## 1. Introduction

[2] Previously published research [*Sommerfeld*, 1914; *Brillouin*, 1914, 1960; *Oughstun and Sherman*, 1988, 1994; *Cartwright and Oughstun*, 2007] in the area of linear pulse propagation through dispersive attenuative Lorentz model dielectrics [*Lorentz*, 1906] has shown that ultrawideband pulses give rise to the evolution of forerunners referred to now as Sommerfeld and Brillouin precursor fields whose space-time properties are a characteristic of the material dispersion. The Brillouin precursor is of particular importance because its peak-amplitude point only decays algebraically with propagation distance *z* > 0 as *z*^{−1/2}, whereas the remainder of the pulse (the Sommerfeld precursor and the signal contribution) decays exponentially with propagation distance [*Oughstun and Sherman*, 1988; *Oughstun*, 1995]. Such phenomena has important physical applications in areas such as remote sensing and biomedical imaging where there exists a trade-off between interrogation distance and image resolution. One aspect of remote sensing which is of considerable interest is the remote sensing of buried or hidden terrestrial objects from orbiting satellites. This then raises the question as to whether or not the ionosphere (a plasma) is capable of supporting the evolution of such a Brillouin precursor. If so, then it may be used to optimize radar imaging, remote sensing, and communication applications through the ionosphere. We consider this question here for a simplified model of the ionosphere that is given by the Drude model [*Drude*, 1900] in which both the Earth's magnetic field and the dependence of the number density of free electrons on altitude are neglected.

[3] The Drude model of a cold uniform plasma in the absence of a magnetic field begins with the equation of motion [*Sturrock*, 1994]

where **E**_{loc} ≈ **E** is the effective local electric field intensity acting on the electron as the driving force, and where *d***v**/*dt* = ∂**v**/∂*t* + **v** · ∇**v** denotes the convective derivative (**v** = *d***r**/*dt*) that may be approximated as *d***v**/*dt* → ∂**v**/∂*t* in the linear approximation considered here. In addition, *γ* ∼ 1/*τ*_{e} is the effective collision frequency, where *τ*_{e} denotes the time interval associated with the mean free path for collisions. Notice that this equation of motion neglects the gravitational force *m*_{e}**g**, where **g** is the acceleration due to gravity, which naturally introduces spatial inhomogeneity in the material response. The phasor solution of the equation of motion (1) then leads to the expression [*Oughstun*, 2006]

for the angular frequency dispersion of the conductivity, where *σ*_{0} ≡ ε_{0}*ω*_{p}^{2}/*γ* is the static conductivity and *ω*_{p}^{2} ≡ *Nq*_{e}^{2}/(ε_{0}*m*_{e}) is the square of the (angular) plasma frequency. For a spatially inhomogeneous medium, the number density *N* of electrons is a function of position. For the ionosphere, this spatial dependence is typically dominated by the dependence on altitude *z* above the Earth's surface (due to the gravitational force), so that *N* ≈ *N*(*z*), where *N*(*z*) is a slowly varying function of *z*.

[4] Even for the spatially homogeneous case where the number density of electrons *N* is approximated as a constant, it is difficult to provide an accurate, closed-form expression of ultrawideband pulse propagation through a lossy plasma because the plasma is both dispersive and attenuative. In that case, each frequency component of the input pulse changes at its own characteristic rate in both phase and amplitude with propagation distance. Previous research has dealt with this difficulty by making simplifying assumptions which then either changes the physical problem or gives incorrect results, or both. For example, *Wait* [1965] assumed the loss to be either negligible or constant. *Hillion* [1997] made approximations to the refractive index in the two extreme cases of low frequency and high frequency. *Lee* [1979a, 1979b] assumed that the input pulse is highly peaked about some carrier frequency that is much greater than the plasma frequency and collision frequency. He then expanded the wave number about this carrier frequency, as is commonly done in the group velocity approximation [*Jones*, 1974]. *Strelkov* [2007] approached this problem in the time domain and made assumptions as to the relationship between the frequency and duration of the pulse. *Dvorak et al.* [1997] did not use any simplifying assumptions on the wave number to study the propagation of a double-exponential pulse, but instead provided an asymptotic extraction technique in order to reduce the number of sample points needed by the fast Fourier transform (FFT) to numerically compute the propagated pulse. Their asymptotic extraction technique equates the high frequency limit of the propagated spectrum to an extraction term given in inverse powers of frequency. They then subtract the high frequency behavior of the pulse from the spectrum, compute its inverse Fourier transform analytically, and then numerically compute the inverse FFT of the remaining spectrum. *Luebbers et al.* [1991] used a numerical finite difference time domain method, with modifications, for use with frequency-dependent materials. In our approach, we make no simplifying assumptions about the pulse spectrum or the material response. We provide an accurate, closed-form, uniform asymptotic approximation to a step function modulated pulse that is propagating through an isotropic collisional plasma. This closed-form expression elucidates the role of the Brillouin precursor in the propagated field when the input signal carrier frequency is below the plasma cutoff.