relative Q-factor of the eigenmodes and the energy of higher harmonics generated by canalising as well as decanalising nonlinear layers. The results demonstrate the possibility to control the scattering and generating properties of a nonlinear structure by means of the excitation intensities.

Preset Field Approximation and Self-consistent Analysis of the Scattering and Generation of Oscillations by a Layered Structure

Nonlinear dielectrics with controllable permittivity are subject of intense studies and begin to find broad applications, for instance in device technology and electronics. Based on a model of resonance scattering and generation of waves on an isotropic nonmagnetic nonlinear layered dielectric structure which is excited by a packet of plane waves, we compare two numerical algorithms for simulating various effects of the fields at multiple frequencies.

Resonant scattering and third-harmonic generation by cubically polarizable grating structures

The paper focuses on the development of a mathematical model of the multifunctional properties of resonant scattering and generation of oscillations by nonlinear, cubically polarizable layered and periodical structures.