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Digest - IEEE Antennas and Propagation Society. International Symposium

Asymptotic description of electromagnetic pulse propagation in active Lorentzian media

Reza Safian, Mo Mojahedi, Costas D. Sarris

Published Jan 1, 2006• 0 citations
Publisher

Abstract

The asymptotic description of the propagation of a step-modulated signal of fixed angular frequency omegacinto the half-space z > 0 that is occupied by a single-resonance active Lorentzian medium, which is a classical model of an inverted two-level atom, has been presented. The steepest descent method as an asymptotic technique provides the detailed time evolution of the pulse propagation in a linear, temporally dispersive active Lorentzian medium. The analysis and the numerical results show the relation of the location of the saddle points and the topography of the phase function in the complex omega-plane with the transient portion of the pulse that can be used to control the transient response of the medium

DOI
10.1109/aps.2006.1710685

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The asymptotic description of the propagation of a step-modulated signal of fixed angular frequency omegacinto the half-space z > 0 that is occupied by a single-resonance active Lorentzian medium, which is a classical model of an inverted two-level atom, has been presented. The steepest descent method as an asymptotic technique provides the detailed time evolution of the pulse propagation in a linear, temporally dispersive active Lorentzian medium. The analysis and the numerical results show the relation of the location of the saddle points and the topography of the phase function in the complex omega-plane with the transient portion of the pulse that can be used to control the transient response of the medium
In a causally dispersive medium the signal arrival appears in the dynamical field evolution as an increase in the field amplitude from that of the precursor fields to that of the steady-state signal. The interrelated effects of phase dispersion and frequency dependent attenuation and/or amplification alter the pulse in such a fundamental way that results in the appearance of precursor fields. Although superluminal group velocities have been found in various dispersive media, the pulse "front" and associated precursors will never travel faster than c , and hence these are the vehicles through which relativistic causality is preserved. While many rigorous studies of wave propagation and associated abnormal group velocities in passive Lorentzian media have been performed, the corresponding problem in active media has remained theoretically unexplored. This problem is addressed in the present paper, by employing the steepest descent method for the determination of the response of an active Lorentzian medium to a step modulated pulse. The steepest descent method provides a detailed description of the propagation of the pulse inside the dispersive medium in the time domain. Moreover, the evolution of the saddle points illuminates the relation between the medium parameters and the temporal evolution of the propagating pulse within the medium. Hence, useful physical insights are obtained and the interesting differences between the passive and active case are deduced.
The uniform asymptotic description of the propagation of an input rectangle-modulated harmonic signal of angular frequency ω c and initial pulse width T into the half-space z ᐳ 0 that is occupied by a single resonance Lorentz medium is described. The asymptotic description is developed by representing the input rectangular pulse as the difference between two Heaviside unit step-function modulated signals that are separated in time by the initial pulse width T. This representation clearly shows that the resultant pulse distortion in the dispersive medium is solely due to the Sommerfeld and Brillouin precursor fields that are associated with the leading and trailing edges of the input pulse. The dynamic evolution of the pulse with increasing propagation distance z ᐳ 0 is described for both long and very short initial pulse widths T. In both cases it is shown that the pulse distortion becomes severe when the propagation distance z is such that the precursor fields associated with the trailing edge of the pulse interfere with the precursor fields associated with the leading edge.
To obtain a more complete understanding of the effects of temporal dispersion on the propagation dynamics of optical pulses in a physically realistic medium, we have examined the classical problem of dispersive pulse propagation in a Lorentz model medium that is characterized by two separate resonance frequencies. Both an input delta-function pulse and an input unit step-function modulated signal with carrier frequency ωc are considered. In each case the dynamic evolution of the propagated field is described in terms 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 field. The appearance of a new precursor field, separate from the well-known Sommerfeld and Brillouin precursor fields, is shown to appear when the two resonance frequencies of the medium are sufficiently separated. Finally, the signal velocity that is characteristic of the field contribution oscillating at the applied carrier frequency ωc is obtained. These results are illustrated with numerical solutions that depict the entire evolution of the propagated field.
The uniform asymptotic description of the propagation of an input rectangle-modulated harmonic signal of fixed angular frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$ and initial pulse width T into the half-space z>0 that is occupied by a single-resonance Lorentz medium is presented. The asymptotic description is developed by representing the input rectangular pulse as the difference between two Heaviside-unit-step-function-modulated signals that are separated in time by the initial pulse width T. This representation clearly shows that the resultant pulse distortion in the dispersive medium is primarily due to the Sommerfeld and Brillouin precursor fields that are associated with the leading and trailing edges of the input pulse. The dynamical pulse evolution with increasing propagation distance z>0 is completely described for both long and very short initial pulse widths T. In both cases it is shown that the pulse distortion becomes severe when the propagation distance z is such that the precursor fields associated with the trailing edge of the pulse interfere with the precursor fields associated with the leading edge. Finally, the asymptotic theory clearly shows that the main body of the pulse propagates with the signal velocity in the dispersive medium.
An asymptotic approach is utilized in order to obtain a unified description of the propagated field dynamics due to an input Gaussian-modulated harmonic wave of arbitrary initial pulse width in a linear, causally dispersive gain medium, described by the single resonance Lorentz model. The asymptotic method of analysis is applied on the unified, exact integral representation of the propagated field, which is characterized by a unified complex phase function that depends upon the input field and gain medium parameters as well as upon the propagation distance in the medium. In order to apply the asymptotic method, an analysis of the evolution of the saddle point locations, which depend upon the dispersive properties of the gain medium, the temporal width, and the carrier frequency of the input Gaussian pulse as well as upon the propagation distance and of the topography of the real part of the unified phase function in the complex ω plane, must be performed. Upon the subsequent numerical application of the asymptotic method, the predictions of the unified asymptotic approach are found to be in exceptional agreement with the respective results of a purely numerical experiment for all considered initial pulse widths and lead to a unified model of Gaussian pulse propagation in a gain Lorentzian medium. According to this model, the propagated field is composed of pulse components, each being due to the asymptotic contribution of a respective relevant saddle point of the unified phase function. The instantaneous angular frequency of oscillation and the stationary point(s) of the envelope of each such pulse component are then obtained from the real and imaginary parts, respectively, of the corresponding relevant saddle point as it evolves in the complex ω plane. This theoretical approach may then yield particularly useful physical insights into attosecond pulse propagation.
The uniform asymptotic description of the propagation of an input Gaussian modulated harmonic signal of fixed angular frequency ω c and initial pulse width T into the half-space z > 0 that is occupied by a single resonance Lorentz medium is described. The asymptotic theory shows that the propagated field can be expressed solely in terms of a generalized Sommerfeld and Brillouin precursor field structure. If the applied signal frequency ω c is either below or near the medium resonance frequency, the propagated field is found to be dominated by the generalized Brillouin precursor field, while if ω c is well above the medium resonance frequency it is dominated by the generalized Sommerfeld precursor field. The resultant pulse distortion in the dispersive medium is shown to be due solely to the manner in which the initial Gaussian pulse envelope spectrum modifies the precursor field amplitude. Finally, the frequency dependence of the pulse velocity in the single resonance Lorentz medium is described.