Computer simulation of soliton propagation and switching

Using the finite-difference time-domain method, we model the propagation of spatial optical solitons having two orthogonal electric field vector components, and the scattering of such solitons by compact subwavelength air holes (i.e., abrupt dielectric discontinuities in the direct paths of the solitons). Our propagation and scattering studies assume a realistic glass characterized by a three-pole Sellmeier linear dispersion, an instantaneous Kerr nonlinearity, and a dispersive Raman nonlinearity. An unexpected spatial soliton scattering phenomenon is observed: the coalescence of the scattered electromagnetic field into a propagating lower-energy spatial soliton at a point many tens of wavelengths beyond the scattering air hole. Overall, our computational technique is general, and should permit future investigations and design of devices exploiting spatial soliton interactions in background media having submicrometer air holes and dielectric and metal inclusions.

We present first-time calculations from the time-domain vector Maxwell's equations of spatial optical soliton propagation and mutual deflection, including carrier waves, in a 2-D homogeneous Kerr-type nonlinear dielectric. The nonlinear Schrodinger equation predicts that two co-propagating, in-phase spatial solitons remain bound to each other, executing a periodic separation. This disagrees with our new extensively tested finite-difference time-domain (FD-TD) solution of Maxwell's equations. FD-TD shows that co-propagating in-phase spatial solitons become unbound, i.e. diverge to arbitrarily large separations, if the ratio of soliton beamwidth to wavelength is order 1 or less. Not relying upon paraxial approximations or analogies to temporal soliton interactions, FD-TD appears to be a robust means of obtaining detailed models of the interaction of sub-picosecond pulsed light beams in nonlinear media directly in the space-time domain. >

The one-particle type temporal soliton exists by maintaining a balance between dispersive linear contributions on the one hand and non-linear effects on the other. The linear contributions occur from processes such as group velocity and polarization mode dispersion. The nonlinear features occur from Kerr, or power law non-Kerr behavior. In addition, a variety of perturbations, such as damping, Brillouin scattering, and Raman effects exist to alter the simple soliton solution. In this paper, we review the propagation of temporal solitons in power law non-Kerr media. This is developed through the higher nonlinear Schroedinger's equation (HNLSE). Also, the fundamentals of multiple-scales are presented that will be used to yield quasi-stationary solitons when perturbations are present. In waveguides, the one-particle type spatial soliton exists by maintaining a balance between the linear propagational diffraction and non-linear self-focusing, while possibly being subjected to a variety of perturbations. Here, we use a spatial optical soliton solution to the nonlinear Schroedinger equation in an inhomogeneous triangular refractive index profile as a small index perturbation to illustrate the oscillation property within a two dimensional waveguide. We determine, from the motion of spatial soliton, its effective acceleration, period of oscillation, and compare results with the Gaussian refractive index profile. Such spatial solitons behave as point masses existing in a Newtonian gravitational potential hole.

The propagation of spatial solitons is systematically investigated in nonlocal nonlinear media with an imprinted transverse periodic modulation of the refractive index. Based on the variational principle and the infinitesimal approximation of Maclaurin series expansion, we obtain an analytical solution of such nonlocal spatial solitons and an interesting result that the critical power for such solitons propagation is smaller than that in uniform nonlocal self-focusing media. It is found that there exist thresholds in modulation period and lattice depth for such solitons. A stable spatial soliton propagation is maintained with proper adjustment of the modulation period and the lattice depth.

Spatial solitons were investigated in periodically poled KTiOPO4 under conditions of second harmonic generation and many new features associated with soliton generation in quasi-phase-matched samples were observed .The effects on the second harmonic generation tuning curves of the beam narrowing mechanisms responsible for soliton generation were found to be dramatic. It was demonstrated experimentally that the mutual collapse of the fundamental and harmonic beams is a complex phenomenon.

Soliton propagation in a directional fiber coupler is investigated. By using Fourier series analysis technique (FSAT), the effects of radii ratio and intermodal dispersion on soliton switching have been investigated.

Although tightly focused input beams for efficient second harmonic (SH) generation (SHG) are usually avoided because of material damage, this is exactly the condition needed for the generation of spatial solitons (QSS) based on quadratic (χ(2)) nonlinearities.[1,2] These QSS’es are mutually self-trapped fundamental (FW) and harmonic (SH) optical beams in which the mutual beam narrowing due to photon exchange incipient in up and down conversion balances diffraction. The generation of quadratic solitons significantly affects the SHG process, for example by placing an upper limit on the amount of fundamental that can be converted to SH, by increasing the bandwidth of the process etc.[3] Of particular interest are periodically poled ferroelectric media which are an important limit of non-critical phase-matching. In fact, periodically poled LiNbO3 (PPLN) has been shown to have the lowest soliton threshold of 1.5 GW/cm2 reported to date [4]. That pioneering work established the existence of quadratic solitons in periodically poled media but neither this work, or any other work on QSSes has addressed the impact of QSS on the SHG tuning curves.

An efficient technique to compensate the effects of absorption losses in the propagation of spatial solitons is analytically and numerically validated. This technique is based on the amplifying properties of certain kinds of tapered nonlinear Kerr coefficients. The analytical demonstration of its applicability is presented here for the first time.

Generation of spatial solitons with noncollinear excitation beams was experimentally investigated for type II second-harmonic generation in KTP. Spatial switching at a distance of 220microm and steering in a 330-microm range were demonstrated. Changes in soliton behavior induced by modification of the phase-matching condition (and) or by an imbalance of the inputs at the fundamental frequency have been characterized.

The new approach to the wavelet analysis for the solutions of the homogeneous wave equation in three spatial dimensions is presented. The approach is based on the ideas suggested by G. Kaiser but has different implementation and has some advantages versus the known approach. A new physical wavelet for this wavelet analysis is also presented, with the brief discussion of its main properties. The wavelet analysis has become widely used during the last twenty years and it has a lot of applications nowadays. However most of them are in the field of the numerical processing of the experimental data, digital images, astronomical, geophysical and medical data and other applications of that kind (see, for example, [1], [2]). The amount of the results in the application of the methods of the continuous wavelet analysis to the solutions of the differential equations is not large. In particular the continuous wavelet analysis for the solutions of the three-dimensional homogeneous wave equations with a constant wave speed was first developed by G. Kaiser in his book [3]. He suggests a method for decomposition of the solutions of the wave equation in terms of the localized solutions of the same equation based on the analytic signal transform and on the theory of the analytic functions of several variables. The wavelet for such decomposition was also suggested, and the class of such wavelets was named ’physical wavelets’. The sort of the wavelet analysis developed by Kaiser is close to the holomorphic wavelet transform (see, for example, [2]). However, this approach may be found unfamiliar by the people who deal with the wavelet analysis within the framework of the signal and image processing. The aims of this paper are as follows. First we develop the wavelet analysis for the solutions of the wave equation not involving the analytic signal transform. The ideas, which we base on, were suggested by Kaiser in [3], however their implementation here differs from his approach. The method we use is intrinsic to the common continuous wavelet transform and we hope will be more familiar to the people who work in the area of the signal and image processing. Our approach also provides some advantages in comparison to that, suggested by Kaiser. We enlarge the class of solutions which can be used as the mother wavelets for the analysis. The second aim is to find a new physical wavelet for our method, i.e., to find the solution of the wave equation which will be an admissible wavelet. The new wavelet is constructed by means of the field of point sources and of proxy wavelets using the technique suggested by G. Kaiser. This new spherically symmetric physical wavelet has good properties such as exponential localization in both the coordinate and the Fourier

<abstract><p>Photoacoustic tomography (PAT) is a novel and rapidly developing technique in the medical imaging field that is based on generating acoustic waves inside of an object of interest by stimulating non-ionizing laser pulses. This acoustic wave was measured by using a detector on the outside of the object it was then converted into an image of the human body after several inversions. Thus, one of the mathematical problems in PAT is reconstructing the initial function from the solution of the wave equation on the outside of the object. In this study, we consider the fractional wave equation and assume that the point-like detectors are located on the sphere and hyperplane. We demonstrate a way to recover the initial function from the data, namely, the solution of the fractional wave equation, measured on the sphere and hyperplane.</p></abstract>

The equal-width wave (EW) equation is a model partial differential equa- tion for the simulation of one-dimensional wave propagation in media with nonlinear wave steepening and dispersion processes. The background of the EW equation is reviewed and this equation is solved by using an advanced numerical method of lines with an adaptive grid whose node movement is based on an equidistribution princi- ple. The solution procedure is described and the performance of the solution method is assessed by means of computed solutions and error measures. Many numerical solutions are presented to illustrate important features of the propagation of a solitary wave, the inelastic interaction between two solitary waves, the breakup of a Gaussian pulse into solitary waves, and the development of an undular bore.

Theoretical study of reorientation and torque of liquid crystal molecules under influence of external electric field and experimentally generation of spatial optical soliton beam and getting a sharp switching in chiral nematic liquid crystal

It is shown that the propagation of steady-state spatial solitons in a bismuth titanium oxide (Bi12TiO20) photorefractive crystal is strongly affected by the presence of concomitant natural optical activity. We assume that the Bi12TiO20 crystal is (110) cut and that the external bias field is along the [001¯] axis. Such orientation introduces self-focusing in one polarization of an optical wave front, whereas it has no effect whatsoever on the component orthogonal to it. Under these conditions, the equations that describe the coupled evolution of the two polarizations of an optical planar beam are derived. Our numerical study of the interaction dynamics shows that, even though the optical beam experiences periodic diffraction in one of the polarization components, it can still recover significantly at particular distances within the crystal. Moreover, it is initially capable of maintaining its solitonlike form over a certain distance of propagation. Relevant examples are provided.

We present a scattering attenuation model based on the statistical wave propagation theory in random media. It is suitable for the weak wavefield fluctuation regime and has practically no restriction in the frequency domain. The presented formulas allow to quantify scattering attenuation in complex geological regions using simple statistical estimates from well-log data. This knowledge is important for further petrophysical interpretations of reservoir rocks. To test our theory we perform numerical simulations of seismic wave propagation in 3-D elastic random media using a finite-difference solution of the elastodynamic wave equation. From the synthetic seismograms we determine the scattering attenuation (Q −1) with help of spectral decay methods. We find good agreement of the frequency-dependent Q values and the theoretical predictions.