Arbitrary Dispersion Research Articles

Dynamic theory of deformable solids with quasiparticle excitations in the presence of electromagnetic fields

A full self-consistent set of equations is deduced to describe the kinetics and dynamics of charged quasiparticles (electrons, holes, etc.) with arbitrary dispersion law in crystalline solids subjected to time-varying deformations. The set proposed unifies the nonlinear elasticity theory equation, a Boltzmann kinetic equation for quasiparticle excitations, and Maxwell's equations supplemented by the constitute relations. The kinetic equation used is valid for the whole Brillouin zone. It is compatible with the requirement for periodicity in k space and contains an essential new term compared to the traditional form of the Boltzmann equation. The theory is exact in the frame of the quasiparticle approach and can be applied to metals and semiconductors, as well as to other crystalline solids including quantum crystals and low-dimensional lattice structures. Instructive examples concerning the form of the Fokker-Plank equation as well as the pinning of effective magnetic induction lines in deformable metals are considered.

Lie-transform averaging in nonlinear optical transmission systems with strong and rapid periodic dispersion variations

Using Lie-transform techniques, we derive higher-order corrections to the path-averaged model governing evolution of dispersion-managed solitons in the spectral domain. The result holds in the case of arbitrary (including moderate and strong) dispersion. The general theory is illustrated by deriving the exact formulas + for a specific symmetric dispersion map.

Arbitrary dispersion control of ultrashort optical pulses with acoustic waves

Acousto-optic programmable dispersive filters (AOPDF) can compensate in real time for large amounts of group-delay dispersion. This feature can be used in chirped-pulse amplification femtosecond laser chains to compensate adaptively for dispersion. An analytical expression relating the group delay at the output of the AOPDF to the input acoustic signal is obtained with coupled-wave theory in the case of collinear and quasi-collinear bulk acousto-optic interactions and also in the case of planar waveguides and optical fibers. With this relation, the acoustic signal that will induce an arbitrary group-delay variation with frequency can be easily obtained. Numerical simulations are shown to support the principle of arbitrary group-delay control with an AOPDF.

Hall and Nernst effects in some models of anisotropic metals

General expressions for the weak-field Hall and Nernst coefficients are found for a metal with an arbitrary electron dispersion. It is shown that, contrary to the free-electron model, the signs of the Hall conductivity, the Nernst coefficient, and thermopower in real metal may differ depending on the derivative of the electron density of states on the Fermi level. Pecularities of Hall and Nernst effects are discussed for two cases: a quasi-two-dimensional metal and a conductor at the edge of the electronic topological transition.

Calculation of second harmonic beam patterns for ultrasonic sources of arbitrary shape

Ultrasonic transducers with nonaxisymmetric radiating surfaces are common in both industrial and medical applications. This presentation describes a numerical technique for calculating the second harmonic field produced in the field radiated by a transducer of arbitrary shape. The angular spectrum of the source is calculated with a two-dimensional Fourier transform, a weighted integral over the source spectrum is evaluated, and the inverse transform of the result yields the second harmonic pressure in any plane parallel to the source plane. Arbitrary absorption and dispersion are taken into account. The method is similar to that used by Alais and Hennion [Acustica 43, 1–11 (1979)] to evaluate the field of a parametric array. In the present theory, however, we do not make use of the parabolic approximation. Beam steering and focusing are introduced by appropriate phasing of the source function. Calculations of radiation patterns produced by rectangular and other sources are presented.

Path-averaged chirped optical soliton in dispersion-managed fiber communication lines

We present a comprehensive path-average theory of dispersion-managed (DM) optical pulse. Applying complete basis of the chirped Gauss-Hermite orthogonal functions, we derive a path-average propagation equation in the time domain and present an analytical description of the breathing dynamics of the chirped DM soliton. This theory describes both self-similar evolution of the central, energy-containing core and accompanying nonstationary oscillations of the far-field tails of an optical pulse propagating in a fiber line with an arbitrary dispersion map. In the case of a strong dispersion management the DM soliton is well described by a few modes in this expansion, justifying the use of a Gaussian trial function in the previously developed variational approach. Suggested expansion in the basis of chirped Gauss-Hermite functions presents a regular way to describe soliton properties for arbitrary dispersion map and to account for the effect of practical perturbations (filters, gratings, noise an so on) on the dynamics of the ideal DM soliton. We also present path-averaged propagation model in the spectral domain that could be useful for multichannel transmission applications. Theoretical results are verified by numerical simulations.

Modeling of pulsed finite-amplitude focused sound beams in time domain.

A time-domain numerical model is presented for simulating the finite-amplitude focused acoustic pulse propagation in a dissipative and nonlinear medium with a symmetrical source geometry. In this method, the main effects responsible in finite-amplitude wave propagation, i.e., diffraction, nonlinearity, and absorption, are taken into account. These effects are treated independently using the method of fractional steps with a second-order operator-splitting algorithm. In this method, the acoustic beam propagates, plane-by-plane, from the surface of a highly focused radiator up to its focus. The results of calculations in an ideal (linear and nondissipative) medium show the validity of the model for simulating the effect of diffraction in highly focused pulse propagation. For real media, very good agreement was obtained in the shape of the theoretical and experimental pressure-time waveforms. A discrepancy in the amplitudes was observed with a maximum of around 20%, which can be explained by existing sources of error in our measurements and on the assumptions inherent in our theoretical model. The model has certain advantages over other time-domain methods previously reported in that it: (1) allows for arbitrary absorption and dispersion, and (2) makes use of full diffraction formulation. The latter point is particularly important for studying intense sources with high focusing gains.

Dispersion-Managed Solitons and Optimization of the Dispersion Management

We overview recent progress in dispersion-managed (DM) fiber optic communications. Wavelength-division-multiplexing transmission of a DM soliton (or more general return-to-zero (RZ) formatted data) is an attractive way to realize middle- and long-distance ultra-high-capacity fiber communication systems. We present a theory of the DM optical soliton and a simple basic theory of the general DM RZ transmission. Two ordinary differential equations for the root-mean-square pulse width and chirp (momentum equations) describe the fast (during compensation period) evolution of the DM pulse. Applying chirped Gauss–Hermite orthogonal functions we derive a path-averaged propagation equation governing both the shape of the DM soliton and the slow (average) evolution of any chirped DM pulse. We describe the breathing dynamics of the self-similar core and oscillating tails of the DM optical pulse propagating in a fiber line with an arbitrary dispersion map. Based on the developed theory we describe the basic system principles, the design, and the optimization rules for DM fiber links. We demonstrate how to determine the energy enhancement of the DM soliton and optimal (chirp-free) points for launching of the signal, and how to evaluate the characteristics of a carrier signal for specific system parameters. DM solitons in systems with in-line filtering and Bragg gratings are also studied. Analytical results are illustrated by numerical simulations for a number of specific dispersion maps actively used in practice.

Particle cracking in discretely reinforced materials with the voronoi cell finite element model

In this paper, a two dimensional small deformation Voronoi Cell finite element model is developed to model damage in heterogeneous materials by particle cracking or splitting. It incorporates a ductile elastic–plastic matrix material containing an arbitrary dispersion of brittle elastic inclusions. Inclusions may be of arbitrary shapes and sizes. Damage initiation by micro-cracking or splitting of the inclusions is assumed to follow a maximum principal stress theory or Rankine criterion. Complete particle cracking or splitting is assumed at the onset of damage. The model accounts for progressive particle fracture in complex microstructures without any user interference. Various microstructural morphologies are simulated to verify the effectiveness of the model.

Self-similar core and oscillatory tails of a path-averaged chirped dispersion-managed optical pulse

We describe the breathing dynamics of the self-similar core and the oscillating tails of a dispersion-managed (DM) soliton. The path-averaged propagation equation governing the shape of the DM soliton in an arbitrary dispersion map is derived. The developed theory correctly predicts the locations of the dips in the tails of the DM soliton. A general solution of the propagation equation is presented in terms of chirped Gauss-Hermite orthogonal functions.

High-order dispersion effects in solitary mode-locked lasers: side-band generation

Analytical expressions for spectral positions, temporal width and energy of dispersive waves generated by arbitrary high-order dispersion in mode-locked solitary lasers are derived, being in good agreement with experiments. This knowledge may be helpful for improving the design of dispersion compensation systems and minimizing the energy drain into side-bands.

On the carrier contribution to the elastic constants in ultrathin films of bismuth under quantizing magnetic field

We present a simple analysis of the carrier contribution to the elastic constants in ultrathin films of Bi under magnetic quantization in accordance with McClure-Choi, Cohen, Abrikosov, Lax and parabolic ellipsoidal band models in the presence of spin, broadening and temperature respectively. It appears that the said contribution increases with increasing electron concentration and decreasing film thickness in various manners. The McClure and Choi model shows this more significantly than other models. We have also suggested an experimental method of determining the said contribution in degenerate materials having arbitrary dispersion laws.

Interchange instabilities in a partially ionized plasma

In a partially ionized plasma, the charged particle population is coupled to the neutrals through charge exchange, ionization and recombination. An interchange instability is identified in which the driving factor is the neutral pressure gradient and the dominant collisional process is charge exchange. Regimes are considered in which the mean free path for neutral collisions with ions λni is small in comparison to the macroscopic length scales. The instability is analogous to a gravitational interchange mode with an effective gravity geff=νinVn where νin is the collision frequency of ions with neutrals. The neutral flow Vn=−(Mniνin)−1dpn/dx results from a balance between the neutral pressure gradient and collisional friction with the ions. An arbitrary kyλni dispersion relation is derived using fluid equations to describe the ions and the Boltzmann equation to describe the neutrals. This dispersion relation contains viscous and inertial effects and is substantially altered in the presence of realistic parallel wavelength due to a coupling between the unstable interchange mode and a stabilizing shear Alfvén mode. Reasonable conditions under which the modes may exist are examined and the possible relevance to divertor plasmas is considered.

Fast spectral algorithm for modeling focused sound beams in a highly nonlinear regime

Numerical modeling of high-intensity focused sound beams is important for medical applications of ultrasound. The KZK equation, which may be modified to include relaxation in the time domain or arbitrary absorption and dispersion in the frequency domain, is widely used to model ultrasound beams in tissue. Numerical solutions are available in either the time or frequency domain, but they become time consuming at high intensities when thin shocks are developed. To speed up calculations, a known high-frequency asymptotic result for shocks can be incorporated in the frequency domain algorithm. In comparison with exclusively numerical schemes, the present approach substantially reduces the number of harmonics and thus the computation time [Yu. A. Pishchal’nikov, O. A. Sapozhnikov, and V. A. Khokhlova, Acoust. Phys. 42, 362–367 (1996)]. Focused cw beams radiated from Gaussian and piston sources are simulated in water and biological tissue. The accuracy of the asymptotic approach is verified via conventional numerical methods of solution. It is shown that the present asymptotic method provides accurate results with a considerable reduction in computation time. In combination with conventional numerical schemes, the developed asymptotic method provides an effective tool for investigating focused sound beams for a wide range of parameters. [Work supported by NIH PO1-DK43881, Fogarty Research Program, and CRDF.]

Dynamics of self-similar dispersion-managed soliton presented in the basis of chirped Gauss-Hermite functions

Applying chirped Gauss-Hermite orthogonal functions we present an analytical description of the breathing dynamics of the chirped dispersion-managed soliton. The theory describes both self-similar evolution of the central, energy-containing core and accompanying nonstationary oscillations of the far-field tails of an optical pulse propagating in a fiber line with arbitrary dispersion map.

A simple theoretical analysis of the Einstein relation in ultrathin films of bismuth under quantizing magnetic field

The Einstein relation for the diffusivity-mobility ratio is studied in ultrathin films of Bi under magnetic quantization in accordance with McClure-Choi, Cohen, Arbikosov, Lax and parabolic ellipsoidal band models in the presence of spin, broadening and temperature, respectively. It appears that the same ratio increases with increasing electron concentration and decreasing film thickness in various manners. The McClure and Choi model shows this more significantly than the other models. We have also suggested an experimental method of determining the Einstein relation in degenerate materials having arbitrary dispersion laws.

The carrier contribution to the elastic constants in small-gap materials

We present a simple theoretical analysis of the carrier contribution to the second and third order elastic constants in nonparabolic materials on the basis of an electron dispersion law by taking into account various anisotropies of the energy band structure within the framework of k⋅p formalism. It is found that the carrier contributions to the elastic constants in n-Cd3As2, InSb, InAs, GaAs, Hg1−xCdxTe, and lattice matched In1−xGaxAsyP1−y increase with the increase of carrier degeneracy in different manners which, depend on the material parameters and band structure. A relationship between the said contributions and the thermoelectric power has been derived for materials obeying arbitrary dispersion laws in the presence of a classically large magnetic field. Our analysis is based on the derivation of a more generalized band structure of the materials which agrees well with the relationship suggested. It is also observed that the second and third order elastic constants increase with the decrease of alloy composition for ternary and quaternary compounds, respectively. The corresponding well-known results for degenerate wide-gap materials can be derived as special cases of our generalized analysis.

On the electronic contribution to the elastic constants in strained quantum wire superlattices of non-parabolic semiconductors with graded structures: Theory and suggestion for experimental determination

We study the electronic contribution to the second- and third-order elastic constants in strained quantum wire superlattices of non-parabolic semiconductors with graded structures and compare the same with the constituent materials, by formulating the appropriate dispersion laws. It is found, taking InSb/GaSb quantum wire superlattice as an example, that the said contributions increase with decreasing thickness and with increasing electron concentration in oscillatory manners together with the fact that the influence of the finite interface width enhances their numerical values. An experimental method is suggested for determining the electronic contribution to the elastic constants in materials having arbitrary dispersion laws. In addition, the well-known results for constituent semiconductors in the absence of stress have also been obtained as special cases of our generalized formulations.

A Simple Theoretical Analysis of the Carrier Contribution to the Elastic Constants of Quantized II-VI Semiconductors

In this paper we study the carrier contribution to the elastic constants in II-VI semiconductors under magnetic quantization, cross-field configuration, quantum wells, electric field aided quantum wells, quantum wells under cross-field configuration, inversion layers, magneto inversion layers, inversion layers under cross-field configuration and quantum well wires, respectively, by deducing the respective carrier energy spectra. It has been found, taking p-CdS as an example, that the carrier contribution to the second-order and third-order elastic constants exhibits various types of oscillations with different physical variables in quantized II-VI structures having quantum confinements in one, two and three dimensions, respectively. The theoretical results are in agreement with the suggested experimental method of determining the said contribution in materials having arbitrary dispersion laws.

On the chance of freak waves at sea

When deep-water surface gravity waves traverse an area with a curved or otherwise variable current, the current can act analogously to an optical lens, to focus wave action into a caustic region. In this region, waves of surprisingly large size, alternatively called freak, rogue, or giant waves are produced. We show how this mechanism produces freak waves at random locations when ocean swell traverses an area of random current. When the current has a constant (possibly zero) mean with small random fluctuations, we show that the probability distribution for the formation of a freak wave is universal, that is, it does not depend on the statistics of the current, but only on a single distance scale parameter, provided that this parameter is finite and non-zero. Our numerical simulations show excellent agreement with the theory, even for current standard deviation as large as 1.0 m s−1. Since many of these results are derived for arbitrary dispersion relations with certain general properties, they include as a special case previously published work on caustics in geometrical optics.