Optical Solitons

Agrawal, Govind P.

doi:10.1016/b978-0-12-145135-6.50007-x

Cited by 6 publications

(9 citation statements)

References 135 publications

(188 reference statements)

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“…Theorem 3.1. For any r 3, there exists s 0 (r) > 0 and for each s > s 0 there exist s > 0, C s > 0 and a continuous function N (r) s : B(0, s ) → R + , 1 Remark that in the position-momentum variables, q j = 1/ √ 2(z j + z j ) and p j = i/ √ 2(z j − z j ), j ∈ N , the reversibility hypothesis reads as H (q, p) = H (q, −p), i.e. H even in the momenta (cf [9]).…”

Section: Resultsmentioning

confidence: 99%

Quasi-invariant modified Sobolev norms for semi linear reversible PDEs

Faou¹,

Grébert²

2010

Nonlinearity

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We consider a general class of infinite dimensional reversible differential systems. Assuming a non resonance condition on the linear frequencies, we construct for such systems almost invariant pseudo norms that are closed to Sobolev-like norms. This allows us to prove that if the Sobolev norm of index s of the initial data z 0 is sufficiently small (of order ǫ) then the Sobolev norm of the solution is bounded by 2ǫ during very long time (of order ǫ −r with r arbitrary). It turns out that this theorem applies to a large class of reversible semi linear PDEs including the non linear Schrödinger equation on the d-dimensional torus. We also apply our method to a system of coupled NLS equations which is reversible but not Hamiltonian.We also notice that for the same class of reversible systems we can prove a Birkhoff normal form theorem that in turn implies the same bounds on the Sobolev norms. Nevertheless the technics that we use to prove the existence of quasi invariant pseudo norms is much more simple and direct.

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Section: Resultsmentioning

confidence: 99%

Quasi-invariant modified Sobolev norms for semi linear reversible PDEs

Faou¹,

Grébert²

2010

Nonlinearity

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“…A fundamental feature of nonlinear processes is that they can give rise to the exchange of energy among electromagnetic fields with different frequencies. In this regards, nonlinear processes can be divided in two classes [ 24 , 25 , 26 ]:…”

Section: Fundamental Phenomenamentioning

confidence: 99%

An Introduction to Nonlinear Integrated Photonics Devices: Nonlinear Effects and Materials

Sirleto

Righini

2023

Micromachines

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The combination of integrated optics technologies with nonlinear photonics, which has led to the growth of nonlinear integrated photonics, has also opened the way to groundbreaking new devices and applications. Here we introduce the main physical processes involved in nonlinear photonics applications, and we discuss the fundaments of this research area, starting from traditional second-order and third-order phenomena and going to ultrafast phenomena. The applications, on the other hand, have been made possible by the availability of suitable materials, with high nonlinear coefficients, and/or by the design of guided-wave structures, which can enhance the material’s nonlinear properties. A summary of the most common nonlinear materials is presented, together with a discussion of the innovative ones. The discussion of fabrication processes and integration platforms is the subject of a companion article, also submitted for publication in this journal. There, several examples of nonlinear photonic integrated devices to be employed in optical communications, all-optical signal processing and computing, or quantum optics are shown, too. We aimed at offering a broad overview, even if, certainly, not exhaustive. We hope that the overall work could provide guidance for those who are newcomers to this field and some hints to the interested researchers for a more detailed investigation of the present and future development of this hot and rapidly growing field.

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“…Nonlinear conversion is a fundamental aspect of nonlinear photonics because it should allow the realization of efficiently integrated coherent light sources. Several second and third order nonlinear optical phenomena can be involved, which can be placed into two classes: parametric and non-parametric processes [ 1 , 2 , 3 , 4 ]. Second-harmonic generation (SHG), sum-frequency generation (SFG), difference-frequency generation (DFG), third-harmonic generation (THG), and four-wave mixing (FWM) belong to the former class, while stimulated Raman scattering (SRS) and stimulated Brillouin scattering (SBS) belong to the latter one.…”

Section: Introductionmentioning

confidence: 99%

Tunable Raman Gain in Transparent Nanostructured Glass-Ceramic Based on Ba2NaNb5O15 †

et al. 2023

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Stimulated Raman scattering in transparent glass-ceramics (TGCs) based on bulk nucleating phase Ba2NaNb5O15 were investigated with the aim to explore the influence of micro- and nanoscale structural transformations on Raman gain. Nanostructured TGCs were synthesized, starting with 8BaO·15Na2O·27Nb2O5·50SiO2 (BaNaNS) glass, by proper nucleation and crystallization heat treatments. TGCs are composed of nanocrystals that are 10–15 nm in size, uniformly distributed in the residual glass matrix, with a crystallinity degree ranging from 30 up to 50% for samples subjected to different heat treatments. A significant Raman gain improvement for both BaNaNS glass and TGCs with respect to SiO2 glass is demonstrated, which can be clearly related to the nanostructuring process. These findings show that the nonlinear optical functionalities of TGC materials can be modulated by controlling the structural transformations at the nanoscale rather than microscale.

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Optical Solitons

Cited by 6 publications

References 135 publications

Quasi-invariant modified Sobolev norms for semi linear reversible PDEs

Quasi-invariant modified Sobolev norms for semi linear reversible PDEs

An Introduction to Nonlinear Integrated Photonics Devices: Nonlinear Effects and Materials

Tunable Raman Gain in Transparent Nanostructured Glass-Ceramic Based on Ba2NaNb5O15 †

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