Propagation of high intensity light in semiconductors

For photons of energy below the bandgap energy, semiconductors exhibit high transparency and low irradiance light passes freely through the medium. However, for high irradiances of light, as from a laser, the transmission through the semiconductor becomes nonlinear. As the irradiance of the incident light increases, the transmission through the semiconductor decreases through nonlinear absorption as well as from the generation of free carriers during the laser pulse. The propagation of light through this irradiance dependent medium can be described by a set of coupled, inhomogeneous, partial differential equations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite‐difference time‐domain simulation of acoustic propagation in heterogeneous dispersive medium

Accurate modeling of pulse propagation and scattering is a problem in many disciplines (i.e., electromagnetics and acoustics). It is even more tenuous when the medium is dispersive. Blackstock [D. T. Blackstock, J Acoust Soc Am 77 (1985) 2050] first proposed a theory that resulted in adding an additional term (the derivative of the convolution between the causal time‐domain propagation factor and the acoustic pressure) that takes into account the dispersive nature of the medium. Thus deriving a modified wave equation applicable to either linear or nonlinear propagation. For the case of an acoustic wave propagating in a two‐dimensional heterogeneous dispersive medium, a finite‐difference time‐domain representation of the modified linear wave equation can been used to solve for the acoustic pressure. The method is applied to the case of scattering from and propagating through a 2‐D infinitely long cylinder with the properties of fat tissue encapsulating a cyst. It is found that ignoring the heterogeneity in the medium can lead to significant error in the propagated/scattered field. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Analytical studies of soliton pulses along two-dimensional coupled nonlinear transmission lines

A nonlinear network with many coupled nonlinear LC dispersive transmission lines is considered, each line of the network containing a finite number of cells. In the semi-discrete limit, we apply the reductive perturbation method and show that the wave propagation along the network is governed by a two-dimensional nonlinear partial differential equation (2-D NPDE) of Schrödinger type. Two regimes of wave propagation, the high-frequency and the low-frequency are detected. By the means of exact soliton solution of the 2-D NPDE, we investigate analytically the soliton pulse propagation in the network. Our results show that the network under consideration supports the propagation of kink and dark solitons.     

New types of exact solutions for the fourth-order dispersive cubic-quintic nonlinear Schrödinger equation

In this study, we use two direct algebraic methods to solve a fourth-order dispersive cubic-quintic nonlinear Schrödinger equation, which is used to describe the propagation of optical pulse in a medium exhibiting a parabolic nonlinearity law. By using complex envelope ansatz method, we first obtain a new dark soliton and bright soliton, which may approach nonzero when the time variable approaches infinity. Then a series of analytical exact solutions are constructed by means of F-expansion method. These solutions include solitary wave solutions of the bell shape, solitary wave solutions of the kink shape, and periodic wave solutions of Jacobian elliptic function.     

Breathers and multi-soliton solutions for the higher-order generalized nonlinear Schrödinger equation

In this paper, the higher-order generalized nonlinear Schrödinger equation, which describes the propagation of ultrashort optical pulse in optical fibers, is analytically investigated. By virtue of the Darboux transformation constructed in this paper, some exact soliton solutions on the continuous wave (cw) background are generated. The following propagation characteristics of those solitons are mainly discussed: (1) Propagation of two types of breathers which delineate modulation instability and bright pulse propagation on a cw background respectively; (2) Two types propagation characteristics of two-solitons: elastic interactions and mutual attractions and repulsions bound solitons. Those results might be useful in the study of ultrashort optical solitons in optical fibers.     

Asymptotic determination of the formation process of nonlinear distortion of one-dimensional pulses in a layered medium : PMM vol.40, n 6, 1976, pp. 1093–1103

Nonlinear effects in the propagation, reflection, and refraction of one-dimensional pulses in a medium consisting of two layers lying on a half-space are considered and analyzed. Properties of layers and of the half-space are different, and stresses are defined by an expansion in powers of strains. The initial pulse of finite duration is specified in the form of boundary condition at the surface of the external layer either for the deformation or for the dislocation rate, and the problem of wave pattern when the initial pulse amplitude tends to zero,i.e. in the case of small nonlinear effects, is solved.Problem is solved by the method of successive integration of nonhomogeneous linear wave equations, in which the solution of the linear problem is taken as the first approximation and the subsequent approximations are derived by approximating the nonlinear terms with the use of the preceding approximation.     

Nonlinear responses from the interaction of two progressing waves at an interface

For scalar semilinear wave equations, we analyze the interaction of two (distorted) plane waves at an interface between media of different nonlinear properties. We show that new waves are generated from the nonlinear interactions, which might be responsible for the observed nonlinear effects in applications. Also, we show that the incident waves and the nonlinear responses determine the location of the interface and some information of the nonlinear properties of the media. In particular, for the case of a jump discontinuity at the interface, we can determine the magnitude of the jump.     

Numerical simulation of oblique and multidirectional wave propagation and breaking on steep slope based on FEM model of Boussinesq equations

Creating a representative numerical simulation of the propagation and breaking of waves along slopes is an important problem in engineering design. Most studies on wave breaking have focused on the propagation of normal incident waves on gentle slopes. In practice, however, waves on steep slopes are obliquely incident or multidirectional irregular waves. In this paper, the eddy viscosity term is introduced to the momentum equation of the improved Boussinesq equations to model wave dissipation caused by breaking and friction, and a numerical model based on an unstructured finite element method (FEM) is established based on the governing equations. It is applied to simulate wave propagation on a steep slope of 1:5. Parallel physical experiments are conducted for comparative analysis that considered a large number of cases, including those featuring of normal and oblique incident regular and irregular waves, and multidirectional waves. The heights of the incident wave increase for different periods to represent different kinds of waves breaking. Based on examination, the effectiveness and accuracy of the numerical model is verified through a comprehensive comparison between the numerical and the experimental results, including in terms of variation in wave height, wave spectrum, and nonlinear parameters. Satisfactory agreement between the numerical and experimental values shows that the proposed model is effective in representing the breaking of oblique incident regular waves, irregular waves, and multidirectional incident irregular waves. However, the initial threshold of the breaking parameter η_t^(I) takes different values for oblique and multidirectional waves. This needs to be paid attention when the breaking of waves is simulated using the Boussinesq equations.     

Self-Similar Parabolic Optical Solitary Waves

We study solutions of the nonlinear Schrödinger equation (NLSE) with gain, describing optical pulse propagation in an amplifying medium. We construct a semiclassical self-similar solution with a parabolic temporal variation that corresponds to the energy-containing core of the asymptotically propagating pulse in the amplifying medium. We match the self-similar core through Painlevé functions to the solution of the linearized equation that corresponds to the low-amplitude tails of the pulse. The analytic solution accurately reproduces the numerically calculated solution of the NLSE.     

Transient analysis of a potential pulse in a piezoelectric (622) crystal plate

Transient analysis of the wave propagation due to an electric potential pulse concentrated at a point on the boundary of an infinite piezoelectric plate of the crystal class (622) resting on an infinite elastic medium is considered. The electric potential is studied for an exponential pulse and step pulse. Numerical results are obtained for theβ-quartz plate that rests on aluminium half-space.     

The Propagation of Finite-Amplitude Waves in a Model Boundary Layer

Finite-amplitude wave propagation is considered in flows of boundary-layer type when the wavelength is long compared to the boundary layer thickness. In this limit, the evolution of the amplitude is governed by the Benjamin-Ono equation and we have computed the coefficients of its nonlinear and dispersive terms for the specific case of Tietjens's model. The propagation of wave packets is also considered, and it is found that for packets centered about an O(1) wavenumber questions again arise relative to long waves, except that now the packet-induced mean flow is the “long wave.” By introducing an appropriate scaling for the far field and employing multiple scales in the direction transverse to the flow, it is shown how the mean-flow distortion can be made to vanish at infinity.     

Propagation of electromagnetic waves in a nonlinear dielectric slab

The early phases of propagation of a large amplitude electromagnetic disturbance in a nonlinear dielectric slab embedded between two linear media are investigated by the method of characteristics. This disturbance is triggered by the arrival of an electromagnetic shock wave at one of the interfaces. Reflection and transmission of an arbitrary signal when it arrives at one of the slab boundaries is characterized in terms of nonlinear reflection and transmission coefficients for the interface. No restrictions are placed on the form of the constitutive laws of the material comprising the slab. By introducing, for the nonlinear dielectric, a class of model equations, an exact solution to the characteristic equations which describes the interaction of a centered wave with anarbitrary oncoming signal is obtained. Solutions for the electromagnetic fields are derived for the special case in which the incident disturbance interacts with the reflected signal from the slab interface. A particular case of the disturbance propagating across a nonmagnetic slab is also examined.     

Open boundary conditions for hyperbolic equations

A previously developed general procedure for deriving accurate difference equations to describe conditions at open boundaries for hyperbolic equations is extended and further illustrated by means of several examples of practical importance. Problems include those with both incoming and outgoing waves at the boundary, the use of locally cylindrical and spherical wave approximations at each point of the boundary, and nonlinear wave propagation. Reflected waves in all cases are minimal and less than 10^?2 of the incident wave.     

Analysis of the diffraction from periodic chiral structures

Consider a time-harmonic electromagnetic plane wave incident on a biperiodic structure in R³. The periodic structure separates two homogeneous regions. The medium inside the structure is chiral and heterogeneous. In general, wave propagation in the chiral medium is governed by Maxwell's equations together with the Drude-Born-Fedorov (constitutive) equations. In this Note, it is shown that for all but possibly a discrete set of parameters, there is a unique quasiperiodic weak solution to the diffraction problem. Our proof is based on a Hodge decomposition, a compact imbedding result, as well as the Lax-Milgram Lemma.     

Nonlinear Dynamics of a Two-Mode Wave Packet with Strong Linear Intermode

We investigate the dynamics of the wave packet formed by two codirected strongly interacting waves propagating in a medium with cubic nonlinearity. We obtain a soliton solution of the nonlinear Schrödinger equation in the degenerate case where the wave packet is described by a single partial momentum. In the nondegenerate case, we use the variational method to find the equation for the pulse duration, which turns out to be analogous to the equation for the coordinate in the Kepler problem. Solving it, we find the dependences of the pulse duration on the propagation distance in the cases of finite and infinite propagation regimes.     

Simulation of soliton solutions for the propagation of a femtosecond pulse in a medium with a cubic nonlinearity

Soliton solutions are constructed numerically for the problem of propagation of a femtosecond pulse in a medium with a cubic nonlinearity. The problem is posed as an eigenvalue problem with an operator nonlinear in the eigenfunctions. For given values of the propagation parameter we find the real eigenvalue λ and the corresponding eigenvector. This eigenvector is a soliton, i.e., a solution that does not vary in the coordinate of propagation of the light pulse. An algorithm is proposed to find the minimum eigenvalue and the corresponding eigenfunctions that satisfy given conditions. Translated from Prikladnaya Matematika i Informatika, No. 2, pp. 63–68, 1999.     

The complexity of wave propagation of the nonlinear Schrödinger equation with weak periodic external field

A nonlinear Schrödinger equation with external field is obtained for nonlinear optics in a non-homogeneous medium. On the basis of the Melnikov function and KAM theory, the complexity of the wave propagation of the equation is studied in the case of weak periodic external field.     

On a model for optical pulse propagation with saturation nonlinearity

We have suggested a new nonlinear equation for optical pulse propagation in the nonlinear medium with saturation type nonlinearity. The equation can be exactly analysed by Hirota's approach and we have studied the form of explicit one and two solition states.     

The new type of non-polarized symmetric electromagnetic waves in planar nonlinear waveguide

Paper focuses on the propagation of monochromatic nonlinear symmetric hybrid waves in a planar dielectric waveguide filled with nonlinear medium. The wave propagation problem is reduced to a transmission eigenvalue problem. Eigenvalues of the problem depend on an additional parameter and correspond to propagation constant. Using perturbation method, it is theoretically proved the existence of a finite number of isolated eigenvalues and therefore, guide waves. The found guide regime is novel in the theory of nonlinear waveguides. Numerical results are presented.