Instantaneous Nonlinearity Research Articles

Comment on “Optical solitons for 2D-NLSE in multimode fiber with Kerr nonlinearity and its modulation instability”

Most recently, Baber et al. (Optical solitons for 2D-NLSE in multimode fiber with Kerr nonlinearity and its modulation instability, Mod. Phys. Lett. B (2024) 2450341, doi:10.1142/S021798492450341X2450341-1) claimed to have obtained exact solutions of a two-dimensional nonlinear Schrödinger equation with an instantaneous Kerr nonlinearity that governs beam movement within a multimode optical fiber featuring a parabolic index shape, using the Jacobi Elliptic Function Expansion. Additionally, they have analyzed the modulation instability for the underlying model. It is the aim of this comment to point out that all results found in that work and published in the journal Mod. Phys. Lett. B (doi:10.1142/S021798492450341X2450341-1) are all erroneous, from the exact solutions of the considered equation to the modulation instability of the underlying model.

Optical solitons for 2D-NLSE in multimode fiber with Kerr nonlinearity and its modulation instability

This paper deals with the soliton solutions for beam movement within a multimode optical fiber featuring a parabolic index shape. It is considered that a Two-Dimensional Nonlinear Schrödinger Equation (2D-NLSE) with an instantaneous Kerr nonlinearity of the kind can represent the beam dynamics. Nonlinear Multimode Optical Fibers (MMFs) of this kind are gaining popularity because they provide novel approaches to control the spectral, temporal, and spatial characteristics of ultrashort light pulses. We gain the optical soliton solutions for the nonlinear evolution beam dynamics using the Jacobi Elliptic Function Expansion (JEFE) method. The exact analytical solution of Nonlinear Partial Differential Equations (NLPDEs) can be achieved with wide application using the effective JEFE approach. These solutions are obtained in the form of dark, bright, combined dark–bright, complex combo, periodic, and plane wave solutions. Additional solutions for Jacobi elliptic functions, encompassing both single and dual function solutions, have been acquired. This approach is based on Jacobi elliptic functions, which will provide us the exact soliton solutions to nonlinear problems. Additionally, we will analyze the Modulation Instability (MI) for the underlying model. Moreover, we show the physical behavior of the beam propagation in a multimode optical fiber the three-dimensional, two-dimensional, and their corresponding contour plots are dispatched using the different values of parameters.

Processing of nonlinear concrete creep curves using nonlinear optimization methods

The article proposes a method for determining the rheological parameters of concrete based on creep curves at various stress levels using the theory of V.M. Bondarenko. Using the proposed methodology, the experimental data presented in the work of A.V. Yashin is processed. The problem of searching for rheological parameters is posed as a nonlinear optimization problem. The sum of squared deviations of the experimental values of creep strains from the theoretical ones is minimized. The interior point method is used as a nonlinear optimization method. Four different expressions for the creep measure are considered, including the creep measure by N.Kh. Harutyunyan, creep measure by A.G. Tamrazyan, a creep measure in the form of a sum of two exponentials, and McHenry’s creep measure. It has been shown that the best agreement with experimental data is provided by the McHenry’s creep measure. An expression has been selected for the nonlinearity function, which describes the nonlinear relationship between stresses and creep strains. It is shown that the instantaneous nonlinearity of deformation and the nonlinearity that manifests itself over time cannot be described by a single function.

Hybridization of solitons in realistic non-instantaneous fibers

Non-local nonlinearities are an integral part of numerous physical systems. Yet, their dissipative nature makes it hard to identify and verify stable eigenstates rigorously. This work sheds new light onto the existence of temporally non-local solitons – a class of dissipative self-maintaining states in time theoretically predicted to emerge in media with a time-delayed (i.e., non-instantaneous) nonlinear response. Revisiting the non-instantaneous nonlinear Schrödinger equation introduced by Conti et al., I introduce a semi-analytical approach to reveal such non-instantaneous solitons in liquid-type media which exhibit realistic non-exponential response functions featuring both rise and decay times. I investigate the susceptibility of such states to typical perturbations, such as third-order dispersion, instantaneous Kerr nonlinearity, and causality of the material response. Empirical numeric experiments illustrate how non-instantaneous solitons may compete against and even fuse with instantaneous (Kerr-type) solitons to an inseparable hybrid solitary waves. While finding a numerical solution to define the nature of this mixed state remains a challenge, I introduce a phase balance condition and other benchmarks which will allow to identify the occurrence of such states in realistic liquid-core fiber systems in future work. The findings of this study give reason to hypothetize a new type of dissipative solitary state and provide new means to understanding information transport and system dynamics of highly non-instantaneous nonlinear systems, particularly liquid-core optical fibers and waveguides.

Femtojoule femtosecond all-optical switching in lithium niobate nanophotonics

Optical nonlinear functions are crucial for various applications in integrated photonics, including all-optical information processing1, photonic neural networks2,3 and on-chip ultrafast light sources4,5. However, the weak native nonlinearity of most nanophotonic platforms has imposed barriers for such functions by necessitating large driving energies, high-Q cavities or integration with other materials with stronger nonlinearity. Here we effectively utilize the strong and instantaneous quadratic nonlinearity of lithium niobate nanowaveguides for the realization of cavity-free all-optical switching. By simultaneous engineering of the dispersion and quasi-phase matching, we design and demonstrate a nonlinear splitter that can achieve ultralow switching energies down to 80 fJ, featuring a fastest switching time of ~46 fs and a lowest energy–time product of 3.7 × 10−27 J s in integrated photonics. Our results can enable on-chip ultrafast and energy-efficient all-optical information processing, computing systems and light sources.

Using Interleaved Stimulation to Measure the Size and Selectivity of the Sustained Phase-Locked Neural Response to Cochlear Implant Stimulation

We measured the sustained neural response to electrical stimulation by a cochlear implant (CI). To do so, we interleaved two stimuli with frequencies F1 and F2 Hz and recorded a neural distortion response (NDR) at F2-F1 Hz. We show that, because any one time point contains only the F1 or F2 stimulus, the instantaneous nonlinearities typical of electrical artefact should not produce distortion at this frequency. However, if the stimulus is smoothed, such as by charge integration at the nerve membrane, subsequent (neural) nonlinearities can produce a component at F2-F1 Hz. We stimulated a single CI electrode with interleaved sinusoids or interleaved amplitude-modulated pulse trains such that F2 = 1.5F1, and found no evidence for an NDR when F2-F1 was between 90 and 120 Hz. However, interleaved amplitude-modulated pulse trains with F2-F1~40 Hz revealed a substantial NDR with a group delay of about 45 ms, consistent with a thalamic and/or cortical response. The NDR could be measured even from recording electrodes adjacent to the implant and at the highest pulse rates (> 4000 pps) used clinically. We then measured the selectivity of this sustained response by presenting F1 and F2 to different electrodes and at different between-electrode distances. This revealed a broad tuning that, we argue, reflects the overlap between the excitation elicited by the two electrodes. Our results also provide a glimpse of the neural nonlinearity in the auditory system, unaffected by the biomechanical cochlear nonlinearities that accompany acoustic stimulation. Several potential clinical applications of our findings are discussed.

Self-Localization of Light Beams in a Medium with Instantaneous Kerr Nonlinearity Switching

A model is proposed for the nonlinear response of a medium in which the permittivity depending on the field intensity is instantaneously switched when the field amplitude reaches a certain value. The propagation of a self-channeled light beam in a medium with such a nonlinearity and with the same type of the Kerr response (self-focusing or defocusing) is considered. It is shown that when the switching field exceeds the threshold value, a symmetric finite-length region (domain) with different optical properties is formed along the beam channel. Three cases of combination of the nonlinearity signs and the values of the effective refractive index are considered. In the focusing medium, the power of radiation propagating along the interface is higher than in the defocusing medium. In the defocusing medium, there exists a threshold value of the total luminous energy flux, beginning with which self-localization can occur, when the values of the effective refractive index lie between the unperturbed dielectric constants of the medium and of the domain.

Optical bistability with bound states in the continuum in dielectric gratings

We consider light scattering by dielectric gratings supporting optical bound states in the continuum. Due to the presence of instantaneous Kerr nonlinearity the critical field enhancement in the spectral vicinity of the bound state triggers the effect of optical bistability. The onset of bistability is explained theoretically in the framework of the temporal coupled mode theory. As the central result we cast the problem into the form of a singly field-driven nonlinear oscillator. The theoretical results are verified in comparison against full-wave numerical simulations.

Combining FDTD and coupled-mode theory for self-pulsing modeling in coupled nonlinear microring resonators

By means of finite-difference time-domain (FDTD) simulations, the stationary and dynamic responses of coupled optical microring resonators made with a material exhibiting an instantaneous Kerr nonlinearity are numerically investigated. We compare the results with the coupled-mode theory (CMT) and find good agreement. We demonstrate by integrating Maxwell’s equations that this system can show a self-pulsing operation in the normal dispersion regime. Finally, this work shows that FDTD simulations can be combined with a CMT analysis for the purpose of designing optical functions whose operation is based on nonlinear coupled micro-resonators.

A Convolution-Free Finite-Element Time-Domain Method for the Nonlinear Dispersive Vector Wave Equation

In this article, a finite-element time-domain method is presented for the solution of the second-order vector wave equation (VWE) subject to electrically complex materials, including general combinations of linear dispersion, instantaneous nonlinearity, and dispersive nonlinearity. The presented method is novel in that it offers greater geometric flexibility than existing finite-difference methods, incorporates both instantaneous and dispersive nonlinearity, scales to arbitrary dispersive and nonlinear orders, and is simpler, faster, and requires less computational complexity than existing mixed formulations due to the use of edge elements only.

Nonlinear control of coherent absorption and its optical signal processing applications

All-optical data processing continues to attract significant interest as a way to overcome the electronic signal processing bottleneck of fiber telecommunication networks. Nonlinear optical devices such as limiters and saturable absorbers rely on intensity-dependent attenuation of light. However, making such devices using intensity-dependent multiphoton dissipation processes is an issue as these make complete absorption and transmission impossible. Here, we show that nonlinear phase retardation in an optical fiber can control the dissipation of coherent light waves interacting on a thin plasmonic absorber from total absorption to perfect transmission. The fiber’s instantaneous Kerr nonlinearity and the femtosecond coherent absorption time scale make this approach ultrafast. We report proof-of-principle demonstrations of all-optical intensity discrimination, power limiting, pulse restoration, pulse splitting, and signal transfer between carrier wavelengths within a fiber circuit. Our results indicate that nonlinear control of coherent absorption can imitate and outperform saturable and multiphoton absorption in terms of bandwidth and contrast.

A Perfectly Matched Layer for the Nonlinear Dispersive Finite-Element Time-Domain Method

A novel implementation of a perfectly matched layer (PML) is presented for the truncation of finite-element time-domain (FETD) meshes containing electrically complex materials, exhibiting any combination of linear dispersion, instantaneous nonlinearity, and dispersive nonlinearity. Based on the complex coordinate stretching formulation of the PML, the presented technique yields an artificial absorbing layer whose matching condition is independent of material parameters. Moreover, by virtue of only modifying spatial derivatives, the incorporation of the PML into existing solvers for complex media is simple and straightforward. The resulting material-independent PML is incorporated into a nonlinear dispersive method for the vector wave equation, which leverages the $z$ transform and Newton-Raphson techniques to yield an implementation free from recursive convolutions, auxiliary differential equations, and linearizations. This permits the unprecedented truncation and attenuation of nonlinear phenomena, such as spatial and temporal solitons, within the FETD method.

Light Scattering, Absorption, and Refraction due to High-Order Optical Nonlinearities in Colloidal Gold Nanorods

We investigated the nonlinear (NL) optical behavior of aqueous colloidal dispersions of rod-shaped gold nanoparticles (NPs) with different aspect ratios beyond the usual analysis of third-order nonlinearity. By applying the Z-scan technique, it was possible to observe large NL absorption, which is understood by considering the simultaneous instantaneous and thermo-optical nonlinearity contributions. In addition, light-scattering experiments were performed at high intensities, showing a significant contribution of the fifth-order scattering process that depends on the gold nanorod (Au-NR) aspect ratio. Likewise, the Z-scan measurements reveal that the quintic nonlinearity is also contributing to the refractive behavior, a feature known for spherical gold NPs but not reported for Au-NRs so far. The experimental results illustrate a nonlinearity management procedure based on the strong influence of the aspect ratio of Au-NRs in the excitation regime close to the transverse surface plasmon resonance, even kee...

A Convolution-Free Mixed Finite-Element Time-Domain Method for General Nonlinear Dispersive Media

In this paper, a mixed finite-element time-domain (FETD) method is presented for the simulation of electrically complex materials, including general combinations of linear dispersion, instantaneous nonlinearity, and dispersive nonlinearity. Using both edge and face elements, the presented method offers greater geometric flexibility than existing finite-difference time-domain (FDTD) implementations, and in contrast to existing nonlinear FETD methods, also incorporates both linear and nonlinear material dispersions. Dielectric nonlinearity is incorporated into the Crank–Nicolson mixed FETD formulation via a straightforward Newton–Raphson approach, for which the associated Jacobian is derived. Moreover, the dispersion is modeled via the Mobius $z$ transform method, yielding a simpler more general algorithm. The method’s accuracy and convergence are verified, and its capability demonstrated via the simulation of several nonlinear phenomena, including temporal and spatial solitons in two spatial dimensions.

Temporal dynamics of the generator of stimulated otoacoustic emissions

The early emission work of David Kemp investigated temporal properties of click-evoked otoacoustic emissions (CEOAEs). Suppressor clicks were positioned up to 10 ms before or after the evoking (test) click and reduced the emission amplitude. Intriguingly, suppressors that preceded the test click by 1-2 ms were more effective than simultaneously presented suppressors. This observation seems not to support the hypothesis that emission generation operates on the basis of an instantaneous gain/suppression mechanism, for which simultaneously presented suppressors should be most effective. Kemps’ observations thus left the field with an important OAE generation question that inspired several OAE studies. Kemps’ results can be explained on the basis of (i) a non-instantaneous gain mechanism at the OAE generation site, or (ii), complex temporal interactions of basilar-membrane impulse responses (BM IRs) that operate with an instantaneous nonlinearity, but yield the observed non-instantaneous suppression properties. We investigated these dynamics using a nonlinear transmission-line model of the human cochlea and found that maximal CEOAE suppression can occur for preceding clicks even when the cochlear nonlinearity is kept time-invariant, supporting hypothesis (ii). Additionally, we used the frequency-dependence of CEOAE suppression to quantify human BM IR duration and cochlear filter tuning, yielding QERB estimates of 13.8 F[in kHz]0.22.The early emission work of David Kemp investigated temporal properties of click-evoked otoacoustic emissions (CEOAEs). Suppressor clicks were positioned up to 10 ms before or after the evoking (test) click and reduced the emission amplitude. Intriguingly, suppressors that preceded the test click by 1-2 ms were more effective than simultaneously presented suppressors. This observation seems not to support the hypothesis that emission generation operates on the basis of an instantaneous gain/suppression mechanism, for which simultaneously presented suppressors should be most effective. Kemps’ observations thus left the field with an important OAE generation question that inspired several OAE studies. Kemps’ results can be explained on the basis of (i) a non-instantaneous gain mechanism at the OAE generation site, or (ii), complex temporal interactions of basilar-membrane impulse responses (BM IRs) that operate with an instantaneous nonlinearity, but yield the observed non-instantaneous suppression propertie...

Control of the third-harmonic generation efficiency upon interaction of few-cycle waves in nonlinear media

It is shown that the interaction of few-cycle waves propagating at an angle to each other in media with instantaneous cubic nonlinearity may lead to a considerable increase in the third-harmonic generation efficiency without energy redistribution between the intersecting beams with a wide spectrum.

Ultrafast saturable absorption of MoS2 nanosheets under different pulse-width excitation conditions.

The newly raised two-dimensional material MoS2 is regarded as an ideal candidate for saturated absorbers. Here, the open-aperture Z-scan method is used to study the saturation absorption (SA) response of monolayer and multilayer MoS2, considering laser irradiation with different pulse widths. Specifically, in cases of 10ns and 10ps laser pulses, the accumulative nonlinearity [e.g., free carrier absorption (FCA)] coupled with SA is found in both monolayer and multilayer MoS2. However, under a 65fs pulse laser, the instantaneous nonlinearity [e.g., two-photon absorption (TPA)] and the SA effect turn to play a significant role. Additionally, the saturation of both TPA and FCA is observed in MoS2. Importantly, the modulation depth of MoS2 shows different change trends by adjusting the laser pulse width.

Measurement of nonlinear refractive indices of air, oxygen, and nitrogen in capillary by changing the temporal width of short laser pulses

Nonlinear refractive indices of air and its constituents were measured as a function of the pulse width in a capillary used to confine the gas and to increase the interaction length. This study was carried out with nonlinear ellipse rotation measurements using an amplified Ti:sapphire laser system providing pulses at 800 nm, continuously tunable from ∼60 fs up to ∼3 ps. Due to the presence of noninstantaneous orientational response, nonlinearities in molecular gases strongly depend on the pulse width. The pulse width dependence and discrimination between instantaneous and noninstantaneous nonlinearities of air gases were obtained with a simple exponential growth model.

Theoretical estimation of nonlinear optical force on dielectric spherical particles of arbitrary size under femtosecond pulsed excitation

Experimental evidence indicates that high-repetition-rate ultrafast pulsed excitation is more efficient in optical trapping of dielectric nanoparticles as compared with continuous-wave excitation at the same average power. The physics behind the different nature of force under these two excitation conditions remained deceptive until quite recently when it was theoretically explained, in the dipole limit, as a combined effect of (1) repetitive instantaneous momentum transfer and (2) optical Kerr nonlinearity. The role of optical Kerr effect was theoretically studied for larger dielectric spherical particles, in the ray optics limit, also. However, a theoretical underpinning is yet to be established as to whether the effect of optical nonlinearity is omnipresent across different particle sizes, which we investigate here. Using localized approximation of generalized Lorenz-Mie theory, we theoretically analyze the nature of force (and potential) and provide a detailed comparative discussion between this generalized scattering formulation with dipole scattering formulation for dielectric nanoparticles.

Ultrafast nonlinear dynamics of thin gold films due to an intrinsic delayed nonlinearity

Using long-range surface plasmon polaritons light can propagate in metal nano-scale waveguides for ultracompact opto-electronic devices. Gold is an important material for plasmonic waveguides, but although its linear optical properties are fairly well understood, the nonlinear response is still under investigation. We consider the propagation of pulses in ultrathin gold strip waveguides, modeled by the nonlinear Schrödinger equation. The nonlinear response of gold is accounted for by the two-temperature model, revealing it as a delayed nonlinearity intrinsic in gold. The consequence is that the measured nonlinearities are strongly dependent on pulse duration. This issue has so far only been addressed phenomenologically, but we provide an accurate estimate of the quantitative connection as well as a phenomenological theory to understand the enhanced nonlinear response as the gold thickness is reduced. In comparison with previous works, the analytical model for the power-loss equation has been improved, and can be applied now to cases with a high laser peak power. We show new fits to experimental data from the literature and provide updated values for the real and imaginary parts of the nonlinear susceptibility of gold for various pulse durations and gold layer thicknesses. Our simulations show that the nonlinear loss is inhibiting efficient nonlinear interaction with low-power laser pulses. We therefore propose to design waveguides suitable for the mid-IR, where the ponderomotive instantaneous nonlinearity can dominate over the delayed hot-electron nonlinearity and provide a suitable plasmonics platform for efficient ultrafast nonlinear optics.