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Abstract.
We investigate scattering and generation of waves on an isotropic,
nonmagnetic, linearly polarised nonlinear layered cubically
polarisable dielectric structure (a layer in free space filled with
nonlinear medium) excited by a packet of plane waves; analysis is
performed in the domain of resonance frequencies. We consider wave
packets consisting of both strong electromagnetic fields at the
excitation frequency of the nonlinear structure which lead to the
generation of waves, and weak fields at the multiple frequencies which
do not lead to the generation of harmonics but influence scattering
and generation of waves. We show that the propagation of
electromagnetic waves in a nonlinear layer with cubic polarisability
of the medium can be described by an infinite system of nonlinear
boundary-value problems (BVPs). When considering particular nonlinear
effects one can reduce this system to a finite number of problems and
to leave certain terms in the representation of the polarisation
coefficients which characterize the physical problem under
investigation [1-3]. Analysis of quasi-homogeneous electromagnetic
fields of the nonlinear dielectric layered structure make it possible
to derive a condition of phase synchronism of waves. If the classical
formulation of the problem is supplemented by this condition, we
arrive at a rigorous formulation for a system of BVPs with respect to
the components of the scattered and generated fields [2, 3]. This
system is transformed to equivalent systems of nonlinear problems,
namely, to a system of one-dimensional nonlinear Fredholm integral
equations (IEs) of the second kind and to a system of nonlinear BVPs
of Sturm-Liouville type. We obtain sufficient conditions for the
existence and uniqueness of their solutions. Algorithms of numerical
solution to nonlinear problems are based on iterative procedures where
approximate solution is described in terms of solutions to linear
problems with an induced nonlinear permittivity. Analytical
continuation of these linear problems into the region of complex
values of the frequency parameter allows us to formulate and analyze
spectral problems. Namely, we look for eigenfrequencies and the
corresponding eigenfields of the homogeneous linear problems with the
induced nonlinear dielectric permeability in the complex domain of the
spectral parameter. We prove that eigenfrequencies form a discrete
countable set of points with the only possible accumulation point at
infinity and lie on a complex two-sheet Riemann surface.
We develop the following fundamental approach that backgrounds analysis of spectral problems in the complex frequency domain: resonance wave scattering and generation in nonlinear structures are determined, in the frequency domain, by the proximity of the excitation frequencies to the complex eigenfrequencies of the corresponding homogeneous linear spectral problems with the induced nonlinear dielectric permeability of the medium. We present some results of calculations that describe properties of the nonlinear permittivities of the layers as well as their scattering and generation characteristics.
References
[1] Y. V. Shestopalov and V. V. Yatsyk, Diffraction of
electromagnetic waves by a layer filled with a Kerr-type nonlinear
medium, J. of Nonlinear Mathem. Physics, vol. 17, No. 3, pp. 311-335,
2010.
[2] L. Angermann and V. V. Yatsyk, Generation and resonance
scattering of waves on cubically polarisable layered structures. Book
chapter, in L. Angermann (ed.), Numerical Simulations - Applications,
Examples and Theory, InTech, Rijeka/Vienna, Croatia/Austria, pp. 175-212, 2011.
[3] L. Angermann and V. V. Yatsyk, Resonance properties of
scattering and generation of waves on cubically polarisable dielectric
layers. Book chapter, in V. Zhurbenko (ed.), Electromagnetic Waves,
InTech, Rijeka/Vienna, Croatia/Austria, pp. 299-340, 2011.