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In this paper, we investigate the properties of modulation instability (MI) in the fractional Schrödinger equation with cubic–quintic nonlinearity (SECN), particularly focusing on some novel dynamical behaviors different from those in the conventional SECN. First, we apply the linear-stability theory to derive an analytical expression for instability gain in such novel fractional physical system. Just like the case of the conventional SECN, our analysis shows that the dynamical behavior of MI in the fractional SECN still depends on the various combinations of the signs of cubic and quintic nonlinearities. In particular, MI is found to still occur even in the fractional Schrödinger equation (SE) with self-defocusing cubic nonlinearity if the quintic nonlinearity is positive. Moreover, we further find that the Lévy indices related to the fractional effects mainly act to increase not only the fastest growth frequency but also the bandwidth, when compared to the case in the conventional SECN. However, the corresponding maximum gain is completely independent on such Lévy indices. The theoretical predictions are confirmed by the numerical results from the split-step Fourier method. Our findings suggest that the novel fractional systems with cubic–quintic nonlinearity provide some possibilities to control the dynamical behavior of MI and new solitary waves.

In this paper, we investigate the propagation dynamics of circular Airy beams (CAB) in a medium with radial parabolic potential both analytically and numerically. Our results show that CAB will exhibit periodic abruptly autofocusing behavior under the action of radial parabolic potentials. Within every period, we interestingly observe a novel autodefocusing behavior, which manifests that optical peak intensity abruptly decreases by several orders of magnitude. In particular, such unique “autofocusing-to-autodefocusing” effect can lead to the formation of a series of elegant periodic optical bottles having paraboloidal shapes during the whole propagation process. Our findings indicate that CAB is expected to provide a unique tool in optical micromanipulation and optical trapping.

In this paper, we derive an analytical solution of linearly chirped Airy beams (CAB) propagating in strongly nonlocal media with focusing and defocusing nonlinearity. Our analytical results show the periodic evolution pattern of CAB supported by strongly nonlocal focusing nonlinearity is made to be gradually reshaped each other between Airy distribution and conventional standard Gaussian distribution. Meanwhile, we further disclose how initial chirp affects the periodic evolution of optical beam. It is found that we can change ballistic trajectory gradually from conventional parabolic to trigonometric shapes through increasing initial chirp; however the corresponding period is only determined by the degree of nonlocality but independent on the former. Moreover, another interesting property is that initial chirp will lead to a positive or negative transverse displacement of the resulting Gaussian distribution depending on the direction of input beam, but does not affect its shape and longitudinal location. On the other hand, for defocusing nonlinearity, CAB is found to exhibit the conventional nonperiodic self-accelerating behavior, which direction is decided by the initial chirp. Our analytical results agree with the existing results and can be confirmed by numerical prediction.

In this paper, we propose a new family of circular Airy beam (CAB) through introducing a cone angle. We investigate the autofocusing properties of such conical circular Airy beam (CCAB) both analytically and numerically, particularly focusing on the novel behaviors different from the case of the conventional CAB. Under the action of positive cone angles, the autofocusing effect of CCAB is greatly strengthened, and the focal length is remarkably reduced, as compared to the case of conventional CAB. On the other hand, the autofocusing effect may be weakened or even completely eliminated, and the focal length is largely extended under the action of negative cone angles. Therefore, such dramatic enhancement or suppression of abruptly focus can be effectively controlled through adjusting the cone angle. The novel autofocusing properties will make CCAB more powerful in various applied fields including optical trapping and particle manipulation.

In this paper, we first derive a modified two-dimensional non-linear Schrodinger equation including high-order diffraction (HOD) suitable for the propagation of optical beam near the low-diffraction regime in Kerr non-linear media with spatial dispersion. Then, we apply our derived physical model to a designed two-dimensional configuration filled with alternate layers of a left-handed material (LHM) and a right-handed media by employing the mean-field theory. It is found that the periodic structure including LHM may experience diminished, cancelled, and even reversed diffraction behaviours through engineering the relative thickness between both media. In particular, the variational method analytically predicts that close to the zero-diffraction regime, such periodic structure can support stable diffraction-management solitons whose beamwidth and peak amplitude evolve periodically with the help of HOD effect. Numerical simulation based on the split-step Fourier method confirms the analytical results.

In this paper, we investigate the behavior of modulation instability(MI) induced by nonlinear diffraction (NLD) resulting from the vectoral effect in metamaterials with simultaneous Kerr-type electric and magnetic nonlinearity, trying to unfold some novel properties different from those in conventional positive-index media(CM). First, NLD in metamaterials differs from CM mainly in that its value is able to be positive or negative, which provides greater flexibility for us to control the dynamical behavior of MI. Then, we further show that NLD exerts strong influence upon the MI scenario for both defocusing and focusing nonlinear cases, depending on its magnitude and sign. Moreover, we demonstrate NLD can make MI possible even in metamaterials with focusing nonlinearity of negative diffraction case. Finally, we observe NLD under certain limits can open up a new modulation window with an infinite modulation frequency range unseen in conventional MI scenario. Our results denote that NLD in metamaterials can provide us a more powerful strategy to engineer the behavior of MI, as compared with the counterpart in CM.

In this paper, we propose a new kind of circular Airy beam whose spectrum is imposed an initial quadratic phase modulation (QPM). Unlike common abruptly autofocusing beams, we find that such modulated circular Airy beams (MCAB) at image plane, i.e., initial plane, may be morphed into conventional Airy patterns or Bessel-like patterns in spatial regime, depending on the magnitude of QPM. We further investigate the propagation characteristics of the MCAB both analytically and numerically. It is found that the focal intensity, as well as the size of focal spot, do not change with the QPM, but the latter greatly affects the focal trajectory and the focal position of MCAB. In particular, we show that, for the case of Airy pattern, MCAB will exhibit a dual abruptly focusing behaviors having the same focal intensity as well as size of focal spot, where there is one focus behind the image plane, and the another one is in front of it. On the other hand, for the case of Bessel-like pattern, the dual abruptly foci are simultaneously observed behind the image plane for positive QPM but in front of it for negative QPM, thus leading to the formation of an elegant optical bottle with paraboloidal shape. Such novel dual focusing behaviors are expected to be applied in the field of optical trapping and particle acceleration.

In this article, we put forward a novel strategy to realize the management of wave collapse through designing probe–pump configuration where probe wave is assumed to propagate in the positive-index region of metamaterials (MMs), while pump wave is assumed to propagate in the negative-index region. We disclose that cross-phase modulation (XPM) in MMs as a new physical mechanism that can be used to arrest the collapse of probe wave in the positive-index region by copropagating it together with pump wave in the negative-index region. Further, we observe that pump wave will evolve into a ring while probe wave will develop a side lob in the wings during the course of coupled waves propagation, different from the corresponding counterpart in the ordinary positive-index materials (OMs) where they simultaneously exhibit the catastrophic self-focusing behavior. Meanwhile, we also discuss how to control the collapse of probe wave by adjusting intensity-detuned pump wave. Our analysis is performed by directly numeri...

In this paper, we apply the conventional method of linear stability to investigate the dynamical behaviors of spatiotemporal instability in a periodic structure consisting of negative-index materials and positive-index materials. It is found that the high-order diffraction (HOD) originating from spatial dispersion in such physical setting not only leads to the appearance of new instability regimes, but also may exert significant influence on the original instability, depending on the combined signs of both diffraction and dispersion. In the conventional positive diffraction regime, the most notable property is that HOD shrinks the regime of the original instability for the normal dispersion case through suppressing the higher-frequency components growth while it leads to the appearance of new instability regimes in addition to deforming the original instability for the anomalous dispersion case. However, in the negative diffraction regime, HOD may lead to the occurrence of spatiotemporal instability even for the normal dispersion case while it mainly shrinks the regimes of the original instability for the anomalous dispersion case.

In this paper, we put forward a novel method for the management of optical collapse through controlling anomalous nonlinear diffraction (NLD) originating from the vectoral effect in metamaterials with simultaneous cubic electric and magnetic nonlinearity, particularly focusing on the unusual behaviors different from those in ordinary positive-index materials. First, NLD in metamaterials can be positive or negative, which makes it possible for us to offer greatly enhanced design freedom to manipulate the behavior of optical collapse at will. Then, it is further demonstrated that the negative NLD can arrest the catastrophic self-focusing of optical wave; however, the positive NLD will lead to the acceleration of its occurrence, as opposed to the case in ordinary positive-index materials. Our analysis is performed by directly numerically solving the nonlinear Schrödinger equation, as well as by using the modulation theory, both showing consistent results. These findings show that NLD in metamaterials can provide a more powerful tool in controlling the collapse of optical wave when compared with the corresponding case in ordinary positive-index materials.

In this paper, we investigate the dynamical behaviors of the modulation instability (MI) of copropagating optical beams in fractional coupled nonlinear Schrodinger equations (NLSE) with the aim of ...

Spatiotemporal instability (STI) in metamaterials (MMs) with a Kerr-type nonlinear polarization is investigated. A general expression for instability gain, taking account of the effects of arbitrary high-order linear and nonlinear dispersions, is derived. Special attention is paid to the general features of instability induced by nonlinear dispersion effects originating from the combination of dispersive magnetic permeability and nonlinear polarization. It is shown that, just like their linear counterparts, all even-order nonlinear dispersions not only deform the original instability regions, but also may lead to the appearance of new instability regions. However, all odd-order nonlinear dispersions always suppress STI irrespective of their signs, quite different from their linear counterparts which exert no influence on instability. Moreover, we find that, unlike the linear dispersions, the nonlinear dispersions lead to the dependence of gain peak on the spatial modulation frequency. The role of the firs...

We report on the generation of superbroad spectrum bunched noise-like pulses from a passively mode-locked erbium-doped fiber laser, in which the cavity is made of purely anomalous dispersion fibers. The maximum 3 dB spectral bandwidth of the pulse is about 98 nm. We show numerically that the superbroad spectrum of the pulse is caused by the fiber birefringence.

Soliton propagation direction can be engineered in optical fibers in the presence of high-order effects (HOEs). It is well known that Raman effects can decelerate the soliton. Here we investigate the manipulation of the deceleration or acceleration of soliton emitted from Airy pulse whose spectrum is imposed an initial quadratic phase modulation (QPM) in optical fibers in the absence of HOEs. We show that, under the action of the anomalous second-order dispersion (SOD) and Kerr nonlinearity, Airy pulse with QPM is able to emit soliton with acceleration or deceleration depending on whether the QPM is negative or positive and at a rate that is determined by the magnitude of QPM. The reason is that the acceleration behaviors of incident Airy pulse is altered depending on whether SOD and QPM have the same or opposite signs. Our study shows the possibility of controlling and manipulating the soliton propagation and interaction in optical fibers without HOEs, by purposely choosing appropriate QPM parameter of an Airy pulse.

In this paper, we propose a scheme consisting of alternating layers of a slab of a negative-index material and a slab of positive-index material. By employing the mean-field theory, we show that the sign of standard diffraction can be made either positive or negative, or even zero, in this layered structure. Subsequently, we demonstrate that close to the zero-diffraction regime, the high-order diffraction (HOD) strongly affects the dynamical behavior of the transverse instability, just like the corresponding high-order dispersion in optical fibers. Specifically speaking, HOD not only deforms the original instability regions in the positive-diffraction regime with focusing nonlinearity, but also may lead to the appearance of new instability regions even in the negative-diffraction regime with focusing nonlinearity or in the zero-diffraction regime. Meanwhile, we also disclose some novel features of the transverse instability for defocusing Kerr nonlinearity as well as non-Kerr nonlinearity. Our findings denote that the temporal modulation instability induced by high-order dispersion in optical fibers can find its transverse instability counterpart resulting from HOD in such a diffraction-management structure.

The additional dispersive magnetic permeability in metamaterials (MMs) described by the Drude model is able to generate the anomalous self-steepening (SS) effect which can be positive, negative even zero. This provides more great flexibility for us to control ultrashort pulse propagation, when compared with the corresponding case in naturally positive-index materials (PIMs). In this paper, we theoretically investigate the self-accelerating control of Airy pulse through such anomalous SS effect in MMs with a cubic nonlinearity. The emphasis is given to the counterintuitive dynamical behavior different from those in PIMs. It is shown that positive SS in MMs acts to promote the self-acceleration of such pulse and the corresponding spectral features exhibits a broad redshifted behaviors, as it does in PIMs; while the negative SS gives rise to the appearance of the opposite situation. That is to say, the self-acceleration of Airy pulse will be delayed through broadening the spectrum toward blue side. Meanwhile, the effect of the specific truncation coefficient imposed on Airy pulses in MMs also be discussed under the roles of positive and negative SS, respectively. Our findings show that the SS effect in MMs can provide a more powerful tool in controlling the self-accelerating of Airy pulse when compared with the corresponding case in PIMs.

We present a numerical study of the propagation dynamics of accelerating and decelerating truncated Airy pulses (TAPs) in the anomalous dispersion region of optical fibres, with inclusion of the Raman scattering effects, by analysing their cross-correlation frequency resolved optical gating traces. We identify the differences between the evolution dynamics of a decelerating and accelerating TAP. It is shown that the main lobe of the pulse is capable of shedding solitons, which are delayed due to the Raman effects. For the decelerating pulse, however, the soliton is dragged from the original pulse and never meets the input oscillatory Airy tail due to the deceleration of both the pulse and the soliton. The rest of the decelerating pulse rebuilds a new Airy waveform with a stronger degree of truncation compared with that of the incident pulse. For the accelerating TAP, the soliton collides continuously with the tail of the pulse and thus gains further energy by means of their nonlinear interaction. As a consequence, the remaining pulse cannot develop a new Airy waveform. In addition, under the same conditions, the Raman-induced frequency shift of the accelerating TAP is much larger compared with that of the decelerating one.

In this paper, we in detail investigate the dynamic behavior of pulse splitting in metamaterials (MMs) with a Kerr nonlinear polarization, focusing on the anomalous propagation properties associating with the unique and engineerable electromagnetic properties of such materials. The numerical results show that, the pulse symmetric splitting in MMs will occur for the case of defocusing nonlinearity with anomalous dispersion due to the negative refractive index; while it will appear for the case of focusing nonlinearity with normal dispersion in ordinary materials. Moreover, our further analysis shows that, like the case in ordinary materials, the inclusion of self-steepening (SS) effect with positive value gives rise to the asymmetry between the leading and trailing pulses, but the relative magnitudes of the two peaks are reversed, namely, the leading pulse is higher than the trailing pulse. However, the negative SS does opposition.

We present a systematic investigation of ultrashort electromagnetic pulse propagation in metamaterials (MMs) with simultaneous cubic electric and magnetic nonlinearity. We predict that spatiotemporal electromagnetic solitons may exist in the positive-index region of a MM with focusing nonlinearity and anomalous group velocity dispersion (GVD), as well as in the negative-index region of the MM with defocusing nonlinearity and normal GVD. The experimental circumstances for generating and manipulating spatiotemporal electromagnetic solitons can be created by elaborating appropriate MMs. In addition, we find that, in the negative-index region of a MM, a spatial ring may be formed as the electromagnetic pulse propagates for focusing nonlinearity and anomalous GVD; while the phenomenon of temporal splitting of the electromagnetic pulse may appear for the same case except for the defocusing nonlinearity. Finally, we demonstrate that the nonlinear magnetization makes the sign of effective electric nonlinear effect switchable due to the combined action of electric and magnetic nonlinearity, exerting a significant influence on the propagation of electromagnetic pulses.

We report on the observation of central wavelength dynamics in an erbium-doped fiber ring laser by using the nonlinear polarization rotating technique. The evolution of central wavelength with the laser operation state was observed experimentally. Numerical simulations confirmed the experimental observation and further demonstrated that the dynamics of wavelength evolution is due to the combined effects of fiber birefringence, fiber nonlinearity, and cavity filter.

We study both analytically and numerically the propagation dynamics of an initially chirped Airy pulse in an optical fiber. It is found that the linear propagation of an initially chirped Airy pulse depends considerably on whether the second-order dispersion parameter β(2) and chirp C have the same or opposite signs. For β(2)C0, the chirped Airy pulse first undergoes an initial compression phase, then reaches a breakup area as depending on the values of C, and then experiences a lossy inversion transformation such that it continues to propagate with an opposite acceleration. The chirped Airy pulse is always dispersed during propagation in the case of β(2)C0. The impact of truncation coefficient and Kerr nonlinearity on the chirped Airy pulse propagation is also disclosed separately.

The transmission of a one-dimensional left-handed photonic crystal consisting of alternating slabs of two materials with positive and negative refractive index respectively is investigated with the help of transfer matrix method (TTM). The properties of defect modes of the left-handed photonic crystal were disclosed. Both the number and the location of defect modes strongly depend on the property and composition of defects. The defect modes resulted from negative defect often appears in the zero average refractive index (ZARI) gap, while those resulted from positive defect usually appears in the Bragg gap. The defect modes, no matter in the Bragg gap or in the ZARI gap, exhibit degeneracy and split as the distance between the two defects decreases. Due to the coupling of the two defects, the split of defect mode does not appear when two different kinds of defect are introduced.

We investigate modulation instability (MI) in a nonlinear oppositely directed coupler with a quadratic nonlinearity, where one channel is made from a positive-index material (PIM) and another channel is fabricated from a negative-index material (NIM), trying to identify the different MI properties from those in a conventional parametric gap system with grating. Both the analytic continuous wave (CW) solutions and dispersion relation are obtained. By using standard linear instability we in detail discuss how the ratio of the backward to forward propagating wave’s power, the phase mismatch, and the coupled coefficient influence the dynamical behavior of MI. Large stable regions are found if the fundamental harmonics (FH) falls in the normal dispersion when compared to the case in the conventional Bragg grating with a quadratic nonlinearity where the CW solutions are unstable in most cases. In addition, we also observe the large stable regimes of the CW solutions even when the coupled strength for the second harmonics (SHs) is weaker than that of the FH. These findings suggest that the oppositely directed coupler with a NIM channel provides more ways to manipulate the MI and soliton.

The dynamic propagation of the finite-energy Airy pulse (FEAP) is studied numerically in the presence of higher-order effects, including Raman scattering, self-steepening (SS), and third-order dispersion (TOD). It is shown that the Raman-induced frequency shift (RIFS) of a FEAP can be tailored by varying the truncation coefficient of the FEAP. In particular, the combined effects of Raman scattering and SS (or TOD) on the nonlinear propagation of a FEAP are identified. It is found that both TOD and SS effects tend to slow down the RIFS during the nonlinear propagation of a FEAP. In addition, the simultaneous contributions of Raman, TOD, and SS effects to the nonlinear propagation of FEAP are further discussed. Compared to the conventional symmetric pulses, such as Sech pulses, the FEAP can generate a broadening spectrum that is extended toward the blue side, in addition to the conventional red-shifted components. These results demonstrate that the FEAP can be used for supercontinuum generation and broadband sources.

Controllable and dispersive magnetic permeability in negative-index materials (NIMs) offers greatly enhanced design freedom to alter the linear and nonlinear properties. This makes it possible for us to control the propagation of ultrashort pulses at will. In this paper, we will investigate the self-focusing of ultrashort pulses associated with anomalous nonlinear self-steepening (SS) effect and anomalous space–time focus (linear SS) effect in bulk NIMs with a nonlinear electric polarization, trying to disclose some unusual behaviors different from those in naturally positive-index materials. It is found that negative SS acts to push the peak intensity toward the trailing edge of a pulse and the resulting spectrum exhibits a broad redshifted trail; however, positive SS leads to the occurrence of the opposite situation in both temporal and spectral domains, that is, the peak intensity will move toward the leading edge of the pulse by shifting the spectrum toward the blue side. In addition, compared with the case in positive-index materials (PIMs), another most notable property of NIMs is that, whether self-focusing of ultrashort pulses will occur at the leading or the trailing edge can be manipulated freely by engineering the SS effect by choosing the size of split-ring resonator circuit elements. Our analysis is performed by directly numerically solving nonlinear Schrodinger equations as well as by using the moment method, both showing consistent results. These findings demonstrate that NIMs can provide us unique opportunities unattainable in PIMs to manipulate ultrashort pulse propagation.