Radio Science

Impulsive plane-wave reflection from a homogeneous plasma half space

Authors


Abstract

[1] The reflected-field impulse response for a plane wave obliquely incident on a lossy plasma half space is determined analytically by the inversion of the frequency-domain reflection coefficient. Both TE and TM incident field polarizations are considered. The resulting expressions involve simple functions and, at most, one or two convolutions of these functions. The simplicity of the expressions allows the temporal behavior of the reflected field to be predicted on the basis of the physical parameters of the plasma. The expressions are validated numerically by comparison to the inverse fast Fourier transform of the frequency-domain reflection coefficient. A comparison to the response of a laboratory plasma shows good agreement.

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 ωp, with a lossless plasma having ν = 0. Assuming a time convention of ejωt, the frequency-dependent permittivity of the plasma may be modeled as [Wait, 1964; Price, 1973]

equation image

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 ωp, and on the angle of incidence.

2. Formulation of the Reflection Coefficient

[7] Consider a time-harmonic plane wave of frequency ω incident from free space onto a plasma half space occupying the region z > 0. The incidence angle measured from the normal to the interface is θ, the permeability of the plasma is μ0, and the permittivity is given by the expression (1). The frequency-domain reflection coefficient, describing the relationship between the incident and reflected fields, is given by [Rothwell and Cloud, 2001]

equation image

Here Z0 is an impedance representing the ratio of tangential electric and magnetic fields at the interface in the free-space region, and is given by

equation image

where η0 = equation image is the intrinsic impedance of free space. The impedance in the plasma medium is

equation image

where η = equation image, kz = equation image, k0 = ωequation image, and k = ωequation image.

3. Time-Domain Reflection Coefficient for TE Polarization

[8] The impulse response for the reflected field, r(t), is the inverse Fourier transform of the frequency-domain reflection coefficient (2). This expression is also called the time-domain reflection coefficient. When a transient incident wave with waveform e(t) is reflected from the interface, the waveform of the reflected wave may be found through the convolution e(t) * r(t). Because of the availability of extensive transform tables, it is convenient to write the reflection coefficient in terms of the Laplace transform variable through the substitution s = .

[9] Substituting the formula for the plasma permittivity (1) into (2), the frequency-domain reflection coefficient can be written for TE polarization as

equation image

where ωc = ωp/cos θ. This representation for the reflection coefficient may be put into a form that can be inverted using tabulated Laplace transform pairs by rationalizing the denominator. Multiplying the numerator and denominator of (5) by equation image gives

equation image

Adding and subtracting the term ωc2/2 allows this expression to be put into the form

equation image

where

equation image
equation image

Note that s1 may be positive real, negative real, or complex, depending on the ratio ν/ωc.

[10] The expression (7) may be rewritten by factoring out the polynomial s(s + v)(s + s1)(s + s2):

equation image

The first two terms in the brackets may be replaced by partial fraction expansions, giving

equation image

Now, using the Laplace transform pairs tabulated by Campbell and Foster [1951], the terms in (11) may be inverted according to

equation image
equation image
equation image
equation image
equation image
equation image

where In(x) is the modified Bessel function of the first kind and u(t) is the unit step function. With these formulas, (11) may be written as

equation image

where

equation image
equation image
equation image

Here, the star (*) indicates the convolution operation.

[11] The derivatives in (18) may be computed directly. Letting

equation image

results in

equation image
equation image
equation image

Using the results of (23)–(25) now gives

equation image

where

equation image
equation image

Next, let

equation image

Carrying out these derivatives produces

equation image
equation image

Here

equation image

[12] Substituting (30) and (31) into (26) gives the solution for the time-domain reflection coefficient

equation image

[13] The expression (33) is composed of completely real quantities as long as v ≥ 2ωc. When v < 2ωc, λ1 becomes imaginary, and expression (33) involves complex terms, although the result is still real. Since it is convenient to deal entirely with real quantities, let

equation image

and consider the identity [Abramowitz and Stegun, 1965]

equation image

where Jn(x) is the ordinary Bessel function of the first kind. Using these expressions, the reflection coefficient becomes

equation image

where

equation image

[14] When the plasma is lossless, and thus ν = 0, the reflection coefficient reduces to the simple expression

equation image

which was derived by Wait [1969a].

4. Time-Domain Reflection Coefficient for TM Polarization

[15] Substituting the formula for the plasma permittivity (1), the frequency-domain reflection coefficient (2) can be written for TM polarization as

equation image

Multiplying the numerator and denominator of (39) by its numerator gives

equation image

where

equation image

and

equation image

Now, define

equation image

In the case θ = 45°, ωc2 − 2ωp2 = 0 and thus

equation image

where N(s) ↔ n(t). If θ ≠ 45°, then

equation image

where C(s) ↔ c(t).

[16] To determine n(t), let

equation image

where

equation image

Then

equation image

Factoring out the polynomial s(ν + s)(s + s1)(s + s2)(s + s3)(s + s4) and using partial fraction expansions as in the TE case yields

equation image

[17] Using the Laplace transform pairs (12)(17) now gives

equation image

where

equation image

Note that f4(t) is identical to f2(t), except that in the former, λ1 has been replaced by λ2. Carrying out the first two sets of derivatives gives the result

equation image

where

equation image

Carrying out the next set of derivatives gives

equation image

where

equation image

[18] The result for n(t) is obtained upon computing the final set of derivatives. The form of this result depends on whether λ1 and λ2 are real or imaginary. Because ωcωp, there are three possible cases:

equation image
equation image
equation image

where

equation image

[19] Lastly, the inverse transform of C(s) must be found. To begin, (42) is written as

equation image

where

equation image

and

equation image

Note that s5 is either positive real, negative real, or complex, while s6 may be positive real, or complex. These values lead to three possible cases for c(t).

[20] CASE A: s5 real and positive, s6 real and positive

equation image

[21] CASE B: s5 real and negative, s6 real and positive

equation image

[22] CASE C: s5 and s6 are complex conjugates

equation image

[23] In the above expressions, K = 1/[ωp2(tan2θ − 1)] and

equation image

Note that case B occurs if and only if θ > 45°. When θ < 45°, case A occurs if θ < tan−1equation image, while case C occurs if θ > tan−1equation image . Note also that to obtain (64), the following transform pair must be used:

equation image

Although c(t) is noncausal under case B, it is found that convolution with n(t) in (45) produces a causal result. See Cossmann et al. [2007].

[24] It can now be seen that there are many possible combinations of the quantities n(t) and c(t). Although the condition θ = 45° has been singled out for special mathematical treatment, it is not considered a separate case, since it can be recovered as the limit for the other cases considered, and does not exhibit any special behavior. Thus there are a total of eight possible combinations. Case B of c(t), which occurs for θ > 45°, may be combined with all three cases for n(t). The same is true for case C of c(t). However, for case A of c(t), case 1 of n(t) is not allowed.

[25] The behavior of r(t) under TM polarization is determined by the amount of oscillation in each of n(t) and c(t). For instance, case 3 yields a formula for n(t) that involves only exponential and modified Bessel functions, and thus shows no oscillation. Similarly, cases A and B of c(t) involve only damped exponential functions, and thus c(t) is also nonoscillatory. Thus the combination of these cases produces a reflected-field waveform r(t) that exhibits no oscillation, as shown in Figure 6 and discussed in the next section. In contrast, cases 1 and 2 involve several ordinary Bessel functions and so n(t) oscillates strongly. When this condition is combined with case C for c(t), which is described using oscillatory trigonometric functions, the result for r(t) is highly oscillatory, as seen in Figure 4 and described in the next section. Other combinations produce an amount of oscillation somewhere in between these extreme cases.

[26] When the plasma is lossless, ν = 0 may be substituted into n(t) to produce

equation image

Similarly, substituting ν = 0 into c(t) gives

equation image

and

equation image

Thus there are two possible cases. For the limiting case of θ = 45°, the reflection coefficient becomes simply

equation image

5. Numerical Results

[27] For TE polarization, the reflection impulse response, described by (33) and (36), depends on incidence angle only through the quantity ωc. Thus when the impulse response is normalized by ωc, and time is described by the normalized parameter ωct, the results are independent of θ. Figure 1 shows the impulse response for a lossless plasma, and for a plasma with a normalized collision frequency of ν/ωc = 0.25, computed using (36). Also shown is the impulse response found by calculating the inverse FFT of the frequency-domain reflection coefficient (5). A good match is seen in each case, thus validating the time-domain expressions. Since the expression for a lossless plasma (38) involves only an ordinary Bessel function, significant oscillations are present in the reflected field. When ν/ωc is increased from zero, the oscillations begin to damp out since the expression (36) includes both an exponential damping function, and convolution with a nonoscillating modified Bessel function. Figure 2 shows the result of further increasing the collision frequency. When ν/ωc approaches unity, the exponential damping is high enough to overwhelm the oscillations, and only a small bump is seen after the initial peak. When ν/ωc passes 2, the time-domain expression switches to (33). Since this expression involves only modified Bessel functions, there is no oscillation at all after the initial peak. Thus the form of the time-domain reflection coefficient is useful for predicting the oscillatory nature of the reflected field.

Figure 1.

Time-domain impulse response for the electric field reflected by a plasma half space. TE polarization, for various values of ν/ωc. Lines, from time-domain expressions; symbols, from frequency-domain expressions.

Figure 2.

Time-domain impulse response for the electric field reflected by a plasma half space. TE polarization, for various values of ν/ωc. Lines, from time-domain expressions; symbols, from frequency-domain expressions.

[28] As pointed out in section 4, for TM polarization many combinations of the expressions for n(t) and c(t) are possible. Several results are shown in Figures 3–6, corresponding to most of the possible combinations. These results are sorted by the value of the normalized collision frequency, ν/ωp. Since the expressions of n(t) and c(t) involve θ explicitly, different values for the incidence angle are examined for each value of ν/ωp.

Figure 3.

Time-domain impulse response for the electric field reflected by a plasma half space. TM polarization, ν/ωp = 0, for various values of θ. Lines, from time-domain expressions; symbols, from frequency-domain expressions.

Figure 4.

Time-domain impulse response for the electric field reflected by a plasma half space. TM polarization, ν/ωp = 0.25, for various values of θ. Lines, from time-domain expressions; symbols, are from frequency-domain expressions.

Figure 5.

Time-domain impulse response for the electric field reflected by a plasma half space. TM polarization, ν/ωp = 2.2, for various values of θ. Lines, from time-domain expressions; symbols, from frequency-domain expressions.

Figure 6.

Time-domain impulse response for the electric field reflected by a plasma half space. TM polarization, ν/ωp = 4, for various values of θ. Lines, from time-domain expressions; symbols, from frequency-domain expressions.

[29] Figure 3 shows the impulse response for TM plane-wave reflection from a lossless plasma. An incidence angle of θ = 30° corresponds to a combination of n(t) given by (68) with c(t) given by (69). An incidence angle of θ = 45° gives a result that is described by the special-case formula (71). Finally, θ = 60° combines n(t) from (68) with c(t) from (70). Since n(t) in each case involves only ordinary Bessel functions, and since there is no exponential damping, the reflection is highly oscillatory. It can be seen that in this lossless case, as the incidence angle is increased, the oscillations become more rapid, and also decrease in amplitude at a faster rate. For θ < 45°, the ordinary Bessel functions are convolved with an oscillatory trigonometric function (69), while for θ > 45° the convolution is with a nonoscillating exponential function (70). Note that although c(t) is noncausal in the case of θ > 45°, the convolution n(t) * c(t) is causal. Also note that the time-domain results compare well with the results obtained using the FFT, thus validating the TM impulse response expressions.

[30] Figure 4 shows the impulse response for TM polarization with a small collision frequency of ν/ωp = 0.25. Here, an incidence angle of θ = 30° corresponds to a combination of Case 1 of n(t) and Case C of c(t) (call this case 1-C). An incidence angle of θ = 45° gives a result that is described by Case 1 of n(t) and the special-case formula (44) (call this case 1-S). Finally, θ = 60° combines Case 1 of n(t) with Case B of c(t) (case 1-B). It can be seen that as loss is added, the oscillation is damped out compared to the lossless case. However, since case 1 of n(t) involves many ordinary Bessel functions, the oscillations are still pronounced.

[31] Figure 5 shows the impulse response for TM polarization with a collision frequency of ν/ωp = 2.2. Now an incidence angle of θ = 30° corresponds to Case 2-C, an incidence angle of θ = 45° gives a result that is described by Case 2-S, and θ = 60° corresponds to Case 2-B. Because case 2 of n(t) still involves many ordinary Bessel functions, there are still some oscillations in the reflected field. However, these oscillations are highly damped because of the larger damping constant.

[32] Lastly, Figure 6 shows the impulse response for TM polarization with a collision frequency of ν/ωp = 4.0. Here an incidence angle of θ = 30° corresponds to Case 3-A, an incidence angle of θ = 50° corresponds to Case 3-B, and θ = 70° corresponds to Case 2-B. Since case 3 of n(t) involves only modified Bessel functions, the curves for θ = 30° and θ = 50° show no oscillation at all.

6. Prediction of the Temporal Behavior of the Reflected Field

[33] From the theory sections and the examples given above, it can be seen that the behavior of the reflected field is either oscillatory, being composed of Bessel and trigonometric functions, or nonoscillatory, being composed of exponential and modified Bessel functions. This behavior is easily predicted on the basis of the values of the physical parameters of the plasma and on the angle of incidence.

[34] For TE polarization, the only relevant condition is between the collision frequency and the angle-normalized plasma frequency. When ν > 2ωc the reflected-field waveform is nonoscillatory, and when ν ≤ 2ωc the waveform is oscillatory. When ν = 0, the oscillation is strongest, and when ν = 2ωc the oscillation is weakest. This condition is summarized in Table 1.

Table 1. Oscillatory Properties of Terms in Reflected-Field Waveforms
TermConditionNature of Term
TE: r(t)v > 2ωcnon oscillatory
 v ≤ 2ωcoscillatory
TM: n(t)v > 2ωcnon oscillatory
 v ≤ 2ωcoscillatory
TM: c(t)θ > 45°non oscillatory
 θ < tan−1equation imagenon oscillatory
 45° ≥ θ ≥ tan−1equation imageoscillatory

[35] For TM polarization, the reflected waveform consists of the convolution of two functions: r(t) = n(t) * c(t). Each of these functions may be oscillatory or nonoscillatory, depending on the plasma parameters and the incidence angle. Their convolution will be oscillatory when both are oscillatory, slightly oscillatory when only one is oscillatory, and nonoscillatory when both are nonoscillatory. The term n(t) behaves in an identical manner to the TE reflected field, as shown in Table 1. The term c(t) is slightly more complicated. It is nonoscillatory when either θ > 45° or when θ < tan−1equation image. When 45° ≥ θ ≥ tan−1equation image, the term is oscillatory. This is summarized in Table 1.

7. Comparison to Experiment

[36] As a partial validation of the theory presented in the previous sections, consider the experimental results obtained by Schmitt for a laboratory-generated plasma [Schmitt, 1964]. In his experiment, a short-duration electromagnetic pulse was applied to the input of a coaxial discharge tube, and the reflection of the pulse from the afterglow of the discharge was observed. Since the system supports a quasi-TEM transmission line mode, the pulse reflected by the plasma may be approximated using the results from section 3 or section 4 with the incidence angle set to zero.

[37] Schmitt measured the reflected pulse waveform at a variety of times after the creation of the plasma discharge. As time in the afterglow increases, the plasma frequency of the discharge drops. For a gas pressure of 1 mm Hg, the limitations of the instrumentation are least noticeable at lower plasma frequencies, and Schmitt isolates the result at 700 μsec into the afterglow as a useful example. At this time and pressure, he identifies a plasma frequency of fp = 2.8 × 108 Hz and a collision frequency of ν = 3 × 107 Hz. His measured reflected pulse is reproduced in Figure 7, and shows the oscillatory behavior of a lightly damped reflection. Also shown in the figure is the theoretical reflected waveform computed using (36) with an incidence angle of θ = 0°, and convolved with a model of the incident pulse used by Schmitt. The model incident pulse, e(t), is a Gaussian with a width at half-amplitude of 1.8 ns. The agreement between theory and experiment is quite good, considering the limitations of the experiment which include the excitation of surface waves and the accumulation of charge by the glass walls of the discharge tube.

Figure 7.

Pulse reflection response of a laboratory plasma as measured by Schmitt [1964] compared to theoretical response.

[38] Since relative error is often not a good measure of the similarity between time series (due to random noise, timing and scaling issues, etc.) the correlation coefficient between the data sets shown in Figure 7 is computed using

equation image

Here mx is the mean value of the measured time waveform x, and my is the mean value of the theoretical time waveform y. This coefficient has a maximum value of unity if and only if the two waveforms are identical, subject to an arbitrary amplitude scale, time shift, and constant offset. Any other relationship results in a maximum correlation of less than unity, with the size of the coefficient indicating the similarity between the waveforms. For the data shown in Figure 7, the maximum value of the correlation coefficient is computed to be 0.979, which shows very strong agreement between the theoretical and experimental curves.

8. Conclusion

[39] The reflected-field impulse responses for TE- and TM-polarized plane waves incident from free space onto a plasma half space are obtained in terms of the convolutions of simple functions. Limiting cases for zero loss and 45° incidence angle are determined as special cases. For the TE case, the temporal behavior of the impulse response is found to depend only on the relative collision frequency ν/ωc, with two possible forms for the time-domain expression depending on the value of this parameter. The form of the expression allows the temporal behavior to be predicted: when ordinary Bessel functions are involved, the reflected field is oscillatory, with the oscillations damped by the plasma loss. When only modified Bessel functions comprise the expression, the reflection coefficient shows no oscillation. Numerical results are used to reveal the temporal behavior and to validate the expressions by comparison to the inverse FFT of the frequency-domain reflection coefficient.

[40] The expressions for the impulse response in the TM case are more complicated, being composed of a convolution of two functions, the forms of which depend on both the relative collision frequency ν/ωp, and the incidence angle, θ. Several different combinations of these two functions are possible, and many of these combinations are examined numerically. Similar to the TE case, the temporal behavior of the reflected field may be predicted on the basis of which forms of the expressions are required. When ordinary Bessel functions are involved, the field is oscillatory, although the oscillations may be highly damped when the collision frequency is large. When only modified Bessel functions are needed, the field is not oscillatory, and only shows a single, initial peak followed by a long tail. Again, the time-domain expressions are validated by comparison to the inverse FFT of the frequency-domain reflection coefficient.

[41] An additional validation is made by comparing the theoretical results with the measured response of a pulse reflected by a laboratory plasma. The reflected waveform, which is generated in the afterglow of a coaxial discharge tube, compares well with theoretical prediction.

[42] The approach used in this paper can be extended to determine the pulse response of a homogeneous half space with general dispersion properties. By writing the frequency dependence of the permittivity as the ratio of polynomials, multi-term Debye, Lorentz, and Cole-Cole models may be considered. The inversion of the frequency-domain reflection coefficients is considerably more involved than for the plasma case, and remains the ultimate goal of this research. A preliminary study is given in [Rothwell, 2007b], but a general solution remains to be completed.

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