Ultrawideband pulse propagation through a homogeneous, isotropic, lossy plasma

Authors


Abstract

[1] We investigate a linearly polarized, plane wave electromagnetic step function modulated sine wave pulse traveling through an isotropic, homogeneous, lossy plasma with dielectric permittivity described by the Drude model. The results of this investigation extend the useful frequency domain below the plasma cutoff frequency. An asymptotic method of analysis is used to provide a closed-form approximation to the integral representation of the propagated pulse that is valid for all input carrier frequencies. This closed-form expression is the sum of its three asymptotic component fields: the Sommerfeld precursor, the Brillouin precursor and the signal contribution. These expressions reveal that, because of conductivity, each field component attenuates exponentially with distance at its own characteristic rate and that, for sufficiently large propagation distance, the Sommerfeld precursor will be the dominant contribution to the field. However, a study of the penetration capability of each field component shows that, for a large enough propagation distance with carrier frequencies below cutoff, the Brillouin precursor decays algebraically as z−2 with a minimal exponential attenuation with propagation distance while the Sommerfeld precursor decays at a rate that approaches a z−3/4 algebraic decay. Optimal signal penetration through a finite distance of a lossy plasma medium, for either radar imaging, remote sensing, or communication applications, may then be realized by using an appropriately constructed sequence of Brillouin precursor pulses.

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]

equation image

where ElocE is the effective local electric field intensity acting on the electron as the driving force, and where dv/dt = ∂v/∂t + v · ∇v denotes the convective derivative (v = dr/dt) that may be approximated as dv/dt → ∂v/∂t in the linear approximation considered here. In addition, γ ∼ 1/τe is the effective collision frequency, where τe denotes the time interval associated with the mean free path for collisions. Notice that this equation of motion neglects the gravitational force meg, 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]

equation image

for the angular frequency dispersion of the conductivity, where σ0 ≡ ε0ωp2/γ is the static conductivity and ωp2Nqe2/(ε0me) 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 NN(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.

2. Formulation

[5] We assume the ionosphere to be an unbounded, isotropic, homogeneous, conducting plasma so that both the dependence of the number density of electrons on altitude and the effects of the Earth's magnetic field are ignored. The angular frequency dependence of the relative complex dielectric permittivity of such a plasma may then be described by the Drude model as [Drude, 1900]

equation image

where εc(ω) ≡ ε(ω) + iσ(ω)/ω, with ε(ω) = 1 and σ(ω) denoting the temporal frequency spectra of the relative dielectric permittivity and electric conductivity (given by equation (2)), respectively. Typical values of these parameters for the E layer of the ionosphere are given by Messier [1971] as ωp = π × 107 rad/s and νeff = π × 105 rad/s. In addition, the relative magnetic permeability of the plasma is taken as μ ≡ 1.

[6] Consider a linearly polarized plane wave pulse traveling in the positive z direction. Let the temporal behavior of the electric field component on the plane z = 0 be given by a step function modulated sine wave

equation image

with fixed carrier frequency ωc and initial field strength E0 (V/m), where u(t) denotes the Heaviside unit step function u(t) = 0 for t < 0 and u(t) = 1 for t > 0. The temporal behavior of the propagated electric field component on the plane z > 0 is given exactly by the Fourier integral representation [Oughstun and Sherman, 1994]

equation image
equation image

where a is greater than the abscissa of absolute convergence for E(0, t) and where equation image(ω) ≡ (ω/c)n(ω) is the complex wave number in the dispersive attenuative medium with complex index of refraction n(ω) = equation image. Because we are considering a plane wave pulse, the temporal frequency spectrum equation image(z, ω) of the magnetic field component may be computed directly from the temporal frequency spectrum of the electric field component as equation image(z, ω) = n(ω)/cequation image(z, ω) and so the following analysis is focused on the electric field component alone.

[7] On the basis of Brillouin's asymptotic method of analysis [Brillouin, 1960], define the complex phase function

equation image
equation image

where θ ≡ ct/z is a dimensionless space-time parameter for all z > 0. The integral representation of the propagated electric field component (6) may then be written as

equation image

For space-time values θ ≤ 1 (which correspond to wave speeds greater than the speed of light c in vacuum) the integral appearing in equation (8) is identically zero [Sommerfeld, 1914], as can be shown by direct application of Jordan's lemma [Whittaker and Watson, 1943]. The remainder of this paper is then devoted to the evaluation of the contour integral appearing in equation (8) for all subluminal space-time points θ > 1.

[8] The integral representation of the propagated electric field component given in (8) is ideally suited to analysis by asymptotic expansion techniques as the propagation distance z increases. Here, we employ the saddle point method due to Debye [1909] as modified by Olver [1970]. In this method, the original contour of integration is deformed to pass through the saddle point(s) of the complex phase function ϕ(ω, θ) such that the new contour equation image(ω, θ) remains in the valleys below the saddle point(s) and moves continuously with θ. When this is done, ℜ{equation image} obtains a local maximum along the new contour equation image(ω, θ) at each saddle point and this maximum becomes more pronounced as zequation image. Hence, the integral in equation (8) may be approximated by the value of the integral within separate small neighborhoods about each of the relevant saddle point(s), the accuracy of this approximation increasing in the sense of Poincárè as zequation image, as described by Erdelyi [1956]. The value of this integral within a neighborhood of a specific saddle point is found by expanding the complex phase function ϕ(ω, θ) and the spectral amplitude function equation image(0, ω) about that point, resulting in an integral that can be evaluated in a straightforward manner. However, there are situations in which the saddle point method does not provide a uniform approximation to the integral representation of the propagated pulse. For instance, a saddle point may be an infinite-order saddle point whose real part is located at infinity. In this case, ϕ(ω, θ) does not have a valid Taylor expansion about the saddle point. Another instance is when a saddle point comes within close proximity of a pole of the amplitude function. The amplitude function will then have a Taylor series expansion about the saddle point whose domain is limited by the location of the pole. These situations are commonly encountered when studying electromagnetic pulse propagation through a dielectric material [Oughstun and Sherman, 1994; Oughstun, 2005; Cartwright and Oughstun, 2007] and are also naturally encountered in the Drude model.

3. Complex Phase Function

[9] The saddle point method depends upon the topology of the complex phase function in the complex ω plane, which is determined from the multivalued expression

equation image

where the values ω1,2 are defined as

equation image

There are then four branch points of ϕ(ω, θ) located at ω = 0, ω1, ω2, and at ω = −iνeff. We define one branch of ϕ(ω, θ) using two branch cuts: one extends horizontally from ω1 to ω2 and the other extends vertically from ω = 0 to ω = −iνeff, as illustrated in Figure 1.

Figure 1.

The branch cuts and dynamic evolution of the saddle points for the Drude model.

[10] The saddle points of ϕ are the solutions to the saddle point equation

equation image

where the prime denotes differentiation with respect to ω. There are three solutions to equation (11) that are relevant to the following analysis. Two of these solutions are denoted here by ωD±(θ) and are referred to as the distant saddle points because they evolve in the region ∣ℜ{ωequation image(θ)}∣ > νeff/2 of the complex ω plane, as depicted in Figure 1. This pair of first-order distant saddle points lie in the lower half plane, symmetrically situated about the imaginary axis. The approximate locations of these distant saddle points are found to be

equation image

where we have defined the functions

equation image

and

equation image

for all θ ≥ 1. Notice that in the limit as θ → 1+, their real parts become unbounded

equation image

whereas in the limit as θ → equation image, the distant saddle points approach the outer branch points

equation image

respectively.

[11] The third relevant solution to equation (11), denoted here by ωN(θ) and referred to as the near saddle point, is a first-order saddle point that is located along the positive, imaginary axis at ≈i2 × 106 when θ = 1+, travels down the positive imaginary axis with increasing θ and approaches the branch point located at the origin as θ → equation image. The dynamical behavior of all three saddle points ωD±(θ) and ωN(θ) is depicted in Figure 1.

[12] The saddle point method requires that the original path of integration appearing in equation (8) be deformed through each of the accessible saddle points of the complex phase function ϕ(ω, θ) and that the path remains in the valleys of these saddle points. Because the locations of the saddle points change with θ, the deformed path of integration equation image(ω, θ) will also vary with θ. For the Drude-model conductor, all three saddle points ωN(θ) and ωD±(θ) are accessible for all space-time points equation image > 1. Figure 2 shows an acceptable deformed path equation image(ω, θ) for the space-time point θ = 1.5.

Figure 2.

An acceptable deformed path equation image(ω, θ) that lies in the valleys of the relevant saddle points for the space-time point θ = 1.5.

4. Sommerfeld Precursor

[13] The asymptotic expansion of equation (8) about the symmetric pair of distant saddle points ωD±(θ) for values of θ > 1 yields the dynamical evolution of the Sommerfeld precursor ES(z, t). These two, first-order saddle points are symmetrically located about the imaginary axis and approach the values ωD±(θ) → ±equation imagei2δ as θ → 1+, at which point they are infinite-order saddle points.

[14] The saddle point method cannot be applied to the distant saddle points for values of θ → 1+ because the complex phase function does not have a Taylor series expression that is valid about these saddle points in this limit. A uniform expansion of the Sommerfeld precursor ES(z, t), valid for all θ > 1, is then found through use of the theorem due to Handelsman and Bleistein [1969] with the result

equation image

as zequation image for all θ > 1. Here Jn(ξ) denotes the Bessel function of the first kind of integer order n. The magnitude of the remainder term is bounded as

equation image

for zZ > 0 and θ ≥ 1, where K > 0 is a constant independent of θ and z. The coefficients appearing in (17) are given by

equation image
equation image
equation image
equation image

For values of θ > 1 and bounded away from unity such that (z/c)∣α(θ)∣ ≫ 1, the large argument asymptotic expansion of the Bessel function may be substituted into equation (17). This substitution reduces this uniform expansion given to the nonuniform result obtained by direct application of the saddle point method for space-time points θ bounded above 1.

[15] The Sommerfeld precursor field ES(z, t)/E0 for the step function modulated sine wave E(z, t) with applied signal frequency ωc = 1 × 105 rad/s at an observation distance of z/zd = 5 into the Drude-type plasma is illustrated in Figure 3. Here, zd ≡ [ℜ{equation image(ωc)}]−1 denotes a single absorption depth (at the applied signal frequency ωc) into the plasma. Characteristic of the Sommerfeld precursor is an amplitude that starts at zero when θ = 1, increases rapidly to a peak occurring shortly after θ = 1, and then monotonically decreases with increasing θ. The instantaneous angular frequency of oscillation of the Sommerfeld precursor, defined as the time derivative of the oscillatory phase [Brillouin, 1914, 1960], begins at infinity (when the field amplitude is zero) and then rapidly decreases with increasing equation image, approaching the angular frequency value equation image as θ → equation image [Oughstun and Sherman, 1994; Wyns et al., 1988].

Figure 3.

Dynamic behavior of the Sommerfeld precursor ES(z, t)/E0 for the step function modulated sine wave with applied signal frequency ωc = 1 × 105 rad/s at a relative observation distance z/zd = 5 into the lossy plasma.

5. Brillouin Precursor

[16] The asymptotic expansion of equation (8) about the near saddle point ωN(θ) for values of θ > 1 yields the dynamical evolution of the Brillouin precursor EB(z, t). This first-order saddle point is located along the positive imaginary axis for all θ > 1, asymptotically approaching the origin as θ → equation image. Both the amplitude function equation image(0, ω) and the complex phase function ϕ(ω, θ) have valid Taylor series expansions about the near saddle point ωN(θ) so that the saddle point method may be directly applied with the result

equation image

as zequation image for all θ > 1.

[17] The Brillouin precursor field EB(z, t)/E0 for the step function modulated sine wave E(z, t) with applied signal frequency ωc = 1 × 105 rad/s at a relative observation distance of z/zd = 5 into the Drude-type plasma is illustrated in Figure 4. Characteristic of the Brillouin precursor in a lossy plasma is an amplitude that starts at zero when θ = 1, increases to a peak amplitude whose position is dependent upon propagation distance, and then monotonically decreases with increasing θ. The peak amplitude point of the Brillouin precursor occurs at larger and larger space-time values θ as the propagation distance increases. Notice the long, nonoscillatory tail of the Brillouin precursor in a lossy plasma, which is caused by the asymptotic approach of the near saddle point ωN(θ) to the origin as θ → equation image. This long tail is a characteristic of the Brillouin precursor in a dispersive, conducting medium that does not appear in the Brillouin precursor evolution in a dispersive dielectric.

Figure 4.

Dynamic behavior of the Brillouin precursor EB(z, t)/E0 for the step function modulated sine wave with applied signal frequency ωc = 1 × 105 rad/s at a relative observation distance z/zd = 5 into the lossy plasma.

6. Signal Contribution

[18] The signal contribution Ec(z, t) to the propagated field of the step function modulated sine wave is due to the simple pole singularity at ω = ωc appearing in the integrand of (8). It is assumed here that ωc = 2πfc is real, positive and finite (as all physical frequencies must be). The simple pole located at ωc may influence the value of E(z, t) if either the path equation image(ω, θ) crosses the pole singularity, if either of the saddle points ωequation image(θ) come within close proximity of the pole, or both.

[19] Assume that the original path of integration appearing in (8) is deformed into a path equation image(ω, θ) which, within each small neighborhood about each saddle point, consists of the paths of steepest descent through the relevant saddle points. Let θ = equation images be the space-time point at which the path equation image(ω, θ) crosses the simple pole located at ωc, as defined by the equation

equation image

We then look for the saddle point that satisfies the above equation. Notice that equation image{ϕ(ωN(θ), θ)} = 0 for all θ while equation image{ϕ(ωequation image(θ), 1)} = 0 but then decreases monotonically with increasing θ and is concave up. Because equation image{ϕ(ωc, θ)} decreases linearly with increasing θ, it follows that if equation image{ϕ(ωc, 1)} > 0, the path emanating from the near saddle point ωN(θ) will cross the pole at ωc, otherwise the path emanating from the distant saddle point in the right half plane ωequation image(θ) will cross the pole at ωc. Hence, the space-time point θs at which the path emanating from the appropriate saddle point has crossed the pole is dependent upon the applied carrier frequency ωc. For the material parameters used here, the critical applied carrier frequency is found to be ωs ≈ 2.2 × 106 rad/s. For applied carrier frequencies below ωs, the near saddle point ωN(θ) is used in the analysis of the signal contribution, otherwise, the distant saddle point ωequation image(θ) is used in the analysis of the signal contribution.

[20] Direct application of the saddle point method does not provide a uniform asymptotic expansion of the signal contribution due to both the restricted validity of the Taylor expansion in the neighborhood of the pole and the residue contribution. A uniform asymptotic expansion that accounts for the presence of the simple pole in the integrand of E(z, t) is provided by the theorem due to Felsen [1963], Felsen and Marcuvitz [1959], and later generalized by Bleistein [1966]. This expansion is called the signal contribution, denoted by Ec(z, t), and is given by

equation image
equation image
equation image

as zequation image. Here, ts ≡ (z/cs,

equation image

and

equation image

The sign choices in equation (25) are determined by equation imageSP(θ)}. If equation imageSP(θ)} > 0, then the upper sign choice is to be used, while the lower sign choice is to be used when equation imageSP(θ)} < 0. Note that equation imageSP(θ)} = 0 at θ = θs but also that ΔSP(θ) ≠ 0 for the Drude model, which would correspond to the coalescence of the saddle point and the pole at θ = θs.

[21] The signal contribution Ec(z, t)/E0 for the step function modulated sine wave E(z, t) with applied signal frequency ωc = 1 × 105 rad/s at a relative observation distance of z/zd = 5 into the Drude-type ionospheric plasma is illustrated in Figure 5. Characteristic of the signal contribution is the smooth turn-on of oscillatory behavior at the applied carrier frequency ωc. It should be noted here that the signal contribution consists not only of the oscillatory behavior at the applied carrier frequency (i.e., the addition of the residue), but also consists of the influence of the pole on either the distant or near saddle point contributions. We choose to consider this influence as part of the pole contribution, although it may also be accounted for in either the Sommerfeld or Brillouin precursor, respectively.

Figure 5.

Dynamic behavior of the signal contribution Ec(z, t)/E0 for the step function modulated sine wave with applied signal frequency ωc = 1 × 105 rad/s at a relative observation distance z/zd = 5 into the lossy plasma.

7. Total Field

[22] The asymptotic approximation to the electric field component of the propagated step-modulated sine wave (8) is the sum of the three contributions

equation image

as zequation image. The asymptotic approximation to the step function modulated sine wave E(z, t)/E0 with applied signal frequency ωc = 1 × 105 rad/s at a relative observation distance of z/zd = 5 into the Drude-type plasma is illustrated by the solid curve in Figure 6. As a comparison, the propagated electric field component calculated using a 222-point fast Fourier transform (FFT) with a maximum sampling frequency of fmax = 5× 108 sampled at the Nyquist rate is also plotted as the dashed curve in Figure 6. As discussed by Dvorak et al. [1997], the validity of the FFT result is compromised due to the long tail of the Brillouin precursor and the limited number of sample points used. However, Figure 6 is useful in that it clearly shows both the Brillouin precursor and the signal contribution. The asymptotic approximation to the step function modulated sine wave E(z, t)/E0 with applied signal frequency ωc = 1 × 105 rad/s at a relative observation distances of z/zd = 10 and 200 are illustrated in Figures 7 and 8, respectively. At z/zd = 200 the Sommerfeld precursor is the dominant contribution to the total field, which is not apparent in Figure 8 due to the large θ scale, and so is shown separately in Figure 9. Nevertheless, the Brillouin precursor is certainly comparable in amplitude at this large propagation distance.

Figure 6.

The asymptotic (solid) and numerical (dashed) solutions of the propagated step function modulated sine wave with applied signal frequency ωc = 1 × 105 rad/s at a relative observation distance z/zd = 5 into the lossy plasma.

Figure 7.

The asymptotic solution of the propagated step function modulated sine wave with applied signal frequency ωc = 1 × 105 rad/s at a relative observation distance z/zd = 10 into the lossy plasma.

Figure 8.

The asymptotic solution of the propagated step function modulated sine wave with applied signal frequency ωc = 1 × 105 rad/s at a relative observation distance z/zd = 200 into the lossy plasma. At this propagation distance, the Sommerfeld precursor is the dominant contribution to the field.

Figure 9.

The early time behavior (i.e., the Sommerfeld precursor) of the propagated step function modulated sine wave with applied signal frequency ωc = 1 × 105 rad/s at a relative observation distance z/zd = 200 into the lossy plasma.

8. Peak Amplitude Points

[23] The preceding asymptotic analysis and numerical results clearly show that the propagated electric field of the step-modulated sine wave in a Drude-model conductor is the sum of the Sommerfeld precursor, the Brillouin precursor and the signal contribution. The peak amplitude points of these components using the asymptotic approximations (17), (23) and (25) are now examined in order to determine the penetration capabilities of each field component.

[24] We consider a step function modulated sine wave for three different below-cutoff angular carrier frequencies: ωc = 1 × 104 rad/s, ωc = 1 × 105 rad/s and ωc = ωp. Figure 10 shows the real and imaginary parts of the complex wave number as functions of ω with each of these three carrier frequencies denoted by plus signs. Notice that the imaginary part of equation image(ω) corresponds to the absorption of the pulse. We choose angular carrier frequencies that lie at or below the cutoff frequency ωp because the effects of dispersion are minimal above cutoff and so the precursor fields are inconsequential there. This is seen in Figure 10 (top) in which the solid curve shows the real part of the complex wave number equation image{equation image(ω)} and the dashed curve shows the wave number of free space k0(ω) = ω/c.

Figure 10.

The real and imaginary parts of the complex wave number equation image(ω). The pluses represent the applied frequency cases of ωc = 1 × 104, 1 × 105 rad/s and ωc = ωp used in this paper. The dashed line represents the wave number of free space k0 = ω/c.

[25] The base ten logarithm of the peak amplitude points of the Sommerfeld precursor, the Brillouin precursor and the signal contribution for these cases are plotted in Figure 11 as functions of the base ten logarithm of the propagation distance, z. The solid curves depict the case ωc = 1 × 104 rad/s, the dashed curves depict the case ωc = 1 × 105 rad/s, and the dotted curves depict the case ωc = ωp. The red curves describe the base ten logarithm of the peak amplitude of the Sommerfeld precursor, the blue curves describe the base ten logarithm of the peak amplitude of the Brillouin precursor, and the green curves describe the base ten logarithm of the peak amplitude of the signal contribution. These results show that, at some space-time point that is dependent on the value of the angular carrier frequency, the Sommerfeld precursor becomes the dominant contribution to the field and remains so for all larger space-time points. This result is in keeping with the physical interpretation of a plasma as a high-pass filter. The Sommerfeld precursor is then the high-frequency response of the material to the input pulse. It should be noted here that for the case ωc = ωp, the peak amplitude of the signal contribution, depicted by the dotted green curve, arises from the pole's influence on the distant saddle point ωequation image(θ). Hence, the signal contribution in this case looks like a Sommerfeld precursor with a much reduced amplitude rather than a simple time harmonic oscillation at the applied carrier frequency.

Figure 11.

The base ten logarithm of the peak amplitude points of the Sommerfeld precursor, the Brillouin precursor, and the signal contribution, given by the red, blue, and green curves, respectively, as functions of the base ten logarithm of z. The solid, dashed, and dotted curves depict the cases ωc = 1 × 104 rad/s, ωc = 1 × 105 rad/s, and ωc = ωp, respectively.

9. Discussion

[26] These Drude-model conductivity results are now contrasted with results of pulse propagation through a dispersive attenuative dielectric material. When a step-modulated sine wave propagates through a pure dielectric (i.e., no conductivity) such as a Lorentz or Debye-model material, the Brillouin precursor decays only algebraically as z−1/2. This can be explained in terms of the location of the near saddle point ωN(θ). In a pure dielectric, εc(ω) = ε(ω) does not have a pole at the origin so that the near saddle point travels down the positive imaginary axis, crosses the origin, and continues down the negative imaginary axis into the lower half of the complex ω plane. At the space-time point at which the near saddle point crosses the origin, the attenuation of the Brillouin precursor is algebraic in propagation distance (as z−1/2), not exponential. This is seen in the asymptotic expansion (23) where ϕ(0, θ) = 0, which gives

equation image

and the sole z dependence of the Brillouin precursor arises from the factor

equation image

In contrast, in a purely conducting medium such as the Drude model, the complex permittivity εc(ω) = ε(ω) + iσ(ω)/ω possesses a simple pole at the origin which restricts the range of the near saddle point to the positive imaginary axis. Hence, there is no space-time point at which ϕ(ωN, θ) = 0. This fact is evident in Figure 12 which illustrates the real part of the complex phase function ℜ{ϕ(ω, θ)} as a function of the space-time parameter θ = ct/z at the saddle points ω = ωequation image(θ), ωN(θ) and at the pole ωc, as described by the red, blue and green curves, respectively. Again, the solid, dashed and dotted curves depict the carrier frequency values ωc = 1 × 104 rad/s, ωc = 1 × 105 rad/s and ωc = ωp, respectively. From this analysis, one would assume that all components (and hence, the total field) of the propagated pulse attenuate exponentially. However, this exponential attenuation rate may be sufficiently small to yield an approximate algebraic day over some finite propagation distance interval.

Figure 12.

The real part of the complex phase function evaluated at the distant saddle point (red), near saddle point (blue), and carrier frequencies ωc = 1 × 104 rad/s (solid green), ωc = 1 × 105 rad/s (dashed green), and ωc = ωp (dotted green), respectively, plotted as a function of absorption depth.

[27] If the decay rate of the propagated component is algebraic in propagation distance as Apeak = Bzp where B is a constant, then the attenuation rate of the peak amplitude may also be determined by plotting the base ten logarithm of the peak amplitude point as a function of the base ten logarithm of propagation distance. The decay rate p of the peak amplitude is then the slope of the relation log(Apeak) = log(B) + plog(z). Numerical calculations of the slopes of the red curves given in Figure 11 show that the algebraic decay rate of the Sommerfeld precursor approaches p = −.75 as z increases but remains finite. Numerical calculations of the slopes of the blue curves given in Figure 11 show that, for large enough z, the algebraic decay rate of the Brillouin precursor is p = −2. That is, the peak amplitude points of the Sommerfeld and Brillouin precursors decay as z−3/4 and z−2 for large z, respectively. A possible explanation of these two apparently contradicting results is that, for large propagation distances, the propagated spectrum consists mainly of low and high frequencies (see Figure 10). In turn, the high frequencies give rise to the Sommerfeld precursor in the vicinity of θ = 1, which experiences very little exponential attenuation, while the low frequencies give rise to the Brillouin precursor, which experiences little exponential attenuation for large θ (see Figure 12). This approximate nonexponential, algebraic peak amplitude decay may then be used to advantage over some practical, finite propagation distance by designing a pulse composed of a pair of time-delayed and inverted Brillouin precursors for the ionosphere, as has been done for a dielectric medium by Oughstun [2005]. With this elementary pulse as a building block, coded pulse sequences may then be designed for optimal imaging, communications, and remote sensing applications through the ionosphere when the object properties require that the signal frequency be below the plasma cutoff frequency. Examples include satellite imaging of a terrestrial object obscured by foliage (FOLPEN), by soil (GPR) or underwater.

Acknowledgments

[28] The research presented in this paper was supported by the Air Force Office of Scientific Research (AFOSR) under grant FA9550-07-1-0112.

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