Bifurcation analysis to the Lugiato–Lefever equation in one space dimension

Miyaji, Tomoyuki; Ohnishi, Isamu

doi:10.1016/j.physd.2010.07.014

Cited by 39 publications

(50 citation statements)

References 27 publications

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“…For example, the nonlinear dynamics of the LLE and its bifurcation behavior is extremely rich and still deserves attention, as a full characterization of the existing stationary solutions has not been completed at this date [18,[95][96][97]. This nonlinear analysis is also needed in order to gain further insight in the Brought to you by | MIT Libraries Authenticated Download Date | 5/13/18 10:02 AM wide variety of phase-locking phenomena that are taking place in Kerr combs [49,[98][99][100] and also to optimize the Kerr comb generation process [101].…”

Section: Resultsmentioning

confidence: 99%

Kerr optical frequency combs: theory, applications and perspectives

Chembo

2016

Nanophotonics

137

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Abstract:The optical frequency comb technology is one of the most important breakthrough in photonics in recent years. This concept has revolutionized the science of ultra-stable lightwave and microwave signal generation. These combs were originally generated using ultrafast mode-locked lasers, but in the past decade, a simple and elegant alternative was proposed, which consisted in pumping an ultra-high-Q optical resonator with Kerr nonlinearity using a continuous-wave laser. When optimal conditions are met, the intracavity pump photons are redistributed via four-wave mixing to the neighboring cavity modes, thereby creating the so-called Kerr optical frequency comb. Beyond being energy-efficient, conceptually simple, and structurally robust, Kerr comb generators are very compact devices (millimetric down to micrometric size) which can be integrated on a chip. They are, therefore, considered as very promising candidates to replace femtosecond mode-locked lasers for the generation of broadband and coherent optical frequency combs in the spectral domain, or equivalently, narrow optical pulses in the temporal domain. These combs are, moreover, expected to provide breakthroughs in many technological areas, such as integrated photonics, metrology, optical telecommunications, and aerospace engineering. The purpose of this review article is to present a comprehensive survey of the topic of Kerr optical frequency combs. We provide an overview of the main theoretical and experimental results that have been obtained so far. We also highlight the potential of Kerr combs for current or prospective applications, and discuss as well some of the open challenges that are to be met at the fundamental and applied level.

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Section: Resultsmentioning

confidence: 99%

Kerr optical frequency combs: theory, applications and perspectives

Chembo

2016

Nanophotonics

137

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“…Equations (1) and (3) are relatively simple-looking, non-variational, and realistic enough to account for experimental observations [24,25,28,57,58]. For these reasons, after having been the subject of numerical and experimental investigation, these systems are receiving renewed mathematical attention [31,59]. In this frame, the control parameters are the injection amplitude E I and the detuning parameter θ.…”

Section: Introductionmentioning

confidence: 99%

“…Several methods can be used to derive (5). Miyaji, Ohnishi, and Tsutsumi [59] used the normal form reduction for the Lugiato-Lefever model. We will use the method of multiple-scale, whereby (5) arises as a solvability condition at fifth order of the analysis.…”

Section: Introductionmentioning

confidence: 99%

Localized Turing patterns in nonlinear optical cavities

Kozyreff

2012

Physica D: Nonlinear Phenomena

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The subcritical Turing instability is studied in two classes of models for laser-driven nonlinear optical cavities. In the first class of models, the nonlinearity is purely absorptive, with arbitrary intensity-dependent losses. In the second class, the refractive index is real and is an arbitrary function of the intra-cavity intensity. Through a weakly nonlinear analysis, a Ginzburg-Landau equation with quintic nonlinearity is derived. Thus, the Maxwell curve, which marks the existence of localized patterns in parameter space, is determined. In the particular case of the Lugiato-Lefever model, the analysis is continued to seventh order, yielding a refined formula for the Maxwell curve and the theoretical curve is compared with recent numerical simulation by Gomilà, Firth, and Scroggie [Physica D 227, 70 (2007).]

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“…It has been the subject of intensive mathematical study [11][12][13], and it has been experimentally demonstrated that this model can be applied to arXiv:1210.8210v1 [physics.optics]…”

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confidence: 99%

“…It has been the subject of intensive mathematical study [11][12][13], and it has been experimentally demonstrated that this model can be applied to arXiv:1210.8210v1 [physics.optics] 31 Oct 2012 fiber ring resonators [14]. Unexpected phenomena such has convective instability [15] and excitability [16] are also known occur in this system.…”

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confidence: 99%

Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators

2013

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We establish an exact partial differential equation to model Kerr comb generation in whisperinggallery mode resonators. This equation is a variant of the Lugiato-Lefever equation that includes higher-order dispersion and nonlinearity. This spatio-temporal model, whose main variable is the total intracavity field, is significantly more suitable than the modal expansion approach for the theoretical understanding and the numerical simulation of wide-span combs. It allows us to explore pulse formation in which a large number of modes interact cooperatively. This versatile approach can be straightforwardly extended to include higher-order dispersion, as well as other phenomena like Raman, Brillouin and Rayleigh scattering. We demonstrate for the first time that when the dispersion is anomalous, Kerr comb generation can arise as the spectral signature of dissipative cavity solitons, leading to wide-span combs with low pumping.PACS numbers: 42.62. Eh, 42.65.Hw, 42.65.Sf, 42.65.Tg The development of frequency combs -equidistant frequency lines from a short-pulse laser -revolutionized the measurement of frequencies [1] and has opened up a host of potential applications in fundamental and applied physics, including the measurement of physical constants, the detection of earth-like planets, chemical sensing, the generation, measurement, and distribution of highly accurate time, and the generation of low-phase-noise microwave radiation [2]. Ti:sapphire lasers were used as the original source of frequency combs, but in the past seven years, alternative fiber laser sources have developed [2]. Recently, Del 'Haye, et al. [3] demonstrated that it is possible to use the whispering gallery modes in microresonators in combination with the Kerr effect to generate an equidistant frequency comb that is also referred to as a Kerr comb. Since many applications in both fundamental and applied science would benefit from the small size, simplicity, robustness, and low power consumption of these whispering gallery mode sources, a considerable worldwide effort has gone into understanding and controlling them [4]. In particular, there have been several efforts to develop mathematical models of these sources [5][6][7][8], but all the efforts to date have serious drawbacks.A complete modal expansion has been derived to describe the growth of the Kerr combs from noise [5,6]. This model predicts a cascaded growth in which a primary comb is first generated, which then generates a secondary comb, and later higher order combs. This model is in complete agreement with experiments [5]. However, * E-mail: yanne.chembo@femto-st.fr it is difficult and computationally expensive to use this mode expansion beyond the primary comb generation because the number of modes that must be kept in a calculation grows like the third power of time. Moreover, it is difficult to study pulse formation in this model, since a large number of modes interact cooperatively. Pulse formation plays a critical role in comb generation. It is desirable to find a spatio-tempora...

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Bifurcation analysis to the Lugiato–Lefever equation in one space dimension

Cited by 39 publications

References 27 publications

Kerr optical frequency combs: theory, applications and perspectives

Kerr optical frequency combs: theory, applications and perspectives

Localized Turing patterns in nonlinear optical cavities

Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators

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