We investigate frequency comb generation in the presence of polarization effects induced by nonlinear mode coupling in microresonator devices. A set of coupled temporal Lugiato-Lefever equations are derived to model the propagation dynamics, and an in-depth study is made of the modulational instability of their multistable homogeneous steady-state solutions. It is shown that new kinds of instabilities can occur for co-propagating fields that interact through nonlinear cross-phase modulation. These instabilities display properties that differ from their scalar counterpart, and are shown to result in the generation of new types of incoherently coupled frequency comb states.
We introduce the first principle model describing frequency comb generation in a WGM microresonator with the backscattering-induced coupling between the counter-propagating waves. {Elaborated model provides deep insight and accurate description of the complex dynamics of nonlinear processes in such systems.} We analyse the backscattering impact on the splitting and reshaping of the nonlinear resonances, demonstrate backscattering-induced modulational instability in the normal dispersion regime and subsequent frequency comb generation. We present and discuss novel features of the soliton comb d…
A study is made of frequency comb generation described by the driven and damped nonlinear Schrödinger equation on a finite interval. It is shown that frequency comb generation can be interpreted as a modulational instability of the continuous wave pump mode, and a linear stability analysis, taking into account the cavity boundary conditions, is performed. Further, a truncated three-wave model is derived, which allows one to gain additional insight into the dynamical behaviour of the comb generation. This formalism describes the pump mode and the most unstable sideband and is found to connect t…
Kerr frequency combs from microresonators are now extensively investigated as a potentially portable technology for a variety of applications. Most studies employ anomalous dispersion microresonators that support modulational instability for comb initiation, and mode-locking transitions resulting in coherent bright soliton-like pulse generation have been reported. However, some experiments show comb generation in normal dispersion microresonators; simulations suggest the formation of dark pulse temporal profiles. Excitation of dark pulse solutions is difficult due to the lack of modulational i…
We present an investigation of the modulational instability of partially coherent signals in electrical transmission lines. Starting from the modified Ginzburg-Landau equations and the Wigner-Moyal representation, we derive a nonlinear dispersion relation for the modulational instability. It is found that the effect of signal broadbandness reduces the growth rate of the modulational instability.
We consider plane waves propagating in quadratic nonlinear slab waveguides with nonlinear quasi-phase-matching gratings. We predict analytically and verify numerically the complete gain spectrum for transverse modulational instability, including hitherto undescribed higher order gain bands.
We study theoretically and experimentally the modulational instability of broad optical beams in photorefractive nonlinear media. We demonstrate the impact of the anisotropy of the nonlinearity on the growth rate of periodic perturbations. Our findings are confirmed by experimental measurements in a strontium barium niobate photorefractive crystal.
We study the higher order harmonics of scalar modulational instability in the regime where it arises spontaneously through amplification of vacuum fluctuations. We obtain detailed predictions concerning the detunings, intensities, growth rates and spectral widths of the harmonics. These predictions are well verified by experimental results obtained by propagating high intensity light pulses through optical fibers.
We present a technique that uses noisy broadband pulse bursts generated by modulational instability to probe nonlinear processes, including infrared-inactive Raman transitions, in molecular gases. These processes imprint correlations between different regions of the noisy spectrum, which can be detected by acquiring single shot spectra and calculating the Pearson correlation coefficient between the different frequency components. Numerical simulations verify the experimental measurements and are used to further understand the system and discuss methods to improve the signal strength and the sp…
We study theoretically the propagation of relatively long pulses with ionizing intensities in a hollow-core photonic crystal fiber filled with a Raman-inactive gas. Due to photoionization, previously unknown types of asymmetric self-phase modulation and `universal' modulational instabilities existing in both normal and anomalous dispersion regions appear. We also show that it is possible to spontaneously generate a plasma-induced continuum of blueshifting solitons, opening up new possibilities for pushing supercontinuum generation towards shorter and shorter wavelengths.
Gas-filled hollow-core photonic crystal fiber (PCF) is used for efficient nonlinear temporal compression of femtosecond laser pulses, two main schemes being direct soliton-effect self-compression, and spectral broadening followed by phase compensation. To obtain stable compressed pulses, it is crucial to avoid decoherence through modulational instability (MI) during spectral broadening. Here we show that changes in dispersion due to spectral anti-crossings between the fundamental core mode and core wall resonances in anti-resonant-guiding hollow-core PCF can strongly alter the MI gain spectrum…
We have numerically studied the effect of free-carrier-induced loss and dispersion on the modulational instability (MI) gain at low input powers in silicon-on-insulator (SOI) nanowaveguides with normal and anomalous second-order dispersion. We have shown that the free carriers affect the gain spectra even at low input powers. First time we have reported the gain in normal SOI nanowaveguides even in the absence of higher order dispersion parameters, which is due to the interaction of free-carrier-induced dispersion and nonlinearity. The MI gain in an anomalous SOI nanowaveguide vanishes even at…
Based on the Wigner distribution approach, an analysis of the effect of partial incoherence on the transverse instability of soliton structures in nonlinear Kerr media is presented. It is explicitly shown, that for a Lorentzian incoherence spectrum the partial incoherence gives rise to a damping which counteracts, and tends to suppress, the transverse instability growth. However, the general picture is more complicated and it is shown that the effect of the partial incoherence depends crucially on the form of the incoherence spectrum. In fact, for spectra with finite rms-width, the partial inc…
We study modulational instability in nonlinear arrays of subwavelength metallic nanoparticles, and analyze numerically nonlinear scenarios of the instability development. We demonstrate that modulational instability can lead to the formation of regular periodic or quasi-periodic modulations of the polarization. We reveal that such nonlinear nanoparticle arrays can support long-lived standing and moving oscillating nonlinear localized modes - plasmon oscillons.
It is shown that the properties of the modulational instability of partially coherent waves propagating in a nonlinear Kerr medium depend crucially on the profile of the incoherent field spectrum. Under certain conditions, the incoherence may even enhance, rather than suppress, the instability. In particular, it is found that the range of modulationally unstable wave numbers does not necessarily decrease monotonously with increasing degree of incoherence and that the modulational instability may still exist even when long wavelength perturbations are stable.
We investigate the modulational instability of a mathematical model [based on the model proposed by T. Brabec and F. Krausz, Phys.Rev. Lett. 78, 3282(1997)] appropriate for few cycle optical pulses with pulse duration as short as one carrier oscillation cycle in the context of a standard silica fiber operating at the telecommunication wavelength 1550 nm.
We analyze the modulational instability of nonlinear Bloch waves in topological photonic lattices. In the initial phase of the instability development captured by the linear stability analysis, long wavelength instabilities and bifurcations of the nonlinear Bloch waves are sensitive to topological band inversions. At longer timescales, nonlinear wave mixing induces spreading of energy through the entire band and spontaneous creation of wave polarization singularities determined by the band Chern number. Our analytical and numerical results establish modulational instability as a tool to probe …
We study the modulational instability induced by periodic variations of group-velocity dispersion and nonlinear coefficients in a highly birefringent fiber. We observe, for each resonance order, the presence of two pairs of genuine vector type sidebands, which are spectrally unbalanced between the polarization components for nonzero group-index mismatch, and one pair of balanced sidebands emerging and dominating at increasing group-index mismatch. As the conventional modulational instability manifests itself, it is partially suppressed by the proximity of these new unstable regions.
We investigate theoretically frequency comb generation in a bottle microresonator accounting for the azimuthal and axial degrees of freedom. We first identify a discrete set of the axial nonlinear modes of a bottle microresonator that appear as tilted resonances bifurcating from the spectrum of linear axial modes. We then study azimuthal modulational instability of these modes and show that families of 2D soliton states localized both azimuthally and axially bifurcate from them at critical pump frequencies. Depending on detuning, 2D solitons can be either stable, or form persistent breathers, …
We investigate one-dimensional transverse modulational instability in a non local medium excited with a spatially incoherent source. Employing undoped nematic liquid crystals in a planar pre-tilted configuration, we investigate the role of the spectral broadening induced by incoherence in conjunction with the spatially non local molecular reorientation. The phenomenon is modeled using the Wigner transform.
We show that the use of a dispersion oscillating fiber in passive cavities significantly extend modulational instability to novel high-frequency bands, which also destabilize the branches of the steady response which are stable with homogeneous dispersion. By means of Floquet theory, we obtain exact explicit expression for the sideband gain, and a simple analytical estimate for the frequencies of maximum gain. Numerical simulations show that stable stationary trains of pulses can be excited in the cavity.
We investigate the modulational instability of plane waves in quadratic nonlinear materials with linear and nonlinear quasi-phase-matching gratings. Exact Floquet calculations, confirmed by numerical simulations, show that the periodicity can drastically alter the gain spectrum but never completely removes the instability. The low-frequency part of the gain spectrum is accurately predicted by an averaged theory and disappears for certain gratings. The high-frequency part is related to the inherent gain of the homogeneous non-phase-matched material and is a consistent spectral feature.
We introduce a complete analytical and numerical study of the modulational instability process in a system governed by a canonical nonlinear Schrödinger equation involving local, arbitrary nonlinear responses to the applied field. In particular, our theory accounts for the recently proposed higher-order Kerr nonlinearities, providing very simple analytical criteria for the identification of multiple regimes of stability and instability of plane-wave solutions in such systems. Moreover, we discuss a new parametric regime in the higher-order Kerr response which allows for the observation of seve…
We study on the relations between modulational instability and several well-known nonlinear excitations in a nonlinear fiber, such as bright soliton, nonlinear continuous wave, Akhmediev breather, Peregrine rogue wave, and Kuznetsov-Ma breather. We present a quantitative correspondence between them based on the dominant frequency and propagation constant of each perturbation on a continuous wave background. Especially, we find rogue wave comes from modulational instability under the "resonance" perturbation with continuous wave background. These results will deepen our understanding on rogue w…
The nonlinear optical properties of semiconductors near an excitonic resonance are theoretically investigated by using the macroscopic J-model [Th. Oestreich and A. Knorr, Phys. Rev. B 48, 17811 (1993); ibid. 50, 5717 (1994)] based on the microscopic semiconductor Bloch equations. These nonlinear properties cause modulational instability of long light pulses with large gain, and give rise to a self-induced transmission of short light pulses in a semiconductor. By an example of the latter well studied effect the validity of the used macroscopic model is demonstrated and a good agreement is foun…
We have investigated the modulational instability (MI) in a negative index media (NIM) using a new generalized model describing the pulse propagation in a negative index material embedded into a Kerr medium. We have found that one could control the gain of MI in a NIM by tuning the initial electric or magnetic field amplitudes . Our model successfully recovers previously proposed models to describe pulse propagation in NIMs exhibiting Kerr nonlinearity. Moreover it contains a few additional terms connecting both the electric and magnetic field envelopes in a NIM.
The nonlinear stage of modulational instability in optical fibers induced by a wide and easily accessible class of localized perturbations is studied using the nonlinear Schrodinger equation. It is showed that the development of associated spatio-temporal patterns is strongly affected by the shape and the parameters of the perturbation. Different scenarios are presented that involve an auto-modulation developing in a characteristic wedge, possibly coexisting with breathers which lie inside or outside the wedge.
With reference to spatially non-local nematic liquid crystals, we develop a theory of optical spatial solitons and modulational instability in anisotropic media with arbitrarily large birefringence. Asymmetric spatial profiles and multivalued features are predicted for self-localized light versus walk-off angle. The results hold valid for generic self-focusing birefringent media and apply to large angle steering of individual and multiple self-trapped optical beams.
We present the first experimental investigation of modulational instability in a layered Kerr medium. The particularly interesting and appealing feature of our configuration, consisting of alternating glass-air layers, is the piecewise-constant nature of the material properties, which allows a theoretical linear stability analysis leading to a Kronig-Penney equation whose forbidden bands correspond to the modulationally unstable regimes. We find very good {\it quantitative} agreement between theoretical, numerical, and experimental diagnostics of the modulational instability. Because of the pe…
Light states relevant to the $χ^{(2)}$ comb generation in optical microresonators possess typically dual backgrounds for coupled first-harmonic (FH) and second-harmonic (SH) envelopes. Stability and/or instability of these backgrounds is crucial for realization of stable $χ^{(2)}$ combs and also for an efficient SH generation. We explore the properties of the dual backgrounds and their instability for the cases of FH and SH pumping of the resonator. In contrast to the optical parameteric oscillation, the instability is controlled by a 4th-degree characteristic equation for the increment. Coeff…
We study modulational instability (MI) in optical fibers with random group velocity dispersion (GVD) generated by sharply localized perturbations of a normal GVD fiber that are either randomly or periodically placed along the fiber and that have random strength. This perturbation leads to the appearance of low frequency MI side lobes that grow with the strength of the perturbations, whereas they are faded by randomness in their position. If the random perturbations exhibit a finite average value, they can be compared with periodically perturbed fibers, where Arnold tongues appear. In that case…
We present an analysis of modulational instability of diffractionless waves in a face-centered square lattice of waveguides featuring non-Kerr nonlinearity, which are constituted by a combination of positive and negative refractive indices. The unit cell of the lattice consists of three different waveguides with different optical properties. The dispersion curve of the lattice supports flat bands and thereby the base equations describing the model have particular solutions that correspond to the diffractionless waves propagating along the waveguides. We also observe a unique ramification of no…
We analyze the modulation instability induced by periodic variations of group velocity dispersion and nonlinearity in optical fibers, which may be interpreted as an analogue of the well-known parametric resonance in mechanics. We derive accurate analytical estimates of resonant detuning, maximum gain and instability margins, significantly improving on previous literature on the subject. We also design a periodically tapered photonic crystal fiber, in order to achieve narrow instability sidebands at a detuning of 35 THz, above the Raman maximum gain peak of fused silica. The wide tunability of …
We study, both theoretically and experimentally, modulational instability in optical fibers that have a longitudinal evolution of their dispersion in the form of a Dirac delta comb. By means of Floquet theory, we obtain an exact expression for the position of the gain bands, and we provide simple analytical estimates of the gain and of the bandwidths of those sidebands. An experimental validation of those results has been realized in several microstructured fibers specifically manufactured for that purpose. The dispersion landscape of those fibers is a comb of Gaussian pulses having widths muc…
We study a one-dimensional discrete nonlinear Schrödinger model with hopping to the first and a selected N-th neighbor, motivated by a helicoidal arrangement of lattice sites. We provide a detailed analysis of the modulational instability properties of this equation, identifying distinctive multi-stage instability cascades due to the helicoidal hopping term. Bistability is a characteristic feature of the intrinsically localized breather modes, and it is shown that information on the stability properties of weakly localized solutions can be inferred from the plane-wave modulational instability …
We investigate analytically, numerically, and experimentally the modulational instability in a layered, cubically-nonlinear (Kerr) optical medium that consists of alternating layers of glass and air. We model this setting using a nonlinear Schrödinger (NLS) equation with a piecewise constant nonlinearity coefficient and conduct a theoretical analysis of its linear stability, obtaining a Kronig-Penney equation whose forbidden bands correspond to the modulationally unstable regimes. We find very good {\it quantitative} agreement between the theoretical analysis of the Kronig-Penney equation, num…
In this paper intermodal modulational instability (IM-MI) is analyzed in a multimode fiber where several spatial and polarization modes propagate. The coupled nonlinear Schrödinger equations describing the modal evolution in the fiber are linearized and reduced to an eigenvalue problem. As a result, the amplification of each mode can be described by means of the eigenvalues and eigenvectors of a matrix that stores the information about the dispersion properties of the modes and the modal power distribution of the pump. Some useful analytical formulas are also provided that estimate the modal a…
Nonlinear light propagation in a single-mode micron-size waveguide made of semiconducting excitonic material has been theoretically studied in terms of exciton-polaritons by using an analysis based on macroscopic fields. When a light pulse is spectrally centered in the vicinity of the ground-state Wannier exciton resonance, it interacts with the medium nonlinearly. This optical cubic nonlinearity is caused by the repulsive exciton-exciton interactions in the semiconductor, and at resonance it is orders of magnitude larger than the Kerr nonlinearity (e.g., in silica). We demonstrate that a very…
We vary the time-frequency mode structure of ultrafast pulse-pumped modulational instability (MI) twin beams in an argon-filled hollow-core kagomé-style PCF by adjusting the pressure, pump pulse chirp, fiber length and parametric gain. Compared to solid-core systems, the pressure dependent dispersion landscape brings increased flexibility to the tailoring of frequency correlations and we demonstrate that the pump pulse chirp can be used to tune the joint spectrum of femtosecond-pumped sources. We also characterize the resulting mode content, not only by measuring the multimode second-order cor…
We investigate modulational instability (MI) in asymmetric dual-core nonlinear directional couplers incorporating the effects of the differences in effective mode areas and group velocity dispersions, as well as phase- and group-velocity mismatches. Using coupled-mode equations for this system, we identify MI conditions from the linearization with respect to small perturbations. First, we compare the MI spectra of the asymmetric system and its symmetric counterpart in the case of the anomalous group-velocity dispersion (GVD). In particular, it is demonstrated that the increase of the inter-cor…
We consider chaotic dynamics of a system of two coupled ring resonators with a linear gain and a nonlinear absorption. Such a structure can be implemented in various settings including microresonator nanostructures, polariton condensates, optical waveguides or atomic Bose-Einstein condensates of ultra-cold atoms placed in a circular-shaped trap. From the theoretical point of view this system is attractive due to its modulational instability and rich structure, including various types of spontaneous symmetry breaking, period doubling bifurcations, eventually leading to chaotic regime. It is des…
We study modulational instability (MI) in optical fibers with random group-velocity dispersion (GVD). We consider Gaussian and dichotomous colored stochastic processes. We resort to different analytical methods (namely, the cumulant expansion and the functional approach) and assess their reliability in estimating the MI gain of stochastic origin. If the power spectral density (PSD) of the GVD fluctuations is centered at null wavenumber, we obtain low-frequency MI sidelobes which converge to those given by a white noise perturbation when the correlation length tends to 0. If instead the stochas…
We discuss the modulational instability (MI) of plane waves in nonlocal Kerr media with the sine-oscillation response function, which can model the nematic liquid crystal with negative dielectric anisotropy. The results in the framework of the nonlocal nonlinear Schrödinger equation show that the MI in this system has two extraordinary properties. First, MI exists in both cases of negative Kerr coefficient $(n_{2}<0)$ and positive Kerr coefficient $(n_{2}>0)$. Second, the maximum gain of MI is not affected by the intensity of the plane wave, and the wave number %spatial frequency of the maximu…
We analytically study rogue-wave (RW) solutions and rational solitons of an integrable fifth-order nonlinear Schrödinger (FONLS) equation with three free parameters. It includes, as particular cases, the usual NLS, Hirota, and Lakshmanan-Porsezian-Daniel (LPD) equations. We present continuous-wave (CW) solutions and conditions for their modulation instability in the framework of this model. Applying the Darboux transformation to the CW input, novel first- and second-order RW solutions of the FONLS equation are analytically found. In particular, trajectories of motion of peaks and depressions o…
Optical microcavities supporting exciton-polariton quasi-particles offer one of the most powerful platforms for investigation of rapidly developing area of topological photonics in general, and of photonic topological insulators in particular. Energy bands of the microcavity polariton graphene are readily controlled by magnetic field and influenced by the spin-orbit coupling effects, a combination leading to formation of linear unidirectional edge states in polariton topological insulators as predicted very recently. In this work we depart from the linear limit of non-interacting polaritons an…
We analyze the consequences of dissipative heating in driven Kerr microresonators theoretically and numerically, using a thermal Lugiato-Lefever model. We show that thermal sensitivity modifies the stability range of continuous wave in a way that blocks direct access to broadband frequency-comb forming waveforms, and we propose a deterministic access path that bypasses the thermal instability barrier. We describe a novel thermal instability that leads to thermooptical oscillations via a Hopf bifurcation.
