Dipole and quadrupole nonparaxial solitary waves (original) (raw)
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Nonparaxial elliptic waves and solitary waves in coupled nonlinear Helmholtz equations
Communications in Nonlinear Science and Numerical Simulation
We obtain a class of elliptic wave solutions of coupled nonlinear Helmholtz (CNLH) equations describing nonparaxial ultra-broad beam propagation in nonlinear Kerr-like media, in terms of the Jacobi elliptic functions and also discuss their limiting forms (hyperbolic solutions). Especially, we show the existence of non-trivial solitary wave profiles in the CNLH system. The effect of nonparaxiality on the speed, pulse width and amplitude of the nonlinear waves is analysed in detail. Particularly a mechanism for tuning the speed by altering the nonparaxial parameter is proposed. We also identify a novel phase-unlocking behaviour due to the presence of nonparaxial parameter.
Nonlinear dynamics of femtosecond optical solitary wave propagation at the zero dispersion point
IEEE Journal of Quantum Electronics, 1995
OLITONS are pulse-like waves that propagate in non-S linear dispersive media without any change in shape or intensity due to a deficate balance between the nonlinear and dispersive effects. Solitons belong to a wider class of localized nonlinear traveling waves, the class of the so-called solitary waves [l]. In nonlinear optical fibers, propagation of a great variety of solitary waves, namely, solitons [2], shock waves [3], kink and antikink waves [4], is predicted. The propagation d sokitons and shock waves, which are usually called bright and dark solitons, respectively, in the optical soliton literature [5], [63, has also experimentally been verified [71, 181. Nonlinear pulse prupagation at the zero dispersion point, uxmspoading to zero s8cond-order (or p u p velocity) dispersion, is desirable m optical communication systems because them the power required for generating (bright) solitons is sigRiticaRtly lawer €51, Id]. The corresponding nonlinear evolution equation (NEE), for the complex envelqk of the ekmie field distribution, results directly from the nonlinear Schrzktinger Gqultion (NLS) [5], [6],, by neglecting the S e c o n d d r d i s p i e n term and taking into account the thirdsrder linear dispersion term. This equation has been analyzed, by using numerical techniques, by various groups in the past [91-[13], for the usual case of a positive thirdorder &persim. Ntice that the effects of fiber loss [14] and axial inhomogeneity E 151 have also been examined. Analytical mults, for the case of dark solitons, have been obtained Manusuipt d v e Athens, Greece. IEEE Log Number 9409267.
Physical Review A, 2021
We demonstrate the formation of periodic waves and envelope solitons in dispersive optical media having a Kerr nonlinear response under the influence of third-order dispersion and self-steepening effect. The stability properties of the bright-and dark-soliton solutions are proved using the stability criterion based on the theory of nonlinear dispersive waves. Regimes for the modulation instability of a continuous wave signal propagating inside the dispersive optical medium are also investigated. The results show that the gain spectrum depends crucially on the self-steepening parameter, while the third-order dispersion has no effect on the modulation instability condition. A similarity transformation is presented to reduce the generalized extended nonlinear Schrödinger equation with distributed coefficients which models the pulse evolution in the presence of the inhomogeneities of media to the related constant-coefficient one. The propagation behaviors of self-similar bright and dark solitons are discussed in a periodic distributed fiber system and an exponential dispersion decreasing fiber.
Vectorial solitary waves in optical media with a quadratic nonlinearity
Physical Review E, 1997
We search for self-trapped beams due to the vectorial interaction of two orthogonally polarized components of a fundamental harmonic and a single component of the second harmonic in a quadratically nonlinear medium. The basic set of equations for both the temporal and the spatial case is derived. The resulting two-parameter family of solitary waves is investigated by means of a variational approximation and by direct numerical methods. Several limiting cases and differences to the scalar interaction in quadratic media are discussed. The propagation of stable solitary waves and mutual collisions are simulated and the decay of unstable solitary waves is demonstrated. We predict an internal boundary in the soliton parameter space which separates stable and unstable domains. ͓S1063-651X͑97͒06806-2͔
Optics Communications, 2003
We derive a propagation equation for the pulse envelope of an electromagnetic field in an isotropic nonlinear dispersive media. The equation is first order in the propagation coordinate. We develop expressions valid without any additional assumptions on the form of the nonlinear polarization. Specific results are given for a Kerr-type nonlinear polarization in the form of a truncated nonlinear differential polynomial. We discuss the applicability of the expansion and determine the conditions for its validity; if and only if the counter-propagating wave is negligible is the expansion valid. We take into account a vectorial character of the electromagnetic field and show that it generates corrections of the same order as the nonparaxial terms.
Physical Review A, 2020
We show theoretically that highly dispersive optical media characterized by a Kerr nonlinear response may support the existence of quartic and dipole solitons in the presence of the self-steepening effect. The existence and stability properties of these localized pulses are examined in the presence of all the material parameters. Regimes for the modulation instability of a continuous-wave signal propagating inside the nonlinear medium are investigated and an analytic expression for the gain spectrum is obtained and shown to be dependent on the self-steepening parameter in addition to second-and fourth-order group velocity dispersion parameters. Self-similar soliton solutions are constructed for a generalized nonlinear Schrödinger equation with distributed second-, third-, and fourth-order dispersions, self-steepening nonlinearity, and gain or loss describing ultrashort pulse propagation in the inhomogeneous nonlinear media via the similarity transformation method. The evolutional dynamics of the self-similar structures are investigated in a periodic distributed waveguide system and an exponential dispersion decreasing waveguide.
Self-Action of a Supershort Laser Pulse in a Medium with Normal Group-Velocity Dispersion
2003
We present the results of numerical and analytical analysis of solutions of the three-dimensional (3D) nonlinear Schrödinger equation with hyperbolic spatial operator. Evolution of the system is considered in separate for two types of the initial field: a Gaussian distribution and a hollow-type (tubular or horseshoe) distribution. The effect of the nonlinear dispersion on wave-packet splitting during self-compression toward the system axis is studied. It is shown that additional focusing of Gaussian wave packets takes place in a wide range of the nonlinear-dispersion parameter. This effect results in a noticeable amplitude growth of one of the two secondary pulses formed as a result of the splitting. For hollow-type distributions, we note the formation of moving inhomogeneities and the excitation of secondary wave fields typical of the hyperbolic system. Here, ψ is the complex envelope of the electromagnetic field of a wave E = E NL ψ(z, x, y, τ) exp(iωt − ikz e) propagating along the z axis with group velocity v gr = (k ω) −1 , α = sign[(v gr) ω ] is determined by the sign of the group-velocity dispersion, E NL is the typical nonlinear field, τ = (t − z e /v gr) k 3/2 c 2 /(ω |(v gr) ω | 1/2), k is the wave number, c is the speed of light, ∆ ⊥ = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 , and the dimensionless evolution variable z and the dimensionless transverse coordinates x and y are related to the corresponding dimensional variables z e , x tr , and y tr as follows: z = kz e /2, x = kx tr , and y = ky tr. This equation, formulated for the first time in [1] for electromagnetic wave packets, is actually applicable for description of the dynamics of nonlinear waves of different nature. For example, in the case of a two-dimensional wave packet (∆ ⊥ = ∂ 2 /∂x 2), it determines the dynamics of nonlinear waves on the water surface [2] and the spatial evolution of wave packets in a magnetized plasma with parameters corresponding to saddle segments of the refractive-index surface [3, 4]. In fact, the formation of localized deep-water waves of anomalously large amplitude (the so-called freak waves) is also analyzed within the framework of twodimensional equation (1) [5, 6]. The same equation describes the physical (electron-positron) vacuum [7]. Recently, 3D equation (1) is actively used to study the features of the self-action of ultrashort laser pulses in condensed media and gases with normal group-velocity dispersion ((v gr) ω > 0) [8-14]. In this case, competition between the processes of nonlinear refraction (self-focusing) of a packet in the transverse direction and the longitudinal dispersion spreading is important. In what follows we focus on the case of normal dispersion, i.e., assume that α = 1 in Eq. (1).
Optics Communications, 2004
This paper presents an investigation on linear and nonlinear propagation of sinh-Gaussian pulses in a dispersive medium possessing Kerr nonlinearity. First, the effects of group velocity dispersion and nonlinearity have been treated separately, and then, the dynamic interplay between group velocity dispersion and nonlinearity induced self phase modulation have been discussed. In both normal and anomalous dispersive media, these pulses broaden due to GVD at a much slower rate in comparison to Gaussian pulses. With the increase in the value of sinh factor X 0 , the broadening decreases for both chirped and unchirped pulses. It has been found that the self phase modulated spectra are associated with considerable internal structure. For small value of X 0 , the number of peaks in the spectrum is less, whereas, for large value of X 0 , number of internal peaks is more in comparison to Gaussian pulses. Moreover, number of internal peaks increases with the increase in the value of X 0 . When the pulse power is appropriate, they can propagate as antisymmetric solitons in anomalous dispersive media. Linear stability analysis shows that such solitons are stable. The dynamic behavior of these pulses, when magnitude of nonlinearity and dispersion are not same, has been also discussed.
Optics Communications, 2001
Propagation of extremely short unipolar pulses of electromagnetic field ("videopulses") is considered in the framework of a model in which the material medium is represented by anharmonic oscillators (approximating bound electrons) with quadratic and cubic nonlinearities. Two families of exact analytical solutions (with positive or negative polarity) are found for the moving solitary pulses. Direct simulations demonstrate that the pulses are very robust against perturbations. Two unipolar pulses collide nearly elastically, while collisions between pulses with opposite polarities and a small relative velocity are inelastic, leading to emission of radiation and generation of a small-amplitude additional pulse.