Nonlinear Paraxial Wave Equation Research Articles

Novel optical soliton solutions to nonlinear paraxial wave model

In this paper, we primarily focus on the nonlinear paraxial wave equation in Kerr media, a generalization of the nonlinear Schrödinger equation utilized to characterize the dynamics of optical beam propagation. By using three potent analytical methods, namely, the Sine-Gordon expansion method, the functional variable method, and the Bernoulli ([Formula: see text])-expansion method, numerous novel soliton solutions are derived. These solutions, comprising hyperbolic, trigonometric and exponential functions, represent significant additions to the field of optics. Furthermore, we elucidate the physical characteristics of these novel optical soliton solutions by presenting a series of three-dimensional (3D) and two-dimensional (2D) graphs with appropriate parameter values.

Quasi-matched propagation of ultra-short, intense laser pulses in plasma channels

The propagation of an ultrashort and relativistically intense laser pulse in a preformed plasma channel is investigated. The nonlinear paraxial wave equation describing the laser propagation in the plasma is solved both analytically and numerically. For any arbitrary temporal laser pulse profile with a given power (less then the critical power for self-focusing) and any prescribed transverse density profile in the channel, we determine the laser intensity distribution along the pulse ensuring quasi-matched propagation, neglecting non-paraxial effects. For the case of a Gaussian laser with an initially uniform spot throughout the pulse, we determine the optimal channel depth that minimizes laser evolution (e.g., minimizes spot size oscillations). The analytical and semi-analytical results obtained for both cases in the weakly relativistic regime are presented and validated through comparison with numerical simulations.

A general Z-scan theory

A novel Z-scan theory based on the solution of the nonlinear paraxial wave equation, completed by the Huygens–Fresnel principle is introduced. This theory is valid for the general case, i.e. for thick samples and large nonlinearities including both nonlinear refraction and absorption. In both limiting cases of thin sample and weak nonlinearity, predictions of this model are in good agreement with theories not using parabolic approximation for the beam profile. It is shown that the widely used parabolic approximation leads to inadequate results when evaluating Z-scan measurements.

GeneralizedABCDpropagation for interacting atomic clouds

We present a treatment of the nonlinear matter wave propagation inspired from optical methods, which includes interaction effects within the atom optics equivalent of the aberrationless approximation. The atom-optical ABCD matrix formalism, considered so far for non-interacting clouds, is extended perturbatively beyond the linear regime of propagation. This approach, applied to discuss the stability of a matter-wave resonator involving a free-falling sample, agrees very well with the predictions of the full nonlinear paraxial wave equation. An alternative optical treatment of interaction effects, based on the aberrationless approximation and suitable for cylindrical paraxial beams of uniform linear density, is also adapted for matter waves.

Theoretical analysis of hot image effect from phase scatterer

The nonlinear hot image effect from phase scatterer in a high-power laser system has been investigated theoretically and numerically. According to the Fresnel-Kirchhoff diffraction integral and nonlinear paraxial wave equation, we derive the functional relationship of the intensity of hot image and its location. It is shown that the image intensity is likely several times larger than that from amplitude scatterer, and the location is situated in the equidistant downstream plane of the scatterer in an optical component located upstream from a nonlinear element. Good agreements are shown between the theoretical prediction and the computer codes developed for ns-pulse propagation simulations. The results indicate that the formation of hot image is akin to the real-time, in-line and volume-phase holographic imaging.

Nonlinear light beam propagation in uniaxial crystals: nonlinear refractive index, self-trapping and self-focusing

We derive the nonlinear paraxial wave equation for the propagation of an optical beam in nonlinear anisotropic media with centrosymmetry. As an application of the equation, we obtain the nonlinear refractive index (NRI) in three uniaxial crystals belonging to the symmetry classes 6/mmm of the hexagonal system, 4/mmm of the tetragonal system, and m of the trigonal system, respectively, and consider the self-trapping and self-focusing of the beam propagating in any direction in these crystals. We conclude that NRI, critical power and self-focusing length are all anisotropic (dependent upon the propagation direction) for an extraordinary light but isotropic for an ordinary light, and that there exists an elliptical self-trapping beam for the extraordinary light.

Parametric acoustic array formation in dispersive fluids

The effect of dispersion on parametric arrays formed by Gaussian beams is investigated via solutions of the nonlinear paraxial wave equation. Analytical solutions are obtained by employing the quasilinear approximation; the results are thus restricted to weak nonlinear interactions. Axial field curves, farfield directivity patterns, and three-dimensional field plots are presented for the difference-frequency signal. The combined effects of dispersion, dissipation, and diffraction are considered in detail. Discrepancies with previous work are discussed. Solutions for the sum-frequency and second-harmonic components are also presented. A transformation is given to make the various solutions apply to arrays formed by primaries from a circular piston.

Reflection of finite amplitude sound beams from pressure release surfaces

A recently described numerical procedure for solving Kuznetsov's nonlinear paraxial wave equation [S. I. Aanonsen, M. F. Hamilton, J. Naze Tjotta, and S. Tjotta, 10th ISNA, Kobe (1984)] is used to model finite amplitude sound beams reflected at normal incidence from both finite and infinite pressure release surfaces. The entire wave field is assumed to be progressive, with reflection simulated by multiplication of each harmonic amplitude by an appropriate coefficient. Although reflection from an infinite pressure release surface reduces finite amplitude losses incurred by the fundamental, it is shown that at some distance from the reflector even the harmonics attain higher axial levels than would occur without reflection. Following reflection from a finite surface, all spectral components exhibit a second nearfield structure characterized by the size of the reflector and the frequency of the component. As a result the axial levels of all components are ultimately higher than after reflection from an infin...

Parametric Acoustic Array Formation Via Weak Collinear and Noncollinear Interaction in Dispersive Fluids

Abstract : The effect of dispersion on parametric arrays formed by Gaussian beams is investigated via solutions of the nonlinear paraxial wave equation. Analytical solutions are obtained by employing the quasilinear approximation; the results are thus restricted to weakly nonlinear interactions. Both (i) collinear and (ii) noncollinear interactions of the primary beams are considered. (i) Axial field curves and farfield directivity patterns are presented for the difference-frequency signal resulting from collinear interaction of Gaussian beams.

Axisymmetric parametric radiation—A weak interaction model

In this paper the difference‐frequency field of a a parametric acoustic radiator is approximated via a weak finite‐amplitude solution of the second‐order nonlinear paraxial wave equation (i.e., the nonlinear optical analog established by E. A. Zabolotskaya and R. V. Khokhlov [Sov. Phys.–Acoust. 15, 35–40 (1969)]) obtained by expressing the primary waves of an axisymmetrically excited bifrequency piston projector in terms of a weighted sum of Gauss–Laguerre eigenmodes. By relating the fundamental (i.e., Gaussian) mode in the farfield of the projector to those of equivalent ’’Bessel beams’’ a closed‐form expression for the difference‐frequency field is thus obtained as a function of range. Following a review of the solution’s phenomenological properties comparisons are made of predicted results and experimental data reported in the literature.

A new analytical solution for the difference‐frequency field of an unsaturated parametric acoustic radiator via the “quasi‐optical” analogy

In this paper a weak finite‐amplitude solution for the difference‐frequency field of a parametric acoustic array is derived via the second‐order nonlinear paraxial wave equation [i.e., the nonlinear optical analog established by E. A. Zabolotskaya and R. V. Khokhlov, Soy. Phys. Acoust. 15, 35–40 (1969)] by expressing the primary waves of an axisymmetrically excited bifrequency piston projector as a weighted sum of Gauss‐Laguerre modes. By relating the fundamental (i.e., Gaussian) modes in the farfield of the projector to those of equivalent “Bessel beams” a new closed‐form expression for the difference‐frequency field is thus obtained as a function of range. A review of the solution's phenomenological properties will then be presented, followed by a brief comparison of predicted results and experimental data reported in the literature.