Second-order Nonlinear Dispersion Research Articles

Impact of higher-order effects on modulation instability in negative index materials

In this article, we present a systematic investigation of modulation instability (MI) in negative index materials (NIMs) with higher-order dispersion and nonlinearity. We adopt the method of linear stability analysis and obtain an expression for the instability gain. Special attention is paid to studying the effect of fourth-order dispersion, quintic nonlinearity, first-order nonlinear dispersion, and second-order nonlinear dispersion on the instability gain. It is found that the fourth-order dispersion forms new instability regions and gives new ways to generate solitons or ultrashort pulse trains. Our study also shows that in the presence of higher-order dispersion a switching of the sign of cubic nonlinearity can generate two distinct MI regions. This result may provide a better way to generate different frequency ultra-short pulses as a result of switching of the sign of cubic nonlinearity by tuning the parameters of constituent elements of the metamaterials. Further, we observed that the cooperating cubic and quintic nonlinearities widen the MI by enhancing maximum gain and bandwidth, on the other hand, the competing cubic and quintic nonlinearities suppress the MI considerably.

Impact of higher-order effects on dissipative soliton in metamaterials

We study the influence of higher-order effects such as third order dispersion (TOD), fourth order dispersion (FOD), quintic nonlinearity (QN), self steepening (SS) and second order nonlinear dispersion (SOND) on the dynamics of dissipative soliton (DS) in metamaterials. Considering each higher-order effect as a perturbation to the system and following Lagrangian variational method, we demonstrate stable dynamics of DS as a result of the interplay between different higher-order effects. We also perform numerical analysis to confirm the analytical results.

Cross-phase modulation induced modulation instability in negative index metamaterial with saturable nonlinear response

The influence of nonlinear dispersion (originating from the dispersive permeability as well as from the nonlinear polarization) on cross-phase modulation (XPM) induced modulational instability (MI) in the negative index domain of metamaterials (MMs) with saturable nonlinearity is investigated numerically. The study explores various possible regimes of normal and anomalous group velocity dispersion and their combinations, which the two copropagating waves could experience in the negative refractive index domain of a MM. It has been shown that, under certain conditions, the first-order nonlinear dispersion, i.e., self-steepening (SS) and the second-order nonlinear dispersion (SOND), may lead to a considerable change in the MI gain spectrum. In addition, it is shown that, at higher perturbation frequencies, the SS effect causes considerable structural changes in the MI gain spectrum.

Effects of higher-order nonlinear dispersions on modulational instability in a three-core coupler with negative index material channel and saturable nonlinearity

A theoretical investigation on the impact of high-order nonlinear dispersions of the modulation instability (MI) spectra in the three core triangular oppositely directed coupler with negative index material channel is carried out. A keen attention was paid to find out the influence of self-steepening (SS), intrapulse Raman scattering (TR), second-order nonlinear dispersion (SOND) as well as saturable nonlinearity (SNL) on the MI spectra. Gain spectra has been investigated for both anomalous group velocity dispersion (GVD) and normal GVD regime. Particularly, our results show that in the normal GVD regime, the instability gain exists even if the perturbation frequency (Ω) is zero; and the instability gain at Ω=0 is nil when the dispersion is anomalous. We also find that the magnitude and sign of SOND exert strong influences on MI sideband. Furthermore, by adjusting various parameters such as SS, intrapulse Raman scattering, and SOND, we observe the occurrence of new instability regions. The MI sidelobes, as well as its bandwidth, have also been affected by the SNL. Finally, the SNL may influence considerably the number of wave trains induced by MI. These results can help to understand the generation of soliton-like excitation in nonlinear three-core oppositely directed couplers with a particular attention on a negative-index material channel.

Strong second harmonic generation in LiInX2 (X=Se, Te) chalcopyrite crystals as explored by first-principles methods

Complex first-order linear optical dispersion and complex second-order non-linear optical dispersion of LiInX2 (X = Se, Te) chalcopyrite single crystals were calculated in the framework of the density functional theory using the local density approximation (LDA), general gradient approximation (GGA) and modified Becke-Johnson potential The performed calculations show that these materials posses direct energy band gap (–Γ) of about 1.58 eV (LDA), 1.85 eV (GGA) and 2.04 eV (mBJ) for LiInSe2 and 1.76 eV (LDA), 1.92 eV (GGA) and 2.40 eV (mBJ) for LiInTe2. Comparison with the experimental data unambiguously indicates that the mBJ succeeds by bringing the calculated gaps in close agreement with the measured ones (2.05 eV -LiInSe2 and 2.41 eV -LiInTe2). It has been found that both compounds exhibit negative uniaxial anisotropy and positive birefringence at the static limit. In addition, LiInTe2 exhibits a higher second harmonic generation (SHG) signal than LiInSe2. Moreover, LiInSe2 (LiInTe2) exhibits a SHG signal of about seven (nine) times greater than the corresponding experimental value for the well known KTiOPO4 (KTP) single crystals.

Lead nitrate hydroxide: A strong second-order optical nonlinearity acentric crystal with high laser damage thresholds

A comprehensive theoretical calculation for the complex first-order linear and the second-order non-linear optical dispersion of acentric lead nitrate hydroxide (Pb16(OH)16(NO3)16) single crystals was performed based on the experimental crystallographic data obtained by Chang et al. [Inorg. Chem. 53, 3320–3325 (2014)]. Calculations show an energy band gap of about 3.70 eV, in close agreement to the measured one (3.78 eV). The energy gap value confirms that the Pb16(OH)16(NO3)16 single crystal exhibits an exceptional laser damage threshold. The complex first-order linear optical dispersion helps to get deep insight into the electronic structure and reveals the existence of considerable anisotropy, negative uniaxial anisotropy, and positive birefringence. The calculated second harmonic generation of Pb16(OH)16(NO3)16 at wavelength (λ = 1064 nm) shows a good agreement with the reported measured value. In addition, the microscopic first hyperpolarizability was obtained at the static limit and at the wavelength 1064 nm.

Modulation instability in metamaterials with fourth-order linear dispersion, second-order nonlinear dispersion, and three kinds of saturable nonlinearites

The linearized nonlinear propagation equation and its coefficients, gain spectrum of modulation instability (MI) in metamaterials (MMs) with fourth-order linear dispersion, second-order nonlinear dispersion, and three kinds of saturable nonlinearites, are analytically deduced by utilizing the linear stability analysis and Drude electromagnetic model. Then variations of gain spectra of MI with the normalized angular frequencies and the optical power densities are calculated in real units. In the negative refractive region, two kinds of gain spectra are discovered. The first (second) one is close to (far from) the zero perturbation frequencies and it corresponds to the lower (higher) normalized angular frequencies. Moreover, the second one has higher cutoff frequency, which is obviously beneficial to generation of high-repetition-rate pulse trains. While in the positive refractive region, only the first kind of gain spectra is found. With increase of the optical power densities, the peak gains and the spectral widths of MI increase before decrease, but they vary the most rapidly (slowly) for the exponential (conventional) saturable nonlinearities. The MI characteristics and their corresponding applications can be adjusted by several methods.

Modulation instability with arbitrarily high perturbation frequencies in metamaterials with nonlinear dispersion and saturable nonlinearity

We present an analytical study of modulation instability (MI) in metamaterial (MM) with first-order and second-order nonlinear dispersion for three types of saturable nonlinearity. Apart from the conventional MI gain spectrum, we find two additional MI gain spectra that arise from the second-order nonlinear dispersion in the MM (provided that the group-velocity dispersion does not vanish): one showing a low cutoff frequency but no high cutoff frequency and the other showing no cutoff frequency. Under certain conditions, a threshold power may be required for MI to occur. These MI gain spectra rarely exist in conventional materials and suggest the possibility of observing MI at arbitrarily high perturbation frequencies, which could be explored for the generation of high-repetition-rate ultrashort electromagnetic pulse trains.

Bright and dark solitons in metamaterials obtained by extended F-expansion method

The ultra-short pulse equation in a metamaterial is solved by the extended F-expansion method. The new phenomena and characteristics of solitons, caused by the anomalous self-steepening effect and the second-order nonlinear dispersion in metamaterials, are discussed. The results show that the second-order nonlinear dispersion in the positive-index region may take the place of the linear dispersion to form the bright and dark solitons. Due to the switchable sign of the anomalous self-steepening effect in the positive-index and negative-index regions, the bright and dark solitons separately exist in the anomalous and normal dispersion regions under some specific conditions. The moving directions of the centers of bright and dark solitons can be controlled by the sign of the anomalous self-steepening effect or by the combination of the anomalous self-steepening effect and third-order linear dispersion.

Modulational instability in metamaterials with saturable nonlinearity and higher-order dispersion

Modulational instability (MI) in negative refractive metamaterials with saturable nonlinearity, fourth-order dispersion (FOD), and second-order nonlinear dispersion (SOND) is investigated by using standard linear stability analysis and the Drude electromagnetic model. The expression for the MI gain spectrum is obtained, which clearly reveals the influence of the saturation of the nonlinearity, FOD, and SOND parameters on the temporal MI. The evolution of the MI in negative refractive metamaterials is numerically investigated. Special attention is paid to study the effects of the higher-order dispersion terms on the formation and evolution of the solitons induced by MI. It is shown that as the third-order dispersion term increases, the solitons travel toward the right. Moreover, the magnitude of the FOD term influences considerably the number of wave trains induced by MI.

Modulation instability in metamaterials with saturable nonlinearity

We analyze modulation instability (MI) in metamaterials with saturable and focusing nonlinearity according to the propagation model, including the self-steepening (SS) parameter and the second-order nonlinear dispersion (SOND). A detailed discussion on the common role of saturable nonlinearity, SS, and SOND terms on the MI is presented. It is found that MI is irrespective of the sign of the SS parameter and dependent on the sign of SOND. Generally, SS and saturable nonlinearity suppress the MI generation, and SOND promotes the MI in the anomalous group-velocity dispersion (GVD) region. Moreover, linear stability analysis predicts that MI may occur in both the abnormal GVD and normal GVD regime, even at the zero-GVD point, which is to say that some interesting MI phenomena appear involving the additional excited SOND term induced by the dispersive magnetic permeability. In particular, the saturable nonlinearity changes the generation condition of MI seriously. Finally, the numerical simulation is performed to confirm the theoretical predictions.

General features of spatiotemporal instability induced by arbitrary high-order nonlinear dispersions in metamaterials

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 first-order nonlinear dispersion, namely self-steepening (SS), and second-order nonlinear dispersion (SND) in STI is particularly analyzed to illustratively demonstrate the general results.

Modulation instability scenario in negative index materials

We present an investigation of the critical frequency windows permitting modulation instability in negative index materials. The principal motivation for our analysis stems from the impact of the inevitable presence of the effective dispersive magnetic permeability in addition to the effective dielectric permittivity determining the propagation model for ultrashort pulses in negative index materials. We emphasize the influence of nonlinear dispersion terms, arising out of the combinatorial effect of the dispersive permeability with the nonlinear polarization, over the MI phenomena, the outcome of its development achieved by using linear stability analysis. Gain spectrum investigation has been carried out for both anomalous and normal dispersion regime in the focusing and defocusing cases of nonlinearity and near zero dispersion regime where higher order linear dispersive effects emerge. The results of linear stability analysis have been validated by direct numerical simulation of the governing equation using the split-step Fourier transform method. We show that second-order nonlinear dispersion opens up distinct MI windows with their appropriate conditions and unlike the first-order nonlinear dispersion term, the sign of it has got deep consequences in the modulation instability scenario.

Influence of dispersive magnetic permeability on modulation instability in metamaterials

The influence of dispersive magnetic permeability on propagation of untrashort pulses in metamaterials is mainly in that it leads to the appearance of the pseudo-χ(5), self-steepening (SS) and second-order nonlinear dispersion terms in the propagation equations. In this paper, the role of dispersive magnetic permeability in modulation instability (MI) in metamaterials is identified based on the Drude model. It is found that in the anomalous dispersion regime, the pseudo-χ(5) nonlinear parameter, which is always negative, increases the MI frequency and growth rate, which is opposite to that in ordinary positive-index materials; the SS tends to suppress MI regardless of its sign, while the second-order nonlinear dispersion effect tends to stimulate MI in the positive-index region and suppress MI in the negative-index region. In the normal dispersion regime, in which MI cannot occur in the ordinary materials, MI can occur due to the role of the second-order nonlinear dispersion, suggesting a new way of generating solitons or ultrashort pulse trains in the normal dispersion regime.

Modulation instability induced by nonlinear dispersion in nonlinear metamaterials

We show that modulation instability may occur even in the normal group-velocity dispersion regime, or in the case of no group-velocity dispersion in metamaterials with a nonlinear electric polarization. The physical origin of the modulation instability is the additional second-order nonlinear dispersion effect resulted from the combination of the linear dispersive magnetic permeability with the nonlinear electric polarization. Based on the Drude model, a numerical simulation is performed to confirm the theoretical predictions, and a detailed discussion on the role of the second-order nonlinear dispersion effect in modulation instability in both negative-index and positive-index regions of metamaterial is presented.

Influence of nonlinear dispersion on modulation instability of coherent and partially coherent ultrashort pulses in metamaterials

The combination of dispersive magnetic permeability with nonlinear polarization leads to a series of nonlinear dispersion terms in the propagation equations for ultrashort pulses in metamaterials. Here we present an investigation of modulation instability (MI) of both coherent and partially coherent ultrashort pulses in metamaterials to identify the role of nonlinear dispersion in pulse propagation. The Wigner–Moyal equation for partially coherent ultrashort pulses and the nonlinear dispersion relation for MI in metamaterials are derived. Combining the standard MI theory with the unique properties of the metamaterial, the influence of the controllable first-order nonlinear dispersion, namely self-steepening, and the second-order nonlinear dispersion on both coherent and partially coherent MI, in both negative-index and positive-index regions of the metamaterial for all physically possible cases is analyzed in detail. For the first time to our knowledge, we demonstrate that the role of the second-order nonlinear dispersion in MI is equivalent to that of group-velocity dispersion (GVD) to some extent, and thus due to the role of the second-order nonlinear dispersion, MI may appear in the otherwise impossible cases, such as in the normal GVD regime.

Formation and propagation of dark solitons in metamaterials

We use an extended Tanh-function expansion method to solve the nonlinear equation for ultrashort pulse propagation in matamaterial，and obtain dark solitary solutions in various cases. Further，we study the influence of the controllable self-steepening effect and the second-order nonlinear dispersion on the formation and propagation of dark solitons in metamaterials. It is found that the negative self-steepening effect in metamaterial makes the soliton center move to the leading side，opposite to the moving direction of soliton in ordinary materials in which the self-steepening effect is always positive. Most importantly，due to the role of the second-order nonlinear dispersion，dark solitons can be formed in the medium without linear dispersion or with an anomalous linear dispersion.

Second-Harmonic Generation and Miller's Delta Parameter in a Series of Benzine Derivatives

Second-order nonlinear susceptibilities and refractive index dispersion measurements have been made in six benzene derivative crystals. The nonlinear susceptibility tends to be much greater in crystals with lower energy absorption edges. A simple classical model is discussed in which the enhanced values are due to anharmonic oscillators near the absorption edge.