Formation of localized structures in bistable systems through nonlocal spatial coupling. I. General framework

The regions of existence and stability of dark solitons in the Lugiato-Lefever model with normal chromatic dispersion are described. These localized states are shown to be organized in a bifurcation structure known as collapsed snaking implying the presence of a region in parameter space with a finite multiplicity of dark solitons. For some parameter values dynamical instabilities are responsible for the appearance of oscillations and temporal chaos. The importance of the results for understanding frequency comb generation in microresonators is emphasized.

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“…(9) as evolving in x, the rescaled fast time, from x = −∞ to x = ∞ [9,[17][18][19][20]. Thus DSs correspond to homoclinic orbits of Eq.…”

Section: Spatial Dynamicsmentioning

confidence: 99%

Dark solitons in the Lugiato-Lefever equation with normal dispersion

et al. 2016

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“…where a 4 = 2vd 3 + 1, a 2 = −4I s + 2θ + v 2 , and a 0 = 4I s θ − 3I 2 s − θ 2 + 1 [36,37]. Alternatively, one can perform a linear stability analysis directly on Eq.…”

Section: Additional Oscillatory Tails Lead To Bright Solitonsmentioning

confidence: 99%

Coexistence of stable dark- and bright-soliton Kerr combs in normal-dispersion resonators

2017

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Using the Lugiato-Lefever model, we analyze the effects of third-order chromatic dispersion on the existence and stability of dark-and bright-soliton Kerr frequency combs in the normal dispersion regime. While in the absence of third-order dispersion only dark solitons exist over an extended parameter range, we find that third-order dispersion allows for stable dark and bright solitons to coexist. Reversibility is broken and the shape of the switching waves connecting the top and bottom homogeneous solutions is modified. Bright solitons come into existence thanks to the generation of oscillations in the switching-wave profiles. Temporal oscillatory instabilities of dark solitons are suppressed in the presence of sufficiently strong third-order dispersion, while bright solitons are never found to oscillate in time. As a result of third-order dispersion both bright and dark solitons are found to move with a velocity that depends on their width.

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“…3(b),(c) illustrate how the eigenvalues, and thus the profile of the SW, change with the parameters (θ, u 0 ). See [28,29] for more details on such eigenvalue analysis. The SW generally moves (in the reference frame of the driving field) with a velocity v that depends on the driving amplitude u 0 [see Fig.…”

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confidence: 99%

Origin and stability of dark pulse Kerr combs in normal dispersion resonators

et al. 2016

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We analyze dark pulse Kerr frequency combs in optical resonators with normal group-velocity dispersion using the Lugiato-Lefever model. We show that in the time domain these correspond to interlocked switching waves between the upper and lower homogeneous states, and explain how this fact accounts for many of their experimentally observed properties. Modulational instability does not play any role in their existence. Furthermore, we provide a detailed map indicating where stable dark pulse Kerr combs can be found in parameter space, and how they are destabilized for increasing values of frequency detuning.Optical frequency combs generated in passive Kerr microresonators have attracted a lot of interest in recent years for their potential for on-chip integration of frequency comb technology [1][2][3][4][5][6]. Their applications span arbitrary waveform synthesis [5], telecommunications [7], and ultra-accurate clocks [4]. These so-called "Kerr frequency combs" (KFCs) are obtained by driving a high-Q microresonator with a continuous-wave (cw) laser. This leads to the generation of spectral sidebands through modulational instability (MI) and subsequent cascaded four-wave mixing. Interestingly, much can been learned about KFCs by considering a time domain representation. In fact, most reported KFCs correspond either to extended periodic patterns or to ultrashort pulses known as temporal cavity solitons (CSs), stable or fluctuating [8][9][10][11][12][13][14]. These studies have benefitted from the fact that KFCs can be modeled using a simple mean-field equation, the Lugiato-Lefever equation (LLE) [9,15].The bulk of KFC studies so far deals with microresonators exhibiting anomalous second-order group velocity dispersion (GVD) at the pump wavelength. However, due to the difficulty in obtaining anomalous GVD in some spectral ranges, generation of KFCs from normal GVD microresonators is now also being sought and has recently been achieved experimentally by several groups [16][17][18]. In [18], a full time-domain characterization is reported: the field is found to consist of square dark pulses of different widths -low intensity dips embedded in a high intensity homogeneous background -with a complex temporal structure. These observations match several previous numerical predictions [14,16,[19][20][21] and are in stark contrast with the isolated ultrashort bright localized structures observed with anomalous GVD [8,12].There has been some speculation as to the physical * lendert.gelens@kuleuven.be origin of the temporal structures observed in normal GVD KFCs, which have been called platicons, dark pulse KFCs, or dark CSs [18,21]. To clarify this issue, we present here a detailed bifurcation analysis of dark structures in the LLE with normal GVD, and predict their region of existence and stability. In particular, we clearly show that they are intimately related to so-called switching waves (SWs) -traveling front solutions of the LLE that connect the upper and lower homogenous state solutions. These SWs were studied theoretica...

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Formation of localized structures in bistable systems through nonlocal spatial coupling. I. General framework

Cited by 32 publications

References 61 publications

Dark solitons in the Lugiato-Lefever equation with normal dispersion

Dark solitons in the Lugiato-Lefever equation with normal dispersion

Coexistence of stable dark- and bright-soliton Kerr combs in normal-dispersion resonators

Origin and stability of dark pulse Kerr combs in normal dispersion resonators

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