Soliton Online

This article offers a comprehensive survey of results obtained for solitons and complex nonlinear wave patterns supported by nonlinear lattices (NLs), which represent a spatially periodic modulation of the local strength and sign of the nonlinearity, and their combinations with linear lattices. A majority of the results obtained, thus far, in this field and reviewed in this article are theoretical. Nevertheless, relevant experimental settings are also surveyed, with emphasis on perspectives for implementation of the theoretical predictions in the experiment. Physical systems discussed in the review belong to the realms of nonlinear optics (including artificial optical media, such as photonic crystals, and plasmonics) and Bose-Einstein condensation. The solitons are considered in one, two, and three dimensions. Basic properties of the solitons presented in the review are their existence, stability, and mobility. Although the field is still far from completion, general conclusions can be drawn. In particular, a novel fundamental property of one-dimensional solitons, which does not occur in the absence of NLs, is a finite threshold value of the soliton norm, necessary for their existence. In multidimensional settings, the stability of solitons supported by the spatial modulation of the nonlinearity is a truly challenging problem, for theoretical and experimental studies alike. In both the one-dimensional and two-dimensional cases, the mechanism that creates solitons in NLs in principle is different from its counterpart in linear lattices, as the solitons are created directly, rather than bifurcating from Bloch modes of linear lattices.

Sketches of our wave-energy device with its horizontal axel at the contraction entrance, its three-dimensional buoy in the contraction indicated in yellow/orange, attached to an induction motor, consisting of magnets on the arc moving through the hollow cylindrical coils indicated in yellow, as well as a green and red LED (Color figure online)

Soliton Online

Download Zip 🔥 https://geags.com/2y3CBE 🔥

Computational mesh for \(N_x=10\), \(N_y=15\). The mesh structure in the contraction can be seen in the magnified right-hand-side plot, where the nodes in the contraction are denoted with a red \(\times \) symbol. While our finite-element model can deal with unstructured meshes, our partially structured meshes tend to be faster and more accurate (Color figure online)

Optimisation of the wave-energy device Our splash inspired the creation of a novel wave-energy device and we showed a working, experimental proof-of-principle model, but also developed and derived a new and fully nonlinear mathematical model of the combined water-wave dynamics, the wave-activated buoy, and the magnetic-induction power generator. Essential ingredients of this comprehensive model have been captured in one variational principle to which we a posteriori added dissipative effects of the electrical circuit, coils of the actuator, and LEDs used as the loads. The overall model was subsequently linearised and discretised using a finite-element method in space and time. This (linear) algebraic model was made fully compatible with the variational structure in the conservative and continuum limits. Its compatible, novel, and nontrivial discretisation was augmented with the resistances of the electrical circuit and coils of the induction motor as well as the LED loads. Preliminary simulations of the linear model showed promising results including (suboptimal) convergence and energy transfer between the three components. Finally, we investigated the resonant behaviour of the system as function of wave-frequency and load for a long wave-packet of harmonic waves. Nonlinear modelling, optimisation, and control of the wave-energy device require further exploration, and both the geometry of contraction, mass, and wave-buoy shape could be optimised for a given wave climate. We also aim to explore feedback control as function of contraction geometry, the number of coils of the induction motor, and the total load. In addition, higher order and more accurate spatial and temporal discretisation schemes require exploring.

with constant \(\gamma \equiv 2\pi a^2 \mu N/L\) and underlined dissipative terms. When we ignore the self-induction term \(L_i\dot{I}\) and the Shockley expression for the LEDs in (51), we note that \((R_{\text {c}}+R_i)I=\gamma G(Z)\dot{Z}\); elimination of I then shows that the magnetic force in the vertical momentum equation (51d) acts as a (nonlinear) drag, proportional to \(\dot{Z}\) or W, cf. the linear analogue in [11]. In the absence of the underlined, linear, and nonlinear dissipative terms in (51), the system (51) should be conservative, which will be explored next. In this conservative limit, we first rewrite (51) as:

Physics World represents a key part of IOP Publishing's mission to communicate world-class research and innovation to the widest possible audience. The website forms part of the Physics World portfolio, a collection of online, digital and print information services for the global scientific community.

Soliton microcombs provide a versatile platform for realizing fundamental studies and technological applications. To be utilized as frequency rulers for precision metrology, soliton microcombs must display broadband phase coherence, a parameter characterized by the optical phase or frequency noise of the comb lines and their corresponding optical linewidths. Here, we analyse the optical phase-noise dynamics in soliton microcombs generated in silicon nitride high-Q microresonators and show that, because of the Raman self-frequency shift or dispersive-wave recoil, the Lorentzian linewidth of some of the comb lines can, surprisingly, be narrower than that of the pump laser. This work elucidates information about the physical limits in phase coherence of soliton microcombs and illustrates a new strategy for the generation of spectrally coherent light on chip.

A frequency comb is a laser whose spectrum is composed of equidistant frequency components that are phase locked to a common frequency reference. The phase noise of the constituent optical lines sets a physical limit on the achievable time and frequency stability7,8,9. Significant efforts have been devoted to the systematic understanding of the linewidth of mode-locked lasers and frequency combs based on solid-state10,11, semiconductor12, and fiber lasers13,14. In 2007, a new type of frequency comb source (microcomb) was demonstrated15. Microcombs harness the Kerr nonlinearity and large intensity buildup in a high-Q microresonator cavity. Low-noise coherent states can be attained through the generation of dissipative solitons16,17,18. Unlike in conventional frequency combs based on mode-locked lasers, where the gain originates from stimulated emission in active gain media and the Lorentzian linewidth is partially dictated by spontaneous emission, the gain of soliton microcombs is based on resonantly enhanced continuous-wave-pumped parametric amplification, and the noise caused by spontaneous scattering is very weak. Another important difference is that in microcombs, the pump laser is coherently added to the comb spectrum, and therefore its noise is expected to be transferred equally to all comb lines. Indeed, earlier studies demonstrated that when the microcomb operates in a low-noise state, the comb lines inherit the linewidth of the pump19,20, with lines further away degrading more due to thermo-refractive noise (TRN) in the cavity21,22.

a Soliton microcombs are generated by coupling a continuous-wave laser into a longitudinal mode of a high-Q microresonator. The phase and intensity noise of the pump and shot noise (vacuum fluctuations) cause timing jitter and pulse carrier-envelope offset fluctuations that result into a finite optical linewidth of the frequency comb lines. b Because of the Raman self-frequency shift, variations in pump's laser frequency to the blue side result in an increase of the soliton repetition rate, and vice versa to the red side. The coupling between pump frequency and repetition rate thus results in the existence of a so-called fixed point" in the comb spectrum, that is a comb line that is most resilient to the fluctuations of the pump's frequency noise. This translates into a Lorentzian linewidth distribution with line number whose minimum can be located far away from the pump. c The shot noise and pump intensity noise affect directly the timing jitter of the soliton pulse train, which results in a linewidth distribution symmetrically located around the pump. These three effects together set the lowest achievable Lorentzian linewidth of soliton microcombs.

We begin by considering the contribution of the frequency noise of the pump. The optical frequency of the m-th microcomb line tag_hash_119m is determined by two degrees of freedom, i.e., the frequency of the pump laser tag_hash_120p and the repetition rate of the soliton microcomb tag_hash_121rep,

with the comb line number, m, counted from the pump. The underlying assumption in the elastic-tape model23 is that the noise sources will result in collective fluctuations of the comb lines. According to this, equation (1) indicates that the linewidth of the pump would be faithfully imprinted on all other comb lines if the repetition rate were fixed. However, in soliton microcombs, due to the existence of intrinsic intrapulse Raman scattering24 and dispersive-wave recoil28,29,30, the pump phase noise will also affect the repetition rate. The repetition rate can be written as25

Equation (9) indicates that the relative reduction in linewidth is more prominent for pump lasers with larger Lorentzian linewidths. This observation allows for decreasing the Lorentzian linewidth of a coherent oscillator by performing frequency translation to the quiet mode with the aid of a soliton microcomb, an aspect that is addressed experimentally in the Supplementary Note 2. 2351a5e196