Nonlinear Optical Wave Research Articles

Bloch-Wave Phase Matching of High Harmonic Generation in Solids.

There has been a longstanding doubt that the conversion efficiency of high harmonics in solids is much lower than expected at such a high level of electron density. To address this issue, we revisit the dynamical process of high harmonic generation (HHG) in solids in terms of wavelet interference. We find that the constructive interference among the wavelets has intrinsic consistency with the phase matching of coupled waves in nonlinear optics, which we call Bloch-wave phase matching. Our analysis indicates that most of the wavelets are out of phase and coherently canceled out during the solid HHG process, leading to only a small fraction of excited electrons effectively contributing to HHG. Consequently, the conversion from the excited electron to HHG is fairly low. Moreover, a Bloch-wave phase-matching scheme is proposed and a nearly 3 orders of magnitude enhancement of solid HHG can be achieved by engineering the crystal structures. Our Letter addresses a longstanding doubt and provides a novel idea and theoretical guidance for realizing efficient all-solid-state tabletop ultraviolet light sources.

Giant nonlinear optical wave mixing in a van der Waals correlated insulator.

Optical nonlinearities are one of the most fascinating properties of two-dimensional (2D) materials. While tremendous efforts have been made to find and optimize the second-order optical nonlinearity in enormous 2D materials, opportunities to explore higher-order ones are elusive because of the much lower efficiency. Here, we report the giant high odd-order optical nonlinearities in centrosymmetric correlated van der Waals insulator manganese phosphorus triselenide. When illuminated by two near-infrared femtosecond lasers, the sample generates a series of profound four- and six-wave mixing outputs. The near-infrared third-order nonlinear susceptibility reaches near the highest record values of 2D materials. Comparative measurements to other prototypical nonlinear optical materials [lithium niobate, gallium(II) selenide, and tungsten disulfide] reveal its extraordinary wave mixing efficiency. The wave mixing processes are further used for nonlinear optical waveguide with multicolor emission. Our work highlights the promising prospect for future research of the nonlinear light-matter interactions in the correlated 2D system and for potential nonlinear photonic applications.

Optical soliton solution of the perturbed Biswas-Milovic equation having cubic-quintic-septic law nonlinearity in the presence of spatio-temporal and chromatic dispersion

In this manuscript, we investigate the analytical and soliton solutions of the cubic-quintic-septic law for the perturbed Biswas-Milovic equation, considering spatio-temporal and chromatic dispersions. The perturbed Biswas-Milovic equation with the spatio-temporal and chromatic dispersion terms provides a comprehensive study for describing nonlinear optical wave propagation in optical fiber. We use the wave transformation to reduce the main equation to a nonlinear ordinary differential equation. The transformation of the original equation into a more simplified form aims to attain a more profound comprehension of the fundamental dynamics of the system. We retrieve the analytical solutions of the presented model by implementing the new Kudryashov technique and a subversion of the new extended auxiliary equation approach. Besides, bright, singular, and V-shape soliton structures are represented. By employing powerful analytical techniques, we systematically derive a wide range of soliton solutions. This approach successfully captures diverse soliton types highlighting the novelty of applying the new Kudryashov technique and a subversion of the new extended auxiliary equation method to this complex model. Moreover, we analyze the soliton behavior influenced by various parameters. The analysis of the parameter influences reveals the complicated relationship governing the dynamics of the perturbed Biswas-Milovic model. Furthermore, this manuscript includes the modulation instability analysis for the presented model. Conducting modulation instability analysis for the presented equation enhances our understanding of the system’s stability and dynamics.

The exact solutions to the generalized (2+1)-dimensional nonlinear wave equation

Due to the importance of the nonlinear partial differential equations in applied physics and engineering, many mathematicians and physicists are interesting to the nonlinear partial differential equations. One of the main tasks of studying the nonlinear partial differential equations is to obtain the exact solutions for the nonlinear partial differential equations. In this paper, we study the exact solutions of a generalized (2+1)-dimensional nonlinear partial differential equation. According to the modified hyperbolic function method and the traveling transformation method, the generalized (2+1)-dimensional nonlinear partial differential equation is changed into an ordinary differential equation. By taking the homogeneous balance between the highest order nonlinear terms and the highest order derivative terms, we change the differential equation into an algebraic system. By solving the algebraic system, we obtain the solutions for the nonlinear partial differential equation. At last, some solutions and their figures are given by specific parameters. All the solutions conclude the trigonometric function solutions, the positive hyperbolic function solutions, and the hyperbolic trigonometric function solutions. These solutions exhibit the dynamics of nonlinear waves and the solutions are useful for the study on interaction behavior of nonlinear waves in shallow water, plasma, nonlinear optics and Bose–Einstein condensates.

On the Propagation Model of Two-Component Nonlinear Optical Waves

Currently, two-component integrable nonlinear equations from the hierarchies of the vector nonlinear Schrodinger equation and the vector derivative nonlinear Schrödinger equation are being actively investigated. In this paper, we propose a new hierarchy of two-component integrable nonlinear equations, which have an important difference from the already known equations. To construct the hierarchical equations, we use the monodromy matrix method, as first proposed by B.A. Dubrovin. The method we use consists of solving the following sequence of problems. First, using the Lax operator, we find the monodromy matrix, which is a polynomial in the spectral parameter. More precisely, we find a sequence of monodromy matrices dependent on the degree of this polynomial. Each Lax operator has its own sequence of monodromy matrices. Then, using the terms from the decomposition of the monodromy matrix, we construct a sequence of second operators from a Lax pair. A hierarchy of evolutionary integrable nonlinear equations follows from the conditions of compatibility of the sequence of Lax pairs. Also, knowledge of the monodromy matrix allows us to find stationary equations that are analogs of the Novikov equations for the Korteweg–de Vries equation. In addition, the characteristic equation of the monodromy matrix corresponds to the spectral curve equation of the relevant multiphase solution for the integrable nonlinear equation. Since the coefficients of the spectral curve equation are integrals of the hierarchical equations, they can be utilized to find the simplest solutions of the constructed integrable nonlinear equations. In this paper, we demonstrate the operation of this method, starting with the assignment of the Lax operator and ending with the construction of the simplest solutions.

Shock waves in semi-infinite photonic lattices

We investigate the existence, stability and propagation dynamics of shock waves existing in defocusing saturable media modulated by a semi-infinite photonic lattice. Such surface waves consist of modulationally stable pedestals in bulk media and decaying oscillatory tails in semi-infinite photonic lattices. Due to Bragg-type reflection, they are strongly localized at the edge of the optical lattice. The kink steepness, pedestal height and localization degree can be controlled by propagation constant, saturable degree and lattice depth. Two types of kinks, i.e., “out-of-phase” and two branches of “in-phase” shock waves are revealed. Out-of-phase shock waves are stable in a substantial part of their existence domain. While the lower-branch in-phase waves with low energy flow are stable in almost their whole existence domain, the higher-branch in-phase waves are completely unstable. We thus show the first example of stable in-phase shock waves in nonlinear optics. Our findings provide new insight into the dynamics of semi-localized nonlinear surface shock waves or kink solitons.

Laser nanoprinting of 3D nonlinear holograms beyond 25000 pixels-per-inch for inter-wavelength-band information processing

Nonlinear optics provides a means to bridge between different electromagnetic frequencies, enabling communication between visible, infrared, and terahertz bands through χ(2) and higher-order nonlinear optical processes. However, precisely modulating nonlinear optical waves in 3D space remains a significant challenge, severely limiting the ability to directly manipulate optical information across different wavelength bands. Here, we propose and experimentally demonstrate a three-dimensional (3D) χ(2)-super-pixel hologram with nanometer resolution in lithium niobate crystals, capable of performing advanced processing tasks. In our design, each pixel consists of properly arranged nanodomain structures capable of completely and dynamically manipulating the complex-amplitude of nonlinear waves. Fabricated by femtosecond laser writing, the nonlinear hologram features a pixel diameter of 500 nm and a pixel density of approximately 25000 pixels-per-inch (PPI), reaching far beyond the state of the art. In our experiments, we successfully demonstrate the novel functions of the hologram to process near-infrared (NIR) information at visible wavelengths, including dynamic 3D nonlinear holographic imaging and frequency-up-converted image recognition. Our scheme provides a promising nano-optic platform for high-capacity optical storage and multi-functional information processing across different wavelength ranges.

Applying a transformation-based method to extract optical traveling waves from the Kundu–Mukherjee–Naskar equation

Optical solitons are solitary waves that propagate without changing shape due to a balance between dispersion and nonlinearity in the medium. Therefore, solitary optical waves are important solutions to nonlinear partial differential equations for modeling pulse propagation in optics. Our work derives new solitary wave solutions to the Kundu–Mukherjee–Naskar (KMN) equation, which governs complex nonlinear optical wave phenomena. Using innovative logarithmic transformation-based analytical techniques, various solution forms are obtained and expressed in closed form via elementary functions. The solutions are validated through direct substitution into the original KMN equation. Our new solutions provide fresh perspectives into the intricate soliton landscape described by this model. Since the KMN equation finds use in fiber optic communications, fluid dynamics, and other domains, these findings have broad implications. The methods showcase promising new pathways for unraveling soliton behaviors by fractional- and integer-order nonlinear models alike. Researchers can build upon these techniques to further advance understanding of the profound mathematical structures underlying real-world physical systems.

Nonlinear Plasmonic Meta‐Crystals with Coupling Induced Symmetry Breaking

AbstractSymmetry plays an important role in determining the amplitude and polarizations of nonlinear optical waves occurring in conventional crystals. While the symmetry of crystals is limited by the materials’ synthesis process, it can be easily controlled in photonic meta‐crystals. Both the linear and nonlinear optical properties of the meta‐crystals can be judiciously tailored by the local symmetry of the meta‐atoms. Here, it is demonstrated that the symmetry of the nonlinear meta‐crystals can be artificially manipulated by introducing the coupling between adjacent plasmonic meta‐atoms. By assembling the two groups of plasmonic meta‐atoms with inversion symmetry into a hexagonal boron nitride‐like lattice, the global inversion symmetry of the metasurface is broken. Second harmonic generations due to the coupling‐induced symmetry breaking are experimentally observed from the meta‐crystals. It is also found that the polarization of the second harmonic waves obey certain symmetry selection rules. The proposed methodology, which takes advantage of the coupling effects and the intrinsic nonlinearity of the plasmonic meta‐atoms, can be used to develop a variety of nonlinear nanophotonic devices with multiple functionalities.

Phase properties of several nonlinear optical waves described by rational solutions

Phase plays an essential role in both classical and quantum wave dynamics, which has motivated many scientists to study the phases of many different nonlinear waves. Among those nonlinear waves, the ones described by rational solutions have been given much attention, partly because some of them can describe rogue wave dynamics. We revisit the phase of several well-known one-dimensional rational solutions and clarify that sudden phase inversion processes indeed exist for rogue waves, in contrast to the ones reported before. Moreover, we investigate the phase properties of rogue waves and rational W-shaped solitons in the Hirota and Sasa-Satsuma model, for which some high-order physical effects are considered, i.e., the third-order dispersion delayed nonlinear response and self-steeping effects. The quite distinctive phases are uncovered for the rational W-shaped solitons with similar density profiles in the two models, by limitation analysis on the underlying topological vector potentials of other related nonlinear waves. These results indicate that the underlying topological vector potentials can be used to identify the phases of nonlinear waves, especially for localized waves with abrupt phase jumps.

Excitonic resonances control the temporal dynamics of nonlinear optical wave mixing in monolayer semiconductors

Monolayer semiconductors are an emerging platform for strong nonlinear light–matter interactions that are enhanced by the giant oscillator strength of tightly bound excitons. Little attention has been paid to the impact of excitonic resonances on the temporal dynamics of such nonlinearities, since harmonic generation and optical wave mixing are generally considered instantaneous processes. We find that a significant time difference, ranging from −40 to +120 fs, is necessary between two light pulses for optimal sum-frequency generation (SFG) and four-wave mixing (FWM) to occur from monolayer WSe2 when one of the pulses is in resonance with an excitonic transition. These resonances involve both band-edge A excitons and high-lying excitons that comprise electrons from conduction bands far above the bandgap. Numerical simulations in the density-matrix formalism rationalize the distinct dynamics of SFG and FWM. The interpulse delays for maximal SFG and FWM are governed primarily by the lifetime of the one-photon and two-photon resonant states, respectively. The method therefore offers an unconventional probe of the dynamics of excitonic states accessible with either one-photon or two-photon transitions. Remarkably, the longest delay times occur at the lowest excitation powers, indicating a strong nonlinearity that offers exploration potential for excitonic quantum nonlinear optics. Researchers show that resonant coupling of light pulses with excitonic transitions affects the optimal time difference between pulses for sum-frequency generation and four-wave mixing in monolayer WSe2.

Nonlinear Surface Polaritons near the Interface between a Magneto-Optical Substance and a Nonlinear Metamaterial with a Permittivity Close to Zero

Relevance. Considerable attention of researchers is drawn to nonlinear surface optical waves, as they are promising for application in optical information processing systems, ultra-sensitive sensors, and modern telecommunications components. Purpose. To obtain an exact solution of the Maxwell equation system for TM-polarised nonlinear surface polaritons propagating along the interface between an optically linear magnetic medium and an optically nonlinear medium with saturated nonlinearity. Methods. Analysis of the properties of nonlinear surface polaritons (NSP) and mathematical modelling of the dependence of the permittivity of a nonlinear medium on the intensity are used. Results. TM-polarised nanoparticles propagating along the interface of a magneto-optical medium and a nonlinear optical metamaterial with a permittivity close to zero (ENZ-metamaterial) are studied. For NSP in such a structure, an exact solution of the Maxwell equation system is obtained, considering the saturation effect of optical nonlinearity. On this basis, the dependences of constant propagation on the total energy flux of NSP and energy fluxes in contact media in the structure were investigated: magneto-optical ENZ-metamaterial with negative permittivity/nonlinear self-focusing ENZ-metamaterial. Conclusions. The NSP energy flow is positive in a nonlinear medium and negative in a linear one. As the saturation level decreases, the NSP energy fluxes in the contacting media increase modulo, if the value of the NSP propagation constant (NSPPC) is fixed. The lowest NSPPC value for guiding nonlinearity is obtained when the total energy flow of the NSP becomes zero. As the saturation level decreases, the NSP propagation variable (NSPPV) increases. At a fixed NSPPC, with a decrease in the magnetic permeability, the energy fluxes of NSP in the contacting media increase. When the magnetic permeability decreases to zero and the NSP energy flux is positive, the NSPPV change range narrows

Structural and vibrational spectral investigation on the identification of Non-Linear Optical properties and wave function analyses (electrostatic potential, electron localisation function, localised orbital locator) of 3-Ethoxy Salicilaldehyde

ABSTRACT In the present study, 3-Ethoxy Salicilaldehyde (3ES) was investigated in terms of structural, vibrational spectroscopic and theoretical analyses. All the theoretical calculations were done with the density functional theory method. The natural bond orbital analysis had been done to understand the probable charge transfer interaction in the molecule. The Frontier molecular orbitlas’ molecular electrostatic potential surface has been plotted and the reactive sites of electrophilic and nucleophilic attacks of the molecule have been estimated. The possible vibrations were identified and analysed using the Vibrational Energy Distribution Analysis 4 (VEDA 4) program and Vibrational spectrum. The Fukui function analysis was also used to predict the reactive sites of electrophilic and nucleophilic attacks for the 3ES molecule. Based on electron density, weak interactions of the studied molecules were identified with the aid of RDG analysis. The dipole moment, polarisaility, first-order hyperpolarisability (α 0) and related properties (β, α 0 and Δα) and second-order hyperpolarisabilties were calculated for the identification of non-linear optical activity of the molecule. The second harmonic generation study also has been done for understanding the non-linear optical activity of the molecule.

Vectorial dispersive shock waves on an incoherent landscape

We study numerically and experimentally the impact of temporal randomness on the formation of analog optical blast waves in nonlinear fiber optics. The principle of operation is based on a two-components nonlinear interaction occurring between a partially coherent probe wave co-propagating in a normally dispersive optical fiber together with an orthogonally polarized intense short pulse. The cross-polarized interaction induces a dual phase singularity in the probe profile which leads to the formation of two sharp fronts of opposite velocities. An optical blast wave is then generated and leads to an expanding rarefaction area surrounded by two dispersive shock waves which regularize the shock onto the probe landscape. Here, we focus our study on the impact of randomness in the shock formation. In particular, we show that the lack of coherence in the probe wave acts as a strong diffusive term, which is able to hamper or inhibit the shock formation. Our experimental observations are confirmed by numerical predictions based on a system of two incoherently coupled nonlinear Schrödinger Manakov equations.

Optical solitons of model with integrable equation for wave packet envelope

Abstract We consider the nonlinear fourth-order partial differential equation that can be used for describing solitary waves in nonlinear optics. The Cauchy problem for this equation is not solved by the inverse scattering transform. However we demonstrate that nonlinear ordinary differential equation for description of the wave packet envelope possesses the Painleve property and is integrable. The Lax pair to this nonlinear ordinary differential equation is presented. Using the determinant for the Lax pair matrix, we find the first integrals of a nonlinear ordinary differential equation. The general solution of the fourth-order nonlinear differential equation is given via the ultraelliptic integrals. Special cases of exact solutions for the fourth-order equation are expressed in terms of the Jacobi elliptic sine. Optical solitons of the original partial differential equation are found.

Comparison of Lambert W function with Adomian decomposition method for wave propagation in saturable absorption medium

We present a comparative study to examine the performances of both the Lambert W function method and Adomian decomposition method (ADM) for solving the nonlinear optical beam propagations in a saturable absorption (SA) medium for two distinct SA models. The SA-I model describes a fast saturable absorber (in which the relaxation time is much shorter than the pulse duration of the incident laser beam and no excited state absorption occurs) for a three-level system, while the SA-II model depicts a four-level system or two-photon absorption saturation. We derive the explicit analytic solutions based on the Lambert W function as well as ADM without using thin-sample approximation, and without weak saturation limit. The two different approaches to solve the nonlinear optical wave equation for two distinct SA models are compared and discussed in detail, and show excellent agreement with each other. Finally, to validate the theoretical analysis, we provide some experimental data for nonlinear optical transmission in methyl orange-doped PVA films, revealing excellent agreement with the theory of the SA-II model.

Integrability, multiple-cosh , lumps and lump-soliton solutions to a (2+1)-dimensional generalized breaking soliton equation

In this paper, we focus on investigating a (2+1)-dimensional generalized breaking soliton (gBS) equation with five model parameters, which contains a lot of important nonlinear partial differential equations (PDEs) as its special cases. Firstly, the integrability features of two special cases of the gBS equation are clarified. Secondly, a general method is established to construct solutions formed by a combination of n−cosh and n−cos expressions. The similar results can be generalized to other PDEs which possess the Hirota bilinear forms. Thirdly, by introducing the nonzero seed solution, we obtain the real non-static lumps, lump-soliton solutions and other relevant exact solutions. The results expand the understanding of lump, freak wave and their interaction solutions in soliton theory. Moreover, various graphical analyses on the presented solutions are made to reveal the dynamic behaviors, which gives an essential improvement in the physical realizing of higher-dimensional lump waves in oceanography and nonlinear optics.

New soliton solutions to the perturbed nonlinear Schrödinger equation by exp(− Φ(ξ))-expansion method

The perturbed nonlinear Schrödinger equation (PNLSE), used to investigate the dynamics of wave propagation of light in nonlinear optical fibers and planar wave guides is considered in this article. The model is considered in the presence of full nonlinearity and perturbed terms. By using the state-of-the-art integration scheme, exp(− Φ(ξ))-expansion method, different structures of explicit solutions such as dark, singular, rational and periodic solitary wave solutions are celebrated. All the newly found solutions are discussed with their existence criteria.

Control of dissipative rogue waves in nonlinear cavity optics: Optical injection and time-delayed feedback.

We investigate and review the formation of two-dimensional dissipative rogue waves in cavity nonlinear optics with transverse effects. Two spatially extended systems are considered for this purpose: the driven Kerr optical cavities subjected to optical injection and the broad-area surface-emitting lasers with a saturable absorber. We also consider a quasi-two-dimensional system (the two dimensions being space and time) of a fiber laser describing the complex cubic-quintic Ginzburg-Landau equation. We show that rogue waves are controllable by means of time-delayed feedback and optical injection. We show that without delayed feedback, transverse structures are stationary or oscillating. However, when the strength of the delayed feedback is increased, all the systems generate giant two-dimensional pulses that appear with low probability and suddenly appear and disappear. We characterize their formation by computing the probability distribution, which shows a long tail. Besides, we have computed the significant wave height, which measures the mean wave height of the highest third of the waves. We show that for all systems, the distribution tails expand beyond two times the significant wave height. Furthermore, we also show that optical injection may suppress the rogue wave formation in a semiconductor laser with a saturable absorber.

Quantum plasmon enhanced nonlinear wave mixing in graphene nanoflakes* *Project supported by the National Natural Science Foundation of China (Grant No. 11947007), the Natural Science Foundation of Guangdong Province, China (Grant No. 2019A1515011499), and the Department of Education of Guangdong Province, China (Grant No. 2019KTSCX087).

A distant-neighbor quantum-mechanical method is used to study the nonlinear optical wave mixing in graphene nanoflakes (GNFs), including sum- and difference-frequency generation, as well as four-wave mixing. Our analysis shows that molecular-scale GNFs support quantum plasmons in the visible spectrum region, and significant enhancement of nonlinear optical wave mixing is achieved. Specifically, the second- and third-order wave-mixing polarizabilities of GNFs are dramatically enhanced, provided that one (or more) of the input or output frequencies coincide with a quantum plasmon resonance. Moreover, by embedding a cavity into hexagonal GNFs, we show that one can break the structural inversion symmetry and enable otherwise forbidden second-order wave mixing, which is found to be enhanced by the quantum plasmon resonance too. This study reveals that the molecular-sized graphene could be used in the quantum regime for nanoscale nonlinear optical devices and ultrasensitive molecular sensors.