## 1. Introduction

[2] Electromagnetic pulses have been used to probe the ionosphere since the early days of radio. The distortion of the reflected pulses may be used as a diagnostic tool to understand the physical properties of the ionosphere, and this approach may be applied to laboratory plasmas as well [*Schmitt*, 1964; *Baird*, 1972]. It is thus helpful to have analytic expressions describing the dispersion effects of propagation through, or reflection from, a region containing a simple plasma.

[3] Many researchers have sought an analytic solution for the time-domain plane-wave field reflected from an interface between free space and a plasma region. Work by *Bowhill* [1954], *Wait* [1970], *Gray* [1976], and others [*Hill and Wait*, 1971; *Gray and Bowhill*, 1974a, 1974b, 1974c], modeled the ionosphere as a stratified, or otherwise inhomogeneous, isotropic plasma, and sought the transient reflected field in terms of the plasma parameters. The simpler case of reflection by a uniform plasma half space has also received much attention, with studies undertaken by *Wait* [1964, 1965, 1969a, 1969b], *Price* [1973], *Gray* [1974, 1975], and others [*Chabries and Bolle*, 1971; *Stanic and Okretic*, 1982]. *McIntosh and Malaga* [1980] give a good summary of the activity in this area prior to 1980. The results of these studies either rely on approximations, or express the transient reflected field in terms of inverse transform integrals or infinite series, neither of which is particularly amenable to physical interpretation. Some exact results were obtained for special cases, such as for a lossless plasma [*Wait*, 1969a; *Gray*, 1974].

[4] Over the past several years, the authors have sought analytic expressions for the reflected-field impulse response of a half space with arbitrary frequency dispersion. Such expressions are useful for the diagnosis of the health of layered materials using transient excitations [*Stenholm et al.*, 2003; *Wierzba and Rothwell*, 2006]. A very general model of the permittivity of a dispersive material is given by a ratio of polynomials in the frequency variable *ω*, which may be factored to produce Lorentz, Debye, and Drude series, and to approximate a Cole-Cole material [*Han et al.*, 2006; *Diaz and Alexopoulos*, 1997]. Closed-form, or nearly closed-form expressions for the time-domain reflected field have been obtained by the authors for conducting [*Suk and Rothwell*, 2002a, 2002b], single-term Debye [*Rothwell*, 2007a], and single-term Lorentz [*Cossmann et al.*, 2006, 2007] materials. The techniques developed in those papers are applied here to lossy isotropic plasmas, providing analytic solutions for the reflected-field waveform that have not been obtained previously. Although a purely numerical solution can be constructed by applying the inverse fast Fourier transform (FFT) to the frequency-domain reflection coefficient, no physical insight into the solution is provided, and thus the FFT approach is used here only to verify the expressions obtained analytically.

[5] While the probing of the earth's ionosphere by short pulses continues to receive attention [*Gutman*, 2005] the transient excitation of plasmas has found use in a diverse number of recent applications. These applications include pulse distortion by reflection from a solid-plasma thin film (such as a metal or semi-conductor) [*Bakunov et al.*, 1996], reflection by controllable “plasma mirrors” [*Bulanov et al.*, 2007], and dispersion of electromagnetic pulses propagating through the Martian ionosphere [*Li and Li*, 2004].

[6] The plasma medium considered in this paper is assumed to be nonmagnetized, and is characterized by collision loss *ν* and plasma frequency *ω*_{p}, with a lossless plasma having *ν* = 0. Assuming a time convention of *e*^{jωt}, the frequency-dependent permittivity of the plasma may be modeled as [*Wait*, 1964; *Price*, 1973]

Using this expression, the frequency-domain Fresnel reflection coefficient can be easily determined for TE and TM-polarized incident waves. The inverse transform of the reflection coefficient, which is computed analytically in the following sections, is the reflection impulse response of the plasma medium, describing the temporal behavior of the transient field. This impulse response is written entirely in terms of Bessel and exponential functions, and perhaps one or two convolutions of these functions. Using these expressions, it is easy to predict the temporal behavior of the reflected field based on the plasma parameters *ν* and *ω*_{p}, and on the angle of incidence.