Dynamics of localized waves with large amplitude in a weakly dispersive medium with a quadratic and positive cubic nonlinearity

The dynamics of localized waves is analyzed in the framework of a model described by the Korteweg-de Vries (KdV) equation with account made for the cubic positive nonlinearity (the Gardner equation). In particular, the interaction process of two solitons is considered, and the dynamics of a “breathing” wave packet (a breather) is discussed. It is shown that solitons of the same polarity interact as in the case of the Korteweg-de Vries equation or modified Korteweg-de Vries equation, whereas the interaction of solitons of different polarity is qualitatively different from the classical case. An example of “unpredictable” behavior of the breather of the Gardner equation is discussed.     

Method for superposing reconstructed images from digital holograms of the same object recorded at different distance and wavelength

A method for allowing direct perfect superimposition and comparison of Fresnel-transform reconstructions of digital holograms recorded of the same object at different distances and wavelengths is proposed and demonstrated. The method takes into account the dependence of the reconstruction pixel on the distance and the wavelength by the Fresnel-transform algorithm. The method avoids the need for image scaling at the end of the reconstruction process that usually is performed in multi-wavelength digital holography. Demonstration is given by superposing the numerical reconstructions of holograms recorded at different distances and wavelengths. The method can be potentially very useful for real-time monitoring in biological processes or for recognition and ranging by multiple wavelengths of a target with fast movements or finally for very fast investigation and study of very fast processes.     

Nonlinear equations for quasi-monochromatic waves near a caustic in a dispersive dissipative medium with cubic or quadratic nonlinearity

The behavior of a quasi-monochromatic nonlinear wave near a caustic is considered. Nonlinear ordinary differential equations for a dispersive dissipative medium with a cubic or quadratic nonlinearity are derived. For the latter medium, nonstationary equations describing it near the caustic are presented with allowance for the dissipative dispersive terms. These equations yield ordinary ones for quasi-monochromatic waves. The amplitude of the second harmonic is expressed in terms of the squared amplitude of the first harmonic. The amplitude of the second harmonic, as well as the solution as a whole, increases near the caustic.     

Traveling waves in a nonlinear medium with cubic nonlinearity

Unlike linear nondispersive media, which allow propagation of wave packets of arbitrary forms, nonlinear media admit only certain profiles of traveling waves. Here we examine media with Duffing oscillators, i.e., with bound electrons for which an equilibrium disturbance causes forces proportional to the first and third powers of deviation. We show that the linearly polarized traveling plane waves such media can transmit have profiles modulated as Jacobi elliptic functions. When discussing propagation across an interface between different media, only incidence from the side of the linear medium is considered. Even in this case, to launch a traveling wave in the nonlinear medium, a severe restriction must be imposed on the incident wave’s amplitude.     

Localized polaritons and second-harmonic generation in a resonant medium with quadratic nonlinearity

We derive model equations for the propagation of ultrashort pulses in materials with resonant linear and quadratic nonlinear responses and find approximate soliton solutions describing all-bright and dark-bright polaritons. We report the specific phase matching condition for efficient 2nd harmonic generation, which involves detuning from the resonance. We also demonstrate that the 2nd harmonic emission by the polaritonic pulses can lead to reduction of their group velocity, having zero as a theoretical limit. Our analytical results are supported by numerical simulations.     

Molecules with a permanent dipole moment in a periodic medium with quadratic nonlinearity

Nonlinear optical effects in a one-dimensional periodic medium containing spatially oriented molecules of J aggregates of a dye and silver nanoparticles are studied. It is shown that, in such composite medium, the presence of a permanent dipole moment of an exciton transition of molecules of J aggregates results in the appearance of new nonlinear optical properties.     

Scattering of time-harmonic elastic waves by an elastic inclusion with quadratic nonlinearity

This paper considers the scattering of a plane, time-harmonic wave by an inclusion with heterogeneous nonlinear elastic properties embedded in an otherwise homogeneous linear elastic solid. When the inclusion and the surrounding matrix are both isotropic, the scattered second harmonic fields are obtained in terms of the Green's function of the surrounding medium. It is found that the second harmonic fields depend on two independent acoustic nonlinearity parameters related to the third order elastic constants. Solutions are also obtained when these two acoustic nonlinearity parameters are given as spatially random functions. An inverse procedure is developed to obtain the statistics of these two random functions from the measured forward and backscattered second harmonic fields.     

Effect of the group velocity mismatch and dispersion of nonlinear susceptibility on the modulation instability of electromagnetic waves in a medium with quadratic nonlinearity

Modulation instability of continuous plane waves of the pump beams and the second harmonic that are described by the set of equations for the process of second-harmonic generation with allowance made for the dispersion of nonlinear susceptibility is considered. The case of type I phase matching is analyzed. In the regions of anomalous and normal group velocity dispersion, the instability increments are determined as functions of the frequency and the wave numbers of perturbations. It was found that the difference in the group velocities of the interacting waves exerts the most substantial effect on the evolution of the modulation instability.     

Evolution of acoustic waves in microinhomogeneous media with quadratic elastic nonlinearity and relaxation

A nonlinear wave equation for the velocity “relaxator” is derived in the framework of the rheological model and the corresponding equation of state of a microinhomogeneous medium containing viscoelastic defects with quadratic nonlinear elasticity. The equation is qualitatively analyzed, and numerical solutions to it are presented for a stationary symmetric shock wave and the evolution of initially harmonic waves.     

Transformation of images upon dual-frequency recording and reading at sum frequencies of dynamic χ<Superscript>(2)</Superscript> holograms using spherical reference waves

The transformation properties of images reconstructed by dynamic χ⁽²⁾ holograms recorded with radiation of the sum frequency with the use of spherical reference waves are studied theoretically and experimentally. Notions of fields and zones of locrises as regions of localization of holographic images formed by reference and object beams of different frequencies at arbitrary frequency ratios of these beams and distances from the reference point source to the hologram are defined. The domains of the parameters of the reference source ensuring the formation of real and imaginary images are determined. The reconstruction of real images by dynamic χ⁽²⁾ holograms is demonstrated.     

Second-harmonic generation in the interior of an isotropic medium with quadratic nonlinearity by a focused inhomogeneously polarized pump beam

This paper presents a theoretical study of second-harmonic generation (SHG) by a focused pump beam in the interior of an isotropic medium, experimentally observed earlier. It shows that the spatial dispersion of the quadratic optical response of the substance can be responsible for this nonlinear process even when a macroscopic inversion center is present in the medium. It is established that this effect, which is impossible in the plane-wave approximation, also does not occur when a Gaussian pump beam with uniform distribution of the polarization state of the wave field over the cross section is used, but that the presence in the pump beam of higher transverse modes with polarization different from the fundamental mode can cause an SHG signal to appear. The conditions for a wave to appear at the doubled frequency are found, analytical dependences for its electric field and total power on the propagation coordinate, the degree of focusing, and the other parameters of the problem are obtained, and the requirements on the optimum experimental geometry are formulated. The dependence of the signal-beam power on the wavevector detuning is studied, and it is shown that interference effects can cause the wave at the second harmonic to disappear when a normal dispersion law is obeyed in the region between the fundamental and doubled frequencies. Zh. éksp. Teor. Fiz. 113, 1261–1276 (April 1998)     

Spontaneous soliton formation with a saturated nonlinearity in a weakly nonlinear medium

We have observed the effect of a saturated nonlinearity on the process of spontaneous soliton formation in a weakly nonlinear medium with absorption. The characteristic properties of the problem are investigated on the basis of a model equation designated as a nonlinear Schrodinger equation with small perturbations describing the saturation of the nonlinearity and the absorption. The stability of the process of decay of the initial pulse is demonstrated by computational methods for a few specific cases. An effect described as an oscillation of the amplitude maximum of the formed pulse has been discovered.V. D. Kuznetsov Siberian Physicotechnical Institute, Tomsk University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 73–76, June, 1995.     

Matrix and linear interference patterns in photographic emulsions of transmission holograms recorded at different wavelengths

Matrix and linear interference patterns formed in emulsion layers of transmission holograms are considered. Boundary effects are observed between matrix and linear interference patterns, as well as between a linear interference pattern and the unexposed part of a hologram.     

Dynamic recording of χ<Superscript>(2)</Superscript> holograms with multifrequency object and reference waves

The transformation properties of images formed by inertialess dynamic χ⁽²⁾ holograms recorded by multifrequency reference and object beams are studied theoretically and experimentally. The regular features of the spatial localization and scales of holographic images obtained with different frequency combinations of the reference and object image-forming beams are ascertained. It is shown that, in the general case of an arbitrary variation in the ratio of these frequencies, the reconstructed images of an object point source are localized on a segment of a straight line, which we referred to as the locris. The slope of the locris does not depend on the frequency ratio of the reference and object beams, and its endpoints are specified by the positions of the reference and object sources. The possibility of obtaining of achromatic holographic images for noncollinear multifrequency interactions is demonstrated. The conditions for the noncollinear phase matching of interacting waves in a KTP crystal, as well as for simultaneous attainment of phase matching for selected frequency combinations, are determined. The important role of controlling the shape of wavefronts in experiments with frequency conversion of radiation in the reference beam by stimulated Raman scattering of light is ascertained.     

Real-time holograms generated by second-harmonic cross correlation of object and reference optical wave fields

Three-dimensional holographic images of extended diffusing objects are simultaneously recorded and reconstructed by optical cross correlation in a second-order nonlinear crystal. An interaction geometry in which the phase-matched object and reference fields propagate slightly noncollinearly is particularly convenient for producing these second-harmonic-generated holograms.     

Localized surface waves at the interface between the linear dielectric and photorefractive medium with drift and diffusion nonlinearity

We consider the propagation of the laser beam near the interface between the linear dielectric and photorefractive medium with drift and diffusion nonlinearity. We demonstrated the possibility of the surface waves formation, found the profiles of such surface modes and investigated waveguiding properties of the dielectric-photorefractive medium boundary. The effective particle theory was used to derive an ordinary differential equation that describes the beam center trajectory during the reflection from the interface.     

Reflection of compressional and Rayleigh waves on the edges of an elastic plate with quadratic nonlinearity

Previous ultrasonic studies have demonstrated that measurements of material nonlinearities can provide a means for detecting early signs of fatigue damage using both compressional (P) and Rayleigh (R) surface waves. However, these experimental studies have typically been limited to the direct wave arrival between the source and receiver in simple geometries where no reflection occurs. In particular, the degree of material nonlinearity is often quantified by the ratio of the cumulative amplitude of the first harmonic to that of the fundamental for the direct arrival only. Hence a practical question arises over the interpretation of ultrasonic measurements of material nonlinearities in the presence of reflected nonlinear waves. Thus, this article investigates the reflection problem of P or R waves at the edge of a homogeneous plate with quadratic nonlinearity using both a theoretical formulation, based on perturbation analysis, and direct numerical simulations using a Cellular Automata formulation. The numerical approach is first validated against an existing theoretical formulation for reflecting nonlinear P waves. It is then used to simulate the nonlinear reflection of R waves at a plate's edge for which no closed-form formulation is presently available.