Beam propagation management in a fractional Schrödinger equation

Autofocus properties of astigmatic chirped symmetric Pearcey Gaussian vortex beams in the fractional Schrödinger equation with parabolic potential

Described by the fractional Schrödinger equation (FSE) with the parabolic potential, the periodic evolution of the astigmatic chirped symmetric Pearcey Gaussian vortex beams (SPGVBs) is exhibited numerically and some interesting behaviors are found. The beams show stable oscillation and autofocus effect periodically during the propagation for a larger Lévy index (0 < α ≤ 2). With the augment of the α, the focal intensity is enhanced and the focal length becomes shorter when 0 < α ≤ 1. However, for a larger α, the autofocusing effect gets weaker, and the focal length monotonously reduces, when 1 < α ≤ 2. Moreover, the symmetry of the intensity distribution, the shape of the light spot and the focal length of the beams can be controlled by the second-order chirped factor, the potential depth, as well as the order of the topological charge. Finally, the Poynting vector and the angular momentum of the beams prove the autofocusing and diffraction behaviors. These unique properties open more opportunities of developing applications to optical switch and optical manipulation.

Unitary evolution for a two-level quantum system in fractional-time scenario

The time-evolution operator obtained from the fractional-time Schrödinger equation (FTSE) is said to be nonunitary since it does not preserve the norm of the vector state in time. As done in the time-dependent non-Hermitian quantum formalism, for a traceless non-Hermitian two-level quantum system, we demonstrate that it is possible to map the nonunitary time-evolution operator in a unitary one. It is done by considering a dynamical Hilbert space with a time-dependent metric operator, constructed from a Hermitian time-dependent Dyson map, in respect to which the system evolves in a unitary way, and the standard quantum mechanics interpretation can be made properly. To elucidate our approach, we consider three examples of Hamiltonian operators and their corresponding unitary dynamics obtained from the solutions of FTSE, and the respective Dyson maps.

Generation of random soliton-like beams in a nonlinear fractional Schrödinger equation

We investigate the generation of random soliton-like beams based on the Kuznetsov-Ma solitons in a nonlinear fractional Schrödinger equation (NLFSE). For Lévy index α = 1, the Kuznetsov-Ma solitons split into two nondiffracting beams during propagation in linear regime. According to the different input positions of the Kuznetsov-Ma solitons, the diffraction-free beams can be divided into three different types: bright-dark, dark-bright and bright-bright beams. In the nonlinear regime, the Kuznetsov-Ma solitons can be evolved into random soliton-like beams due to the collapse. The number of soliton-like beams is related to the nonlinear coefficient and the Lévy index. The bigger the nonlinear coefficient, the more beams generated. Moreover, the peak intensity of soliton-like beams presents a Gaussian distribution under the large nonlinear effect. In practice, the evolution of KM soliton can be realized by a plane wave with a Gaussian perturbation, which can be confirmed that they have the similar dynamics of propagation. In two dimensions, the plane wave with a Gaussian perturbation can be evolved into a bright-dark axisymmetric ring beam in the linear regime. Under the nonlinear modulation, the energy accumulates to the center and finally breaks apart into random beam filaments.

Existence, symmetry breaking bifurcation and stability of two-dimensional optical solitons supported by fractional diffraction

We study existence, bifurcation and stability of two-dimensional optical solitons in the framework of fractional nonlinear Schrödinger equation, characterized by its Lévy index, with self-focusing and self-defocusing saturable nonlinearities. We demonstrate that the fractional diffraction system with different Lévy indexes, combined with saturable nonlinearity, supports two-dimensional symmetric, antisymmetric and asymmetric solitons, where the asymmetric solitons emerge by way of symmetry breaking bifurcation. Different scenarios of bifurcations emerge with the change of stability: the branches of asymmetric solitons split off the branches of unstable symmetric solitons with the increase of soliton power and form a supercritical type bifurcation for self-focusing saturable nonlinearity; the branches of asymmetric solitons bifurcates from the branches of unstable antisymmetric solitons for self-defocusing saturable nonlinearity, featuring a convex shape of the bifurcation loops: an antisymmetric soliton loses its stability via a supercritical bifurcation, which is followed by a reverse bifurcation that restores the stability of the symmetric soliton. Furthermore, we found a scheme of restoration or destruction the symmetry of the antisymmetric solitons by controlling the fractional diffraction in the case of self-defocusing saturable nonlinearity.

Metastable soliton necklaces supported by fractional diffraction and competing nonlinearities

We demonstrate that the fractional cubic-quintic nonlinear Schrödinger equation, characterized by its Lévy index, maintains ring-shaped soliton clusters ("necklaces") carrying orbital angular momentum. They can be built, in the respective optical setting, as circular chains of fundamental solitons linked by a vortical phase field. We predict semi-analytically that the metastable necklace-shaped clusters persist, corresponding to a local minimum of an effective potential of interaction between adjacent solitons in the cluster. Systematic simulations corroborate that the clusters stay robust over extremely large propagation distances, even in the presence of strong random perturbations.

Dissipative surface solitons in a nonlinear fractional Schrödinger equation

We study the existence and stability of dissipative surface solitons supported by the nonlinear fractional Schrödinger equation (NLFSE) with an interface between a semi-infinite chirped lattice and a uniform Kerr medium. In such a system, the existence domain of dissipative surface solitons depends on an upper cutoff value of the linear gain coefficient at a fixed nonlinear loss. The results of the linear stability analysis are in good agreement with that of the propagation simulation in a fractional dimension. Stable dissipative surface solitons generally feature low energy and small propagation constants and adapt to a wide range of two-photon absorption. The instability of solitons can be suppressed by increasing the chirp rate of the lattice. Robust nonlinear dissipative surface states can be easily excited by a Gaussian input beam. Similar characteristics of the two-dimensional dissipative surface solitons are also addressed.

Nonlocal solitons in fractional dimensions

We report the existence and stability properties of multipole-mode solitons supported by the nonlinear Schrödinger equation featuring a combination of the fractional-order diffraction effect and nonlocal focusing Kerr-type nonlinearity. We reveal that multipole-mode solitons, including an arbitrary number of peaks, can propagate stably in fractional systems provided that the propagation constant exceeds a certain value, which is in sharp contrast to conventional nonlocal systems under a normal diffraction, where bound states composed of five peaks or more are completely unstable. Thus, we demonstrate, to the best of our knowledge, the first example of nonlocal solitons in fractional configurations.

Localization and Anderson delocalization of light in fractional dimensions with a quasi-periodic lattice

We address the properties of wavepacket localization-delocalization transition (LDT) in fractional dimensions with a quasi-periodic lattice. The LDT point, which is generally determined by the competition between two sub-lattices comprising the quasi-periodic lattice, turns out to be inversely proportional to the Lévy index. Surprisingly, we find that, in the presence of weak structural disorder, anti-Anderson localization occurs, i.e., the introduced disorder results in an increasing of the size of the linear modes. Inclusion of a weak focusing nonlinearity is shown to improve localization. The propagation simulation achieves excellent agreement with the linear and nonlinear eigenmode analysis.

Off-site and on-site vortex solitons in space-fractional photonic lattices

We address the existence and stability of off-site and on-site vortex solitons with a unit topological charge in space-fractional Kerr lattices. In contrast to the reported ordinary Kerr lattices, vortex solitons in the proposed space-fractional lattices are stable only in the intermediate region of propagation constant, and this region widens rapidly with the increase of a Lévy index. Under the same Lévy index, the stability range of on-site vortices is larger than that of off-site ones. In particular, for on-site vortex solitons, the upper edge of the stability range appears where the maximum of soliton power is located, which provides an effective way to identify the stability range of on-site vortices. Our results extend the study of vortex solitons into space-fractional systems and deepen the understanding of Kerr lattices in fractional dimensions.

Transition from the Wave Equation to Either the Heat or the Transport Equations through Fractional Differential Expressions

We present a model that intermediates among the wave, heat, and transport equations. The approach considers the propagation of initial disturbances in a one-dimensional medium that can vibrate. The medium is nonlinear in such a form that nonlocal differential expressions are required to describe the time evolution of solutions. Nonlocality was modeled with a space-time fractional differential equation of order 1 ≤ α ≤ 2 in time, and order 1 ≤ β ≤ 2 in space. We adopted the notion of Caputo for the time derivative and the Riesz pseudo-differential operator for the space derivative. The corresponding Cauchy problem was solved for zero initial velocity and initial disturbance, represented by either the Dirac delta or the Gaussian distributions. Well-known results for the conventional partial differential equations of wave propagation, diffusion, and (modified) transport processes were recovered as particular cases. In addition, regular solutions were found for the partial differential equation that arises from α = 2 and β = 1 . Unlike the above conventional cases, the latter equation permits the presence of nodes in its solutions.

Dynamics of Gaussian beam modeled by fractional Schrödinger equation with a variable coefficient

In the paper, we investigate the propagation dynamics of the Gaussian beam modeled by the fractional Schrödinger equation (FSE) with a variable coefficient. In the absence of the beam's chirp, for smaller Lévy index, the Gaussian beam firstly splits into two beams, however under the action of the longitudinal periodic modulation, they exhibit a periodically oscillating behaviour. And with the increasing of the Lévy index, the splitting behaviour gradually diminishes. Until the Lévy index equals to 2, the splitting behaviour is completely replaced by a periodic diffraction behaviour. In the presence of the beam's chirp, one of the splitting beams is gradually suppressed with the increasing of the chirp, while another beam on the opposite direction becomes stronger and exhibits a periodically oscillating behaviour. Also, the oscillating amplitude and period are investigated and the results show that the former is dependent on the modulation frequency, the Lévy index and the beam's chirp, the latter depends only on the modulation frequency. Thus, the evolution of the Gaussian beam can be well manipulated to achieve the beam management in the framework of the FSE by controlling the system parameters and the chirp parameter.

Composition Relation between Nonlinear Bloch Waves and Gap Solitons in Periodic Fractional Systems

Evolution of beams in nonlinear optical media with a fractional-order diffraction is currently attracting a growing interest. We address the existence of linear and nonlinear Bloch waves in fractional systems with a periodic potential. Under a defocusing nonlinearity, nonlinear Bloch waves at the centers or edges of the first Brillouin zone bifurcate from the corresponding linear Bloch modes at different band edges. They can be constructed by directly copying a fundamental gap soliton (in one lattice site) or alternatively copying it and its mirror image to infinite lattice channels. The localized truncated-Bloch-wave solitons bridging nonlinear Bloch waves and gap solitons are also revealed. We thus prove that fundamental gap solitons can be used as unit cells to build nonlinear Bloch waves or truncated-Bloch-wave solitons, even in fractional configurations. Our results provide helpful hints for understanding the dynamics of localized and delocalized nonlinear modes and the relation between them in periodic fractional systems with an optical nonlinearity.

Surface gap solitons in a nonlinear fractional Schrödinger equation

We address the propagation dynamics of gap solitons at the interface between uniform media and an optical lattice in the framework of a nonlinear fractional Schrödinger equation. Different families of solitons residing in the first and second bandgaps of the Floquet-Bloch spectrum are revealed. They feature a combination of the unique properties of fractional diffraction effects, surface waves and gap solitons. The instability of solitons can be remarkably suppressed by the decrease of Lévy index, especially obvious for solitons in the second gaps. Additionally, we study the properties of multi-peaked solitons in fractional dimensions and find that they can be made completely stable in a wide region, provided that their power exceeds a critical value. Counterintuitively, at a small Lévy index, the instability region shrinks with the increase of the number of soliton peaks.

Optical Bloch oscillation and Zener tunneling in the fractional Schrödinger equation

We demonstrate optical Bloch oscillation (OBO) and optical Zener tunneling (OZT) in the fractional Schrödinger equation (FSE) with periodic and linear potentials, numerically and theoretically. We investigate in parallel the regular Schrödinger equation and the FSE, by adjusting the Lévy index, and expound the differences between the two. We find that the spreading of the OBO decreases in the fractional case, due to the diminishing band width. Increasing the transverse force, due to the linear potential, leads to the appearance of OZT, but this process is suppressed in the FSE. Our results indicate that the adjustment of the Lévy index can effectively control the emergence of OBO and OZT, which can inspire new ideas in the design of optical switches and interconnects.