Justification of the nonlinear Schrödinger equation in spatially periodic media

Busch, Kurt; Schneider, Guido; Tkeshelashvili, Lasha; Uecker, Hannes

doi:10.1007/s00033-006-0057-6

Cited by 47 publications

(96 citation statements)

References 27 publications

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“…The same problems occur if the truncated third-order system is justified with some approximation result similar to that of Ref. [24]. In this work, the NLSE has been justified in the above sense for semi linear wave equations with periodic coefficients.…”

Section: Influence Of Resonancesmentioning

confidence: 60%

Far-off-resonant wave interaction in one-dimensional photonic crystals with quadratic nonlinearity

et al. 2008

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We extend a recently developed Hamiltonian formalism for nonlinear wave interaction processes in spatially periodic dielectric structures to the far-off-resonant regime, and investigate numerically the three-wave resonance conditions in a one-dimensional optical medium with χ (2) nonlinearity. In particular, we demonstrate that the cascading of nonresonant wave interaction processes generates an effective χ (3) nonlinear response in these systems. We obtain the corresponding coupling coefficients through appropriate normal form transformations that formally lead to the Zakharov equation for spatially periodic optical media.

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Section: Influence Of Resonancesmentioning

confidence: 60%

Far-off-resonant wave interaction in one-dimensional photonic crystals with quadratic nonlinearity

et al. 2008

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“…There are a number of mathematical papers proving error estimates for the approximation of the original system by the NLS equation also in case of quadratic nonlinearities. See [3,[5][6][7] for the spatially homogeneous case and [8] for some first results in the spatially periodic case. The following proof is an easy adaption of the one from [4].…”

Section: Validity Of the Approximationmentioning

confidence: 98%

“…semilinear wave equations with a quadratic nonlinearity in case of no resonances [3,4] and in case of resonances [5,6], water wave models [7] and finally wave equations in periodic media [8].…”

Section: Introductionmentioning

confidence: 99%

On the interaction of NLS‐described modulating pulses with different carrier waves

Chirilus‐Bruckner

Schneider

Uecker

2007

Math Methods in App Sciences

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SUMMARYWe give a detailed analysis of the interaction of two modulating pulse solutions of a nonlinear wave equation. These solutions consist of pulse-like envelopes satisfying approximately a nonlinear Schrödinger equation, advancing in the laboratory frame, and modulating underlying wave trains. We improve the bound for the possible shift of the envelopes caused by the interaction of two well-prepared pulses from order O(1) to order O(ε). Thus, we manifest the statement that there is almost no interaction of pulses with different carrier waves.

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“…The case of linear optics in homogeneous PhC was addressed in [9]. Then, in the framework of nonlinear diffractive optics, the author in [8] derived a Nonlinear Schrödinger equation (NLS) as a model for propagation of waves in 1d PhC. This was extended by [4] in a multi-dimensional setting but still for homogeneous PhC.…”

Section: Introductionmentioning

confidence: 99%

Diffractive optics with harmonic radiation in 2d nonlinear photonic crystal waveguide

Lescarret

Schneider

2012

Z. Angew. Math. Phys.

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Abstract. 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 Schrödinger 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. IntroductionPeriodic media such as metallic crystals and semi-conductors have drawn scientists' attention because of their band-gap spectrum which accounts for their strong dispersive properties. In the range of optical wavelengths we find the Photonic Crystals (PhC) whose popularity is partly due to the small sizes and the possibility to slow down the light. We refer the reader to the quite exhaustive report [6] on PhC. Because the spectrum gives a good hint on the way waves propagate, the research on these materials mostly focuses on this aspect (see [13,14,15,17,18]). However, as pointed out in [31] the spectrum alone does not explain the wave resonance and wave radiation in a medium. Such effects are not harmonic in time and are indeed observed in nonlinear media when one lets a wave evolve from an initial state close but not equal to an eigenvalue. With this idea in mind we address the issue of computing and analyzing the propagation of an electromagnetic pulse in 2d, straight, nonlinear PhC waveguides (see Figure 1.1). The guides are made of a PhC from which a finite number of rows is removed. The spatial extent of the pulse envelope may be of the order of a few cells of periodicity of the PhC or much larger along the direction of the guide. We denote by η the periodicity of the cell divided by the spatial length of the pulse envelope. The nonlinear response of the PhC makes possible the existence of solitons whose shape and energy are preserved and is responsible for harmonics (third harmonics in centro-symmetric media) which may radiate part of the energy. We consider this issue in the framework of diffractive optics (see [11]), that 2 Vincent Lescarret and Guido Schneider is, for times of order 1/η 2 for which one can observe the soliton propagating and the third harmonic radiating. The main goal of this study is to mathematically describe this harmonic radiation and the way it disturbs the soliton.This situation is quite academic but brings new results in (mathematical) nonlinear optics for PhC. The case of linear optics in homogeneous PhC was addressed in [9]. Then, in the framework of nonlinear diffractive optics, the author in [8] derived a Nonlinear Schrödinger equation (NLS) as a model for propagation of waves in 1d PhC. This was extended by [4] in a multi-dimensional setting but still for homogeneous PhC.The present analysis relies on the spectral properties of the waveguide which are precisely given in the next sections. In particular we provide a resolution of identity of a class of transverse operators modeling the situation. It is mainly based on A. Fig...

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Justification of the nonlinear Schrödinger equation in spatially periodic media

Cited by 47 publications

References 27 publications

Far-off-resonant wave interaction in one-dimensional photonic crystals with quadratic nonlinearity

Far-off-resonant wave interaction in one-dimensional photonic crystals with quadratic nonlinearity

On the interaction of NLS‐described modulating pulses with different carrier waves

Diffractive optics with harmonic radiation in 2d nonlinear photonic crystal waveguide

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