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Annual Meeting Optical Society of America

Group-velocity description of the arrival of the Brillouin precursor in a dispersive medium with absorption

George C. Sherman, Kurt Edmund Oughstun

Published Jan 1, 1990• 0 citations
Publisher

Abstract

The propagation of short pulses in a dispersive medium can be described in simple physical terms by employing the concept of the group velocity provided the medium is lossless. When the medium is absorbing, the group velocity description breaks down. In a previous paper,1 we presented a similar description employing the velocity of energy of monochromatic waves in place of the group velocity which is valid for a particular dispersive medium with absorption known as the Lorentz medium. That model describes the entire dynamics of the pulse as it propagates with the exception of the early stages of the evolution of the second precursor (otherwise known as the Brillouin precursor). Since that region contains the peak values of the pulse, it is important to include it in the model. The present paper describes the arrival of the Brillouin precursor in terms of the group velocity of a propagating field of the form exp [ωttω z] where ω and kω are real and positive.

DOI
10.1364/oam.1990.wq1

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The propagation of short pulses in a dispersive medium can be described in simple physical terms by employing the concept of the group velocity provided the medium is lossless. When the medium is absorbing, the group velocity description breaks down. In a previous paper,1 we presented a similar description employing the velocity of energy of monochromatic waves in place of the group velocity which is valid for a particular dispersive medium with absorption known as the Lorentz medium. That model describes the entire dynamics of the pulse as it propagates with the exception of the early stages of the evolution of the second precursor (otherwise known as the Brillouin precursor). Since that region contains the peak values of the pulse, it is important to include it in the model. The present paper describes the arrival of the Brillouin precursor in terms of the group velocity of a propagating field of the form exp [ωttω z] where ω and kω are real and positive.
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 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.
The propagation of electromagnetic pulses in a linear, dispersive Lorentz medium with two isolated resonance frequencies is presented within the context of the classical theory due to Sommerfeld and Brillouin. Particular attention is given to the modification of the dynamical field evolution when one proceeds from a single-resonance to a double-resonance medium. The appearance of a new field structure in the precursor field evolution is obtained, and the conditions for its evolution are described. The signal velocity for a pulsed field is obtained and compared with the energy-transport velocity in a double-resonance medium both when the conditions for the evolution of the modified precursor field structure are satisfied as well as when they are not satisfied and it is absent from the total propagated field structure.
This article mainly deals with the propagation of step-modulated light pulses in a dense Lorentz medium at distances such that the medium is opaque in a broad spectral region including the carrier frequency. The transmitted field is then reduced to the celebrated precursors of Sommerfeld and Brillouin, far apart from each other. We obtain simple analytical expressions of the first (Sommerfeld) precursor, whose shape only depends on the order of the initial discontinuity of the incident field and whose amplitude rapidly decreases with this order (rise-time effects). We show that, in a strictly asymptotic limit, the second (Brillouin) precursor is entirely determined by the frequency dependence of the medium attenuation and has a Gaussian or Gaussian-derivative shape. We point out that this result applies to the precursor directly observed in a Debye medium at decimetric wavelengths. When attenuation and group-delay dispersion both contribute to its formation, we establish a more general expression of the Brillouin precursor, containing the previous one (dominant-attenuation limit) and that obtained by Brillouin (dominant-dispersion limit) as particular cases. We finally study the propagation of square or Gaussian pulses, and we determine the pulse parameters optimizing the Brillouin precursor. Obtained by standard Laplace-Fourier procedures, our results are explicit and contrast in their simplicity those derived by the uniform saddle-point methods, from which it is difficult to retrieve our asymptotic forms.
The classical theory of dispersive signal propagation by Sommerfeld and Brillouin proposed in 1914 provided the first proof within the classical Maxwell–Lorentz theory that an electromagnetic signal could not propagate faster than the vacuum speed of light in a causal dielectric, also introducing the signal velocity and the precursor fields (or forerunners). Modern asymptotic theory has extended these results, providing a physically meaningful definition of the signal velocity while showing the critical role that the precursor fields play in the observed pulse dynamics. Their role in defining the observed pulse velocity is presented and compared with the classical group velocity as well as with other pulse velocity measures that yield superluminal results.
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.