Saturable Nonlinear Media Research Articles

Data-driven prediction of spatial optical solitons in fractional diffraction

A quasi-residual physics-informed neural network (QR_PINN) with efficient residual-like blocks, was investigated based on classical physics-informed neural network to solve nonlinear fractional Schrödinger equation and analyze the transmission of spatial optical solitons in saturable nonlinear media with fractional diffraction. A comprehensive verification of stable transmission of various solitons under PT-symmetric potential was carried out using the QR_PINN. In addition, the transmission of spatial optical solitons was studied under simple real potential (stable transmission) and complex Scarf-II potential (unstable transmission). The results show that the QR_PINN can accurately reconstruct the transmission of spatial optical solitons under fractional diffraction. Meanwhile, as the complexity of the potential function increases, the prediction accuracy of the QR_PINN slightly decreases. These results provide a new approach for the application of deep learning in the nonlinear fractional Schrödinger equation.

Interactions of Airy beams in nonlinear media with fourth-order diffraction.

We investigate to the best of our knowledge the first time the interactions of in-phase and out-of-phase Airy beams in Kerr, saturable and nonlocal nonlinear media with fourth-order diffraction using split-step Fourier transform method. Directly numerical simulations show that normal and anomalous fourth-order diffractions have profound effects on the interactions of the Airy beams in Kerr and saturable nonlinear media. We demonstrate the dynamics of the interactions in detail. In nonlocal media with fourth-order diffraction, nonlocality induces a long-range attractive force between Airy beams, leading to the formation of stable bound states of both in-phase and out-of-phase breathing Airy soliton pairs which are always repulsive in local media. Our results have potential applications in all-optical devices for communication and optical interconnects, etc.

Anomalous interaction of Pearcey Gaussian pulse in saturable nonlinear media

The dynamics of the mutual interaction of two temporally separated Pearcey Gaussian (PG) pulses in a saturable nonlinear medium is investigated. According to the temporal separation, relative phase, and saturation coefficient of the two pulses, we observed attraction or repulsion between PG pulses. The attraction leads to the generation of breathing solitons whose period increases linearly with the saturation parameter. By adjusting the pulse separation, the change of the initial interference makes the two pulses transmit without mutual interference. We also find that the repulsion makes the two pulses separate, and the interval decreases with the saturable parameter. We also investigate the impact of quadratic chirp imposed on the input pulse on the interaction dynamics. Our results reveal a novel scenario for PG pulses interaction in the nonlinear Schrödinger equation and provide an alternative mechanism to control breather soliton generation.

Bistable soliton switching dynamics in a PT -symmetric coupler with saturable nonlinearity

We investigate the switching dynamics in a $\mathcal{PT}$-symmetric fiber coupler composed of a saturable nonlinear material as the core. In such a saturable nonlinear medium, bistable solitons may evolve due to the balance between dispersion and saturable nonlinearity, which we extend in the context of the $\mathcal{PT}$-symmetric coupler. Our investigations of power-controlled and phase-sensitive switching show richer soliton switching dynamics than the currently existing conventional counterparts, which may lead to ultrafast and efficient all-optical switching dynamics at very low power owing to the combined effects of $\mathcal{PT}$ symmetry and saturable nonlinearity. In addition to the input power, the relative phase of the input solitons and saturable coefficient are additional controlling parameters that efficiently tailor the switching dynamics. Also, we provide a suitable range of system and pulse parameters that would be helpful for the practical realization of the coupler to use in all-optical switching devices and photonic circuits. Finally, we develop a variational approach to analytically investigate the switching dynamics in such $\mathcal{PT}$-symmetric couplers that excellently predicts the numerical findings.

Collision-induced amplitude dynamics of fast 2D solitons in saturable nonlinear media with weak nonlinear loss

We study the amplitude dynamics of two-dimensional (2D) solitons in a fast collision described by the coupled nonlinear Schrödinger equations with a saturable nonlinearity and weak nonlinear loss. We extend the perturbative technique for calculating the collision-induced dynamics of two one-dimensional (1D) solitons to derive the theoretical expression for the collision-induced amplitude dynamics in a fast collision of two 2D solitons. Our perturbative approach is based on two major steps. The first step is the standard adiabatic perturbation for the calculations on the energy balance of perturbed solitons, and the second step, which is the crucial one, is for the analysis of the collision-induced change in the envelope of the perturbed 2D soliton. Furthermore, we also present the dependence of the collision-induced amplitude shift on the angle of the two 2D colliding solitons. In addition, we show that the current perturbative technique can be simply applied to study the collision-induced amplitude shift in a fast collision of two perturbed 1D solitons. Our analytic calculations are confirmed by numerical simulations with the corresponding coupled nonlinear Schrödinger equations in the presence of the cubic loss and in the presence of the quintic loss.

Interaction of Airy Beams and Soliton in Saturable Nonlinear Medium

Interaction of Airy Beams and Soliton in Saturable Nonlinear Medium

Optical solitons in a saturable nonlinear medium in the presence of an asymmetric complex potential

We report on the existence of families of stable spatial solitons in a saturable nonlinear medium characterized by a refractive index with asymmetric distribution of gain and loss. The properties of the nonlinear modes bifurcating from the eigenvalue of the underlying linear problem are thoroughly investigated. The eigenvalue ranges in the power-eigenvalue diagrams for different gain/loss profiles are inspected. We find that the saturable nonlinearity severely restricts these ranges, as the eigenvalues tend to move quite fast to an asymptotic profile, as power increases. Numerical simulations of the wave equations are carried out and examples of the dynamics of the asymmetric solitons obtained exhibit a remarkable agreement with the analytic stability results.

Formation of dispersive shock waves in a saturable nonlinear medium.

We use the Gurevich-Pitaevskii approach based on the Whitham averaging method for studying the formation of dispersive shock waves in an intense light pulse propagating through a saturable nonlinear medium. Although the Whitham modulation equations cannot be diagonalized in this case, the main characteristics of the dispersive shock can be derived by means of an analysis of the properties of these equations at the boundaries of the shock. Our approach generalizes a previous analysis of steplike initial intensity distributions to a more realistic type of initial light pulse and makes it possible to determine, in a setting of experimental interest, the value of measurable quantities such as the wave-breaking time or the position and light intensity of the shock edges.

1D Solitons in Saturable Nonlinear Media with Space Fractional Derivatives

AbstractTwo decades ago, standard quantum mechanics entered into a new territory called space‐fractional quantum mechanics, in which wave dynamics and effects are described by the fractional Schrödinger equation. Such territory is now a key and hot topic in diverse branches of physics, particularly in optics driven by the recent theoretical proposal for emulating the fractional Schrödinger equation. However, the light‐wave propagation in saturable nonlinear media with space fractional derivatives is yet to be clearly disclosed. Here, such nonlinear optics phenomenon is theoretically investigated based on the nonlinear fractional Schrödinger equation with nonlinear lattices—periodic distributions of either focusing cubic (Kerr) or quintic saturable nonlinearities—and the existence and evolution of localized wave structures allowed by the model are addressed. The model upholds two kinds of one‐dimensional soliton families, including fundamental solitons (single peak) and higher‐order solitonic structures consisting of two‐hump solitons (in‐phase) and dipole ones (anti‐phase). Notably, the dipole solitons can be robust stable physical objects localized merely within a single well of the nonlinear lattices—previously thought impossible. Linear‐stability analysis and direct simulations are executed for both soliton families, and their stability regions are acquired. The predicted solutions can be readily observed in optical experiments and beyond.

Suppressing acceleration of 2D Airy beams in Lieb photonic lattices with saturable nonlinear media

We analyze the evolution of transverse acceleration two-dimensional (2D) Airy beams propagating in Lieb photonic lattices with different photorefractive self-focusing nonlinearity. Numerical simulation shows that a decreased transverse acceleration of Airy beam and even a completely stopped state, the energy localized at the main lobe position, can be reached while the Airy beam is launched into the on-site/off-site of the Lieb photonic lattices by varying the medium nonlinearity. We discover that the transverse acceleration when the main lobe of Airy beam incident to off-site of the Lieb photonic lattice can be suppressed, which need a bias voltage larger than the on-site case when the main lobe of Airy beam incident to off-site of the Lieb photonic lattice. Additionally, the main lobe energy of the Airy beam still converges at the attractive defect lattice sites and spreads out to surrounding lattice sites due to the existence of repulsive defects under saturable nonlinearity.

饱和非线性介质中艾里孤子的产生与传输调控

饱和非线性介质中艾里孤子的产生与传输调控

Modulation of localized solutions in an inhomogeneous saturable nonlinear Schrödinger equation

In this paper we study the modulation of localized solutions by an inhomogeneous saturable nonlinear medium. Throughout an appropriate ansatz we convert the inhomogeneous saturable nonlinear Schrodinger equation in a homogeneous one. Then, via a variational approach we construct localized solutions of the autonomous equation and we present some modulation patterns of this localized structures. We have checked the stability of such solutions through numerically simulations.

Standing waves on a periodic array of circular cylinders with saturable nonlinear media

A standing wave of a periodic dielectric waveguide is a special guided mode with zero Bloch wavenumber. For two-dimension linear and symmetry periodic waveguides, standing waves are protected by symmetry and only exist at a discrete set of frequencies. While for a periodic array of circular cylinders with Kerr nonlinearity, standing waves exist continuously on frequency and there are infinite number of them for each frequency. In this paper, we consider standing waves on a periodic array of circular cylinders made of saturable nonlinear media. We show numerically that the standing waves in Kerr and saturable nonlinear media are mainly different in two ways: (1) their existence about frequency, and (2) the number of them for fixed frequency. A new iterative scheme based on Newton’s method is developed to solve the nonlinear problems. Approximate analytical expressions of the amplitude-frequency relations of standing waves in saturable nonlinear media are obtained by a perturbation analysis when amplitudes are small or large.

Propagation and interactions of Airy-Gaussian beams in saturable nonliear medium

The propagation and interactions of Airy-Gaussian beams in a saturable nonlinear medium are investigated numerically based on the split-step Fourier transform method. We show that the propagation of a single Airy-Gaussian beam in the saturable nonlinear medium can generate breathing solitons under steady state conditions. The generation and propagation of these breathing solitons can be affected by the initial amplitude and the field distribution factor of the single Airy-Gaussian beam. In a certain power range, these breathing solitons propagate along the acceleration direction with a controllable tilted angle. In the domain existing in these breathing solitons and for a given value of the field distribution factor of the single Airy-Gaussian beam, when the initial amplitude of the single Airy-Gaussian beam increases gradually, the periodicity of these breathing solitons becomes from small to larger and the tilted angle of these breathing solitons increases monotonically. When the value of the initial amplitude of the single Airy-Gaussian beam is given, the bigger the value of the field distribution factor of the single Airy-Gaussian beam, the smaller the tilted angle of these breathing solitons. Furthermore, the stability of these breathing solitons has been investigated by using the beam propagation method, and it has been found that they are stable. We find that the propagations of two Airy-Gaussian beams in the saturable nonlinear medium can generate not only soliton pairs but also interactions between two Airy-Gaussian beams. When the two Airy-Gaussian beams interact with each other, it is found that the in-phase Airy-Gaussian beams attract each other and exhibit a single breathing soliton with strong intensity in the beam center and some symmetric soliton pairs with weak intensity near both sides of the beam center. The smaller the interval between the two incident Airy-Gaussian optical components, the stronger the attraction between two Airy-Gaussian beams, and the less the numbers of the soliton pairs. The energies of both the main lobes of two Airy-Gaussian beams and the single breathing soliton increase with the value of the field distribution factor of two Airy-Gaussian beams. On the other hand, the out-of-phase Airy-Gaussian beams repel each other and exhibit only symmetric soliton pairs on both sides of the beam center. Our analysis indicates that the repellant of two out-of-phase Airy-Gaussian beams becomes big when the interval between two incident Airy-Gaussian optical components decreases and the number of the soliton pairs becomes less when the field distributions of two beams are close to the Gaussian distribution.

Dark spatial solitons splitting in logarithmically saturable nonlinear media

We numerically simulate the evolution of the dark-notch-bearing optical beam in the logarithmically saturable nonlinear media based on beam propagation method (BPM). The simulation results indicate that the multiple dark spatial solitons are deep, possible in this type of nonlinear media. The number of multiple dark spatial solitons depends on the width of the dark notch, the initial conditions and the peak intensity of the initial input beam. Under the odd and even initial conditions, the odd and even number sequence of multiple dark spatial solitons can be obtained, respectively. For an input beam with fixed optical intensity, the number of dark solitons increases with the width of the initial input dark notch. The behavior of the multiple dark solitons in this type of media is similar to that in a photorefractive nonlinear crystal.

Dynamics of elliptic breathers in saturable nonlinear media with linear anisotropy

We have introduced a class of dynamic elliptic breathers in saturable nonlinear media with linear anisotropy. Two kinds of evolution behavior for the dynamic breathers, rotations and molecule-like librations, are both predicted by the variational approach, and confirmed in numerical simulations. The dynamic elliptic breathers can rotate even though they have no initial orbital angular momentum (OAM). As the media are linear anisotropic, OAM is no longer conserved, and hence the angular velocity is not constant but a periodic function of the propagation distance. When the linear anisotropy is large enough, the dynamic elliptic breathers librate like molecules. The dynamic elliptic breathers are present in media with not only saturable nonlinearity but also nonlocal nonlinearity; indeed, they are universal in nonlinear media with linear anisotropy.

Interactions of Airy beams, nonlinear accelerating beams, and induced solitons in Kerr and saturable nonlinear media

We investigate numerically interactions between two in-phase or out-of-phase Airy beams and nonlinear accelerating beams in Kerr and saturable nonlinear media in one transverse dimension. We discuss different cases in which the beams with different intensities are launched into the medium, but accelerate in opposite directions. Since both the Airy beams and nonlinear accelerating beams possess infinite oscillating tails, we discuss interactions between truncated beams, with finite energies. During interactions we see solitons and soliton pairs generated that are not accelerating. In general, the higher the intensities of interacting beams, the easier to form solitons; when the intensities are small enough, no solitons are generated. Upon adjusting the interval between the launched beams, their interaction exhibits different properties. If the interval is large relative to the width of the first lobes, the generated soliton pairs just propagate individually and do not interact much. However, if the interval is comparable to the widths of the maximum lobes, the pairs strongly interact and display varied behavior.

Soliton pair generation in the interactions of Airy and nonlinear accelerating beams

We investigate numerically the interactions of two in-phase and out-of-phase Airy beams and nonlinear (NL) accelerating beams in Kerr and saturable NL media, in one transverse dimension. We find that bound and unbound soliton pairs, as well as single solitons, can form in such interactions. If the interval between two incident beams is large relative to the width of their first lobes, the generated soliton pairs just propagate individually and do not interact. However, if the interval is comparable to the widths of the maximum lobes, the pairs interact and display varied behavior. In the in-phase case, they attract each other and exhibit stable bound, oscillating, and unbound states, after shedding some radiation initially. In the out-of-phase case, they repel each other and, after an initial interaction, fly away as individual solitons. While the incident beams display acceleration, the solitons or soliton pairs generated from those beams do not.

Surface defect gap solitons in two-dimensional optical lattices

We investigate the existence and stability of surface defect gap solitons at an interface between a defect in a two-dimensional optical lattice and a uniform saturable Kerr nonlinear medium. The surface defect embedded in the two-dimensional optical lattice gives rise to some unique properties. It is interestingly found that for the negative defect, stable surface defect gap solitons can exist both in the semi-infinite gap and in the first gap. The deeper the negative defect, the narrower the stable region in the semi-infinite gap will be. For a positive defect, the surface defect gap solitons exist only in the semi-infinite gap and the stable region localizes in a low power region.

Surface defect gap solitons in one-dimensional dual-frequency lattices and simple lattices

We report the existence and stability of surface defect gap solitons in one-dimensional semi-infinite dual-frequency lattices and semi-infinite simple lattices with a defect between them in the saturable nonlinear medium. By changing the defect intensity, defect solitons can exist in different gaps. For a positive defect, solitons only exist in the semi-infinite gap and can be stable in the low power region. With an increase of the negative defect depth, defect solitons will exist in the semi-infinite gap, first gap, and the second gap.