Dispersive Pulse Propagation Research Articles

Covariance and Uncertainty Principle for Dispersive Pulse Propagation

We develop the concept of covariance for waves and show that it plays a fundamental role in understanding the evolution of a propagating pulse. The concept clarifies several issues regarding the spread of a pulse and the motion of the mean. Exact results are obtained for the time dependence of the covariance between position and wavenumber and the covariance between position and group velocity. We also derive relevant uncertainty principles for waves.

Spatiotemporal mode decomposition of ultrashort pulses in linear and nonlinear graded-index multimode fibers

We develop a spatiotemporal mode decomposition technique to study the spatial and temporal mode power distribution of ultrashort pulses in long spans of graded-index multimode fiber, for different input laser conditions. We find that the beam mode power content in the dispersive pulse propagation regime can be described by the Bose–Einstein law, as a result of the process of power diffusion from linear and nonlinear mode coupling among nondegenerate mode groups. In the soliton regime, the output mode power distribution approaches the Rayleigh–Jeans law.

Chirped dispersive pulse propagation of arbitrary initial width: a saddle point perspective

The propagated field dynamics in chirped Gaussian pulse propagation of arbitrary initial width in a linear, causally dispersive, Lorentz-type dielectric are derived, validated, and elaborated. The performed asymptotic analysis relies on a two-term series expansion around the saddle points of the unified phase. As I show, the dynamics of each relevant saddle point are mapped into the characteristics of a respective pulse component contributing to the total propagated field. The accuracy of the description is verified upon comparison with depicted numerical results. This asymptotic approach provides unique insight that is accurate and valid from the quasi-monochromatic to the sub-cycle pulse regimes.

Weakly dispersive modal pulse propagation in the North Pacific Ocean

The propagation of weakly dispersive modal pulses is investigated using data collected during the 2004 long-range ocean acoustic propagation experiment (LOAPEX). Weakly dispersive modal pulses are characterized by weak dispersion- and scattering-induced pulse broadening; such modal pulses experience minimal propagation-induced distortion and are thus well suited to communications applications. In the LOAPEX environment, modes 1, 2, and 3 are approximately weakly dispersive. Using LOAPEX observations it is shown that, by extracting the energy carried by a weakly dispersive modal pulse, a transmitted communications signal can be recovered without performing channel equalization at ranges as long as 500 km; at that range a majority of mode 1 receptions have bit error rates (BERs) less than 10%, and 6.5% of mode 1 receptions have no errors. BERs are estimated for low order modes and compared with measurements of signal-to-noise ratio (SNR) and modal pulse spread. Generally, it is observed that larger modal pulse spread and lower SNR result in larger BERs. [Work supported by ONR.]

Weakly dispersive modal pulse propagation in the North Pacific Ocean

The propagation of weakly dispersive modal pulses is investigated using data collected during the 2004 long-range ocean acoustic propagation experiment (LOAPEX). Weakly dispersive modal pulses are characterized by weak dispersion- and scattering-induced pulse broadening; such modal pulses experience minimal propagation-induced distortion and are thus well suited to communications applications. In the LOAPEX environment modes 1, 2, and 3 are approximately weakly dispersive. Using LOAPEX observations it is shown that, by extracting the energy carried by a weakly dispersive modal pulse, a transmitted communications signal can be recovered without performing channel equalization at ranges as long as 500 km; at that range a majority of mode 1 receptions have bit error rates (BERs) less than 10%, and 6.5% of mode 1 receptions have no errors. BERs are estimated for low order modes and compared with measurements of signal-to-noise ratio (SNR) and modal pulse spread. Generally, it is observed that larger modal pulse spread and lower SNR result in larger BERs.

Inversion of dispersive GPR pulse propagation in waveguides with heterogeneities and rough and dipping interfaces

Abstract We investigate the influence of interface roughness, heterogeneous media, and dipping layers on the inversion of dispersive GPR pulse propagation in a surface waveguide, using 3D FDTD modelling. For both broadside and endfire source–receiver configurations, we calculated responses for different interface roughnesses, heterogeneities in dielectric properties, and dipping interfaces. For increasing roughness and heterogeneity, increased backscatter energy is visible in the data. The use of multiple source–receiver offsets to calculate the phase-velocity spectrum produced a relatively good signal-to-noise ratio. For low interface roughness the medium properties could be reasonably well reconstructed. For the largest interface roughness, significant diffracted energy was present and the medium properties could not be reliably reconstructed. For models having stochastic relative permittivity variations with a Std up to 15% and correlation lengths between 0.1 and 0.5 m, the model parameters could still be relatively well reconstructed. For models with a dipping interface, the shot and countershot configuration clearly indicate lateral changes. Single-mode inversions return a wide range of medium property values, whereas combined TE-TM inversion return permittivity values that are remarkable close to the e 1 and e 2 values for both the shot and countershot configuration. In general, the interface roughness and heterogeneous media have a relative small influence on the inversion results. This is probably due to the use of multi-offset data for calculating the phase-velocity spectrum which is reducing the random noise.

Time-frequency and position-wavenumber acoustic signal analysis.

The spectrogram, which is a plot of the spectral intensity of a signal over time, has been widely used in acoustic signal processing because many signals, such as speech, animal vocalizations, music, and the sonar backscatter from elastic objects, have frequencies that change over time, which convey important information about the signal or source from which it originated. Since the development of the spectrograph at Bell Laboratories in the 1940s, more modern methods for time-frequency analysis, such as the Choi–Williams and Zhao–Atlas–Marks distributions, have been developed, which overcome some of the limitations of the spectrogram and, in particular can show time-frequency detail in the signal that is obscured by the spectrogram. We will discuss these methods and the general area of time-frequency acoustic signal analysis with examples drawn from a variety of applications. A particular focus will be made on showing how time-frequency analysis, and also position-wavenumber analysis, can be used to formulate and gain physical insights into dispersive pulse propagation. We will also comment on and illustrate the use of wavelets for time-frequency analysis. [Work supported by ONR 321US.]

Pulse propagation and classification in time/frequency and position/wave number phase space.

In dispersive pulse propagation, different frequencies travel at different velocities, and, hence, the pulse is nonstationary in that it changes over time and distance. Joint phase space representations, such as the Wigner distribution and the spectrogram, are often employed to study nonstationary signals, and, hence, it is natural to consider their application to dispersive propagation. We consider position/wave number and time/frequency representations of a pulse propagating with dispersion and damping. We give the Wigner distribution of the pulse at a particular time/position in terms of the Wigner distribution of the initial pulse. Approximations of this result are derived, which are accurate and remarkably simple to apply, as compared to the stationary phase approximation. The approximations provide insights as well, in that they show how each point in phase space propagates at the group velocity, and lead to new features for classification, based on local phase space moments, that are invariant to t...

Exact and approximate moments for dispersive pulse propagation

We derive exact moments for pulse propagation in a dispersive medium. These moments are not only inherently interesting but clarify the validity of a recently proposed approximation scheme for wave propagation. The approximation method for pulse propagation is based on the Wigner position‐wave/number representation and is very accurate, easy to apply, and moreover is physically illuminating. In particular one obtains the evolved approximate Wigner distribution from the initial Wigner distribution by a simple linear translation in phase space. Propagation with damping is also taken into account. We will show that the reason for the high accuracy of the approximation is that the important low order moments are exactly given by the approximation and that these low order moments preserve very well the basic shape of the pulse. Moreover, now that we understand why the approximation method works well, the approximation can be systematically improved. We give a number of specific examples of exactly calculable moments to illustrate the method and we compare exact and approximate moments. The work of PL is supported by The Research of LC was supported by AFOSR and the work of PL is supported by ONR (N00014‐06‐1‐0009)

Local characteristics of dispersive pulse propagation

As a wave propagates, its shape and other physical characteristics may change, depending on the initial wave and the propagation environment. In this talk, the effects of structural dispersion on the spatial and temporal spreading of a propagating pulse wave, and on the spectral characteristics of the pulse over time, are considered. These effects are quantified by obtaining the local spatial, temporal, and spectral moments of the pulse. General results for any dispersion relation, as well as the particular case of a two-plate waveguide, are presented. [Work supported by ONR.]

Practical measurement of femtosecond optical pulses using time-resolved optical gating

We report a practical experimental implementation of time-resolved optical gating (TROG) using a dispersive pulse (DP) propagation geometry and a two-photon absorption nonlinear detector. A measurement of 50 fs pulses from a self-mode-locked Ti:sapphire laser is demonstrated and a general procedure for calibrating the dispersion axis of the TROG trace is presented. A complete iterative pulse retrieval algorithm is described and the limitations of the DP-TROG technique are discussed.

Failure of the group-velocity description for ultrawideband pulse propagation in a causally dispersive, absorptive dielectric

The accuracy of the group-velocity description of dispersive pulse propagation in a double-resonance Lorentz model dielectric is shown to decrease monotonically as the propagation distance increases, whereas the accuracy of the asymptotic description increases monotonically as the propagation distance increases above a single absorption depth in the medium at the pulse carrier frequency.

Hybrid numerical–asymptotic code for dispersive-pulse propagation calculations

A fast, hybrid numerical–asymptotic algorithm for the calculation of the dynamical field evolution that is due to an arbitrary input pulse in any linear dispersive, attenuative medium is presented. The algorithm combines the numerical efficiency of the fast-Fourier-transform algorithm over a finite frequency domain that encompasses all the medium resonances with the accuracy and analytical efficiency of the asymptotic description that fully accounts for the high-frequency structure above the highest resonance that is characteristic of the dispersive medium.

Approximate analytic solution for the spatiotemporal evolutionof wave packets undergoing arbitrary dispersion

We apply expansion methods to obtain an approximate expression in terms of elementary functions for the space and time dependence of wave packets in a dispersive medium. The specific application to pulses in a cold plasma is considered in detail, and the explicit analytic formula that results is provided. When certain general initial conditions are satisfied, these expressions describe the packet evolution quite well. We conclude by employing the method to exhibit aspects of dispersive pulse propagation in a cold plasma, and suggest how predicted and experimental effects may be compared to improve the theoretical description of a medium's dispersive properties.

Gaussian Pulse Propagation in a Dispersive, Absorbing Dielectric.

The modified asymptotic description of dispersive Gaussian pulse propagation, which is uniformly valid in the initial pulse envelope width, is shown to reduce to the energy velocity description when the propagation distance becomes sufficiently large in a Lorentz model dielectric. This then resolves the apparent controversy between the modern asymptotic description upon which the energy velocity description is based and the classical group velocity description of Gaussian pulse propagation and related experimental results.

Noninstantaneous, finite rise-time effects on the precursor field formation in linear dispersive pulse propagation

The effects of a noninstantaneous finite rise time on the transient field phenomena associated with dispersive pulse propagation are considered by use of the asymptotic description of the propagated plane-wave field in a single-resonance Lorentz model dielectric. The asymptotic description is presented for both an input hyperbolic tangent modulated signal with initial pulse envelope u(t) = [1 + tanh(βt)]/2 and an input raised cosine envelope signal with initial pulse envelope u(t) = 0 for t ≤ 0, u(t) = [1 − cos(βt)]/2 for 0 ≤ t ≤ π/β, and u(t) = 1 for t ≥ π/β. In both cases the parameter β is indicative of the rapidity of the initial rise time of the signal f(t) = u(t)sin(ωct) with input carrier frequency ωc. In the limit as β → ∞ both of these initial envelope functions approach the unit step function. The dynamical evolution of the propagated field is described by means of the dynamics of the saddle points in the complex ω plane that are associated with the complex phase function appearing in the integral representation of the propagated field and their interaction with the simple pole singularities of the spectrum ũ(ω − ωc) of the initial pulse envelope function. The analysis shows that the Sommerfeld and Brillouin precursor fields that are characteristic of the propagated field that is due to an input unit step function modulated signal will persist nearly unchanged for values of the rise-time parameter β of the order of δ or greater, where δ is the damping constant of the Lorentz model dielectric. As β is decreased below δ, the precursor fields become less important in the overall field structure and the field becomes quasi-monochromatic.

Precursor propagation in dispersive media from short-rise-time pulses at oblique incidence

The derivation and the application of a numerical method designed to study the propagation of pulsed electromagnetic fields into dispersive dielectric materials are reported. A Fourier series based methodology, appropriate for a useful class of pulse train incident signals, is presented and utilized to study the dynamics of dispersive pulse propagation in a half-space constructed of a Debye or a Lorentz model medium. Computed solutions are presented showing precursor propagation excited by pulses obliquely incident upon a planar boundary to the medium. The results show that oblique incident pulses upon a Lorentz medium can excite different types of precursor, propagating in unique directions within the half-space.

Transients in chiral media with single-resonance dispersion: comment

The recent paper by Zablocky and Engheta [ J. Opt. Soc. Am. A10, 740 ( 1993)] on the propagation of a unit-step-function-modulated electromagnetic signal in a linear, causal, dispersive chiral medium with single-resonance dispersion in the limit of zero material damping is found to be only partially valid in that singular limiting case. Of particular interest is their use of an overly restrictive definition of signal arrival in such a lossless dispersive medium. The correct, modern asymptotic description of signal arrival in dispersive pulse propagation is readily available in the literature [ J. Opt. Soc. Am. B5, 817 ( 1988)] and is carefully reviewed here for a lossy dispersive dielectric as described by the single-resonance Lorentz model. This general definition provides an unrestricted description of signal arrival in the limit of zero damping that yields results equivalent to those of Zablocky and Engheta in regions of normal dispersion.

Measurement and modeling of dispersive pulse propagation in drawn wire waveguides

An analytical model of dispersive pulse propagation in semi-infinite cylinders due to transient axially symmetric end conditions has been experimentally investigated. Specifically, the dispersive propagation of the first axially symmetric longitudinal mode in thin wire waveguides, which have ends in butt contact with longitudinal piezoelectric ultrasonic transducers, is examined. The method allows for prediction of a propagated waveform given a measured source waveform, together with the material properties of the cylinder. Alternatively, the source waveform can be extracted from measurement of the propagated waveform. The material properties required for implementation of the pulse propagation model are determined using guided wave phase velocity measurements. Hard tempered aluminum 1100 and 304 stainless steel wires, with 127, 305, and 406 μm diam, were examined. In general, the drawn wires were found to behave as transversely isotropic media.

Dispersive pulse propagation and group velocity

The propagation of pulses in a dissipative medium is investigated both theoretically and experimentally. The theoretical work is based on the dissipative dispersion integral, and the measurements are made in an air‐filled periodic waveguide (i.e., the dispersion is Bloch wave dispersion). The dispersion integral is considered in the context of a sequence of characteristic pulse duration distances. The pulse propagates without distortion up to the smallest characteristic distance, and thereafter undergoes a new variety of distortion as it encounters each subsequent characteristic distance. Several new solutions of the dispersion integral that exhibit a variety of novel propagation properties are found. Pulses that shift in frequency as they propagate, accelerate as they propagate, and propagate at near‐infinite group velocity are found analytically and verified experimentally. [Work supported by ONR.]