Resonant Scattering and Generation of Waves

Electromagnetic phenomena in a space–time domain \(\mathbb {R}^4_>:=\mathbb {R}^3\times (0,\infty )\) can be governed by the system of macroscopic Maxwell’s differential equations where the Gaussian unit system is used (see, for example, Born and Wolf in Principles of Optic, Pergamon Press, Oxford, 1970, [1, Sect. 1.1.1], Landau et al. in Electrodynamics of Continuous Media, Elsevier Butterworth-Heinemann, Oxford, 1984, [2, Chap. IX]). Here, \({\mathbf E },\) \({\mathbf H },\) \({\mathbf D },\) \({\mathbf B }:\,\mathbb {R}^4_>\rightarrow \mathbb {R}^3\) denote the unknown vector fields of electric and magnetic field intensity, electric and magnetic induction, respectively, c is a positive constant—the velocity of light. The function \(\rho :\,\mathbb {R}^4_>\rightarrow \mathbb {R}\) and the vector field \({\mathbf J }:\,\mathbb {R}^4_>\rightarrow \mathbb {R}^3\) are called the electric charge density and the electric current density, respectively. These macroscopic quantities are obtained by averaging rapidly varying microscopic quantities over spatial scales that are much larger than the typical material microstructure scales. Details of the averaging procedure can be found in standard electrodynamic textbooks, for instance, in Jackson, Classical Electrodynamics, Wiley, New York, 1999, [3].

In this chapter, we demonstrate the existence and uniqueness of a so-called weak solution of the semilinear boundary value problem (1.66) and (1.68). In contrast to the more physically inspired notation of the previous chapter, here we make use of a notation which is more convenient for mathematical purposes.

In this chapter, we show how the problem (1.47), (C1)–(C4) can be reduced to finding solutions of a system of one-dimensional nonlinear integral equations w.r.t. the components \(u_n(z),\) \(n=1,2,3,\) \(z\in \mathscr {I}^\text {cl},\) of the fields scattered and generated in the nonlinear layer. Here, we give a derivation of these equations, which is more formally than in the papers[1–7], and which extends the results of the works [8, 9] to the case of excitation of the nonlinear structure by the plane-wave packets (1.51).

It is well known from both the theory and experiments that the scattering and generation characteristics of dielectric structures depend on the relation of the frequencies of excitation to certain problem-inherent (spectral) frequencies. Therefore it is important to be able to estimate or to compute (at least approximately) such problem-inherent parameters. A typical approach is to convert the nonlinear problem under consideration into a linear problem by means of appropriate linearization arguments and then to investigate the resulting linear problems. For instance, in the case of small amplitudes of the incident field, the neglect of nonlinear terms leads to a linear operator equation which can be analyzed spectrally. Here we will make use of a more sophisticated approach which is also related in a rather natural way to the problem to solve the system of Hammerstein integral equations (3.17).

The finite element method is based on the weak formulation (2.5). We consider \(N\in \mathbb {N},\) \(N\ge 2,\) nodes \(\left\{ z_j\right\} _{j=1}^N\) such that \(-2\pi \delta =:z_1<z_2<\ldots<z_{N-1}<z_N:=2\pi \delta ,\) and define the subintervals \(\mathscr {I}_j:=\left( z_j,z_{j+1}\right) \) with the lengths \(h_j:=z_{j+1}-z_j\) and the parameter \(h:=\max _{j\in \{1,\ldots ,N-1\}}h_j.\) Then, for \(j\in \{1,\ldots ,N\}\) we introduce the basis functions \(\psi _j:\;\mathscr {I}^\text {cl}\rightarrow \mathbb {R}\) by the formula and the corresponding linear spaces

The numerical solution of the system of nonlinear Hammerstein integral equations of second kind (3.17) is based on the so-called Nyström method, where the integrals are approximated by appropriate quadrature rules. As the result of this method, a nonlinear system of complex algebraic equations arises. Analogously to the finite element method described in Sect. 5.1, we consider \(N\in \mathbb {N},\) \(N\ge 2,\) nodes \(\left\{ z_{j,N}\right\} _{j=1}^N\) such that \(-2\pi \delta =:z_{1,N}<z_{2,N}<\cdots<z_{N-1,N}<z_{N,N}:=2\pi \delta \) and the subintervals \(\mathscr {I}_{j,N}=\left( z_{j,N},z_{j+1,N}\right) \) with the lengths \(h_{j,N}=z_{j+1,N}-z_{j,N}.\) Then, given a continuous function \(v:\;\mathscr {I}^\text {cl}\rightarrow \mathbb {C},\) a numerical integration scheme for the integral can be defined by a quadrature rule where the coefficients \(\nu _{j,N}\in \mathbb {R}\) are known.

In this chapter, we present a selection of results from our fund of numerical experiments. We methodically focus on the approach based upon the numerical solution of the system (3.17) of nonlinear Hammerstein integral equations, as described in Chaps. 3 and 6. The finite element method from Chap. 5 has also been implemented and successfully tested (Angermann, Yatsyk, Int J Electromagn Waves Electron Syst 13(12), 15–30, 2008, [1]) (Hoff, Numerische Simulation der Oberwellengeneration in nichtlinearen elektromagnetischen Diffraktionsproblemen. Diploma thesis (supervisor: L. Angermann), Department of Mathematics, Clausthal University of Technology, 2014, [2]). Since the obtained approximations to the solution of the boundary value problem (1.66) were largely comparable to the numerical results for the system (3.17) and thus did not provide any other (or even new) findings, we have omitted a similarly detailed description and discussion of the finite element results. Nevertheless, we have included a few comments on these results.

We presented a mathematical model, its analysis, and computational simulations for the problem of resonance scattering and generation of waves by an isotropic, nonmagnetic, nonlinear, plate-like, layered, dielectric structure that is excited by one- or two-sided acting packets of plane waves. This model essentially extends the model proposed earlier in Yatsyk, International Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED-2009), 2009, [1], Angermann and Yatsyk, Int. J. Electromagn. Waves Electron. Syst. 15(1):36–49, 2010, [2], where only the case of normal incidence of a one-sided acting wave packet has been investigated. The involvement of the condition of phase synchronism into the boundary conditions of the problem allowed us to eliminate this restriction. Here, the incident wave packet may fall onto the nonlinear layered structure under an arbitrary angle, and the excitation from the other side of the structure is included. The wave packets under consideration consist of both strong electromagnetic fields at the excitation frequency of the nonlinear structure (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 the process of scattering and generation of waves by the nonlinear structure). The investigations are performed in the domain of resonance frequencies (Shestopalov and Yatsyk, J. Nonlinear Math. Phys. 17(3):311–335, 2010, [3], Angermann and Yatsyk, Numerical Simulations – Applications, Examples and Theory, pp. 175–212. InTech, Rijeka, 2011, [4], Angermann and Yatsyk, Electromagnetic Waves, pp. 299–340. InTech, Rijeka, 2011, [5]), where both the radio (Chernogor, Nonlinear Radiophysics. V.N. Karazin Kharkov National University, Kharkov, 2004, [6]) and optical (Miloslavsky, Nonlinear Optics. V.N. Karazin Kharkov National University, Kharkov, 2008, [7]) frequency ranges are of interest.