Nonlinear photonic crystals: I. Quadratic nonlinearity

Abstract. Using Bloch waves to represent the full solution of Maxwell's equations in periodic media, we study the limit where the material's period becomes much smaller than the wavelength. It is seen that for steady state fields, only a few of the Bloch waves contribute to the full solution. Effective material parameters can be explicitly represented in terms of dyadic products of the mean values of the nonvanishing Bloch waves, providing a new means of homogenization. The representation is valid for an arbitrary wave vector in the first Brillouin zone.

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“…Bloch decomposition. The Bloch eigenmodes are defined from the following eigenvalue problem [2,8,7,21]:…”

Section: Compact Resolvent With a Materialmentioning

confidence: 99%

A Floquet--Bloch Decomposition of Maxwell's Equations Applied to Homogenization

Sjöberg¹,

Engström²,

Kristensson³

et al. 2005

Multiscale Model. Simul.

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“…The linear part ε (1) E of the polarization determines the properties of the underlying linear medium governed by corresponding linear Maxwell equations, whereas the nonlinear part αP NL (E) is responsible for nonlinear effects. The parameter α scales the nonlinear component of the medium to the linear one, and, as in [5], we assume that α is small, which corresponds to the assumption of smallness of the initial excitation currents.…”

Section: E = E (R T) = E I(r·k−ω(k)t) Vmentioning

confidence: 99%

“…Mathematically consistent approach to the same problem requires one to take into account that real waves propagating in a nonlinear medium, being solutions to the nonlinear Maxwell equation, are neither time nor space harmonic. This consistent approach is based on an appropriate asymptotic analysis (see [5], [6]) yielding, in particular, a satisfactory interpretation of the frequency dependent susceptibilities.…”

Section: E = E (R T) = E I(r·k−ω(k)t) Vmentioning

confidence: 99%

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Multilinear spectral decomposition for nonlinear Maxwell equations

Babin¹,

Figotin²

2002

Partial Differential Equations

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Abstract. We study the frequency form, popular in physics, of Maxwell equations for nonlinear spatially inhomogeneous dielectric media. Using wavepackets based on eigenmodes of the underlying linear dielectric medium we develop a mathematically consistent interpretation of the frequency form of nonlinear Maxwell operators. In particular, we construct the operator of nonlinear polarization based on frequency dependent susceptibilities. To this end we use a multilinear variant of the spectral decomposition of selfadjoint operators.

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“…Remark 3.3 (dispersion relations symmetry) The symmetry condition (2.5) on the dispersion relations naturally arise in many physical problems, for example Maxwell equations in periodic media, see [2]- [4], [6], or when L (k) originates from a Hamiltonian. We would like to stress that this symmetry conditions are not imposed to simplify studies but rather to take into account fundamental symmetries of physical media.…”

Section: Condition 32 (Linear Part)mentioning

confidence: 99%

Nonlinear Dynamics of a System of Particle-Like Wavepackets

Babin

Figotin

Instability in Models Connected With Fluid Flows I

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This work continues our studies of nonlinear evolution of a system of wavepackets. We study a wave propagation governed by a nonlinear system of hyperbolic PDE's with constant coefficients with the initial data being a multi-wavepacket. By definition a general wavepacket has a well defined principal wave vector, and, as we proved in previous works, the nonlinear dynamics preserves systems of wavepackets and their principal wave vectors. Here we study the nonlinear evolution of a special class of wavepackets, namely particle-like wavepackets. A particle-like wavepacket is of a dual nature: on one hand, it is a wave with a well defined principal wave vector, on the other hand, it is a particle in the sense that it can be assigned a well defined position in the space. We prove that under the nonlinear evolution a generic multi-particle wavepacket remains to be a multi-particle wavepacket with a high accuracy, and every constituting single particle-like wavepacket not only preserves its principal wave number but also it has a well-defined space position evolving with a constant velocity which is its group velocity. Remarkably the described properties hold though the involved single particle-like wavepackets undergo nonlinear interactions and multiple collisions in the space. We also prove that if principal wavevectors of multi-particle wavepacket are generic, the result of nonlinear interactions between different wavepackets is small and the approximate linear superposition principle holds uniformly with respect to the initial spatial positions of wavepackets.

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Nonlinear photonic crystals: I. Quadratic nonlinearity

Cited by 20 publications

References 55 publications

A Floquet--Bloch Decomposition of Maxwell's Equations Applied to Homogenization

A Floquet--Bloch Decomposition of Maxwell's Equations Applied to Homogenization

Multilinear spectral decomposition for nonlinear Maxwell equations

Nonlinear Dynamics of a System of Particle-Like Wavepackets

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