Distortion of Pulse Signals During Resonant Reflection from Meteor Trails

E. K. Nemirova

(Presented by Professor V. N. Kessenikh, Doctor of Physico-Mathematical Sciences)

Introduction

In radar sounding of meteor trails, in addition to pulse distortions caused by the finite bandwidth of the receiving-indicating path, one may expect the presence of distortions arising directly upon reflection of a high-frequency pulse from a column of ionized gas formed in the atmosphere during the flight of a meteoric body.

Fig. 1. a) Parallel polarization. b) Transverse polarization

One of the causes of distortions may be the dispersive properties of the plasma. From this point of view, the case of resonant reflection is of greatest interest. The existence of resonance for transverse incidence of a radio wave on a meteor trail (Fig. 1) was predicted by Herlofson in 1951 [1]. He showed that the reflection coefficient for transverse incidence will have a maximum if the dia-

meter of the column is small in comparison with the wavelength, and the electron density in some part of the column is somewhat above the critical value.

A number of later works are devoted to the problem of resonant reflection; among them the works of Kaiser, Kloss, and Clegg [2, 3] are of greatest interest.

The theory of Kaiser and Kloss [2] proceeds from the assumption that the meteor trail has a cylindrical form and slowly expands owing to diffusion. Its radius remains small in comparison with the length of the incident wave. The problem of finding the reflection coefficient from a meteor trail is reduced to finding the field near a dielectric cylinder with dielectric permittivity

\[ \varepsilon(r)=1-\frac{4\pi e^2 n(r)}{\varkappa^2 m c^2}, \tag{1} \]

placed in a homogeneous electrostatic field. The solution of the electrodynamic problem is thus replaced by the solution of an electrostatic problem. This approximate method makes it possible to simplify the solution considerably and at the same time gives quantitative results very close to the results obtained by direct integration of Maxwell’s equations.

For a dielectric cylinder in an electrostatic field the following equation holds:

\[ \operatorname{div}(\varepsilon \operatorname{grad} V)=0. \tag{2} \]

Solving this equation, written in cylindrical coordinates, by the method of separation of variables, we obtain the solution in the form

\[ V(r,\varphi)=\sum_{0}^{\infty} V_m(r)\sin m\varphi, \]

where \(V_m\) must satisfy the equation

\[ -\frac{d}{dr}\left(\varepsilon(r)r\frac{dV_m}{dr}\right)=\frac{m^2\varepsilon(r)V_m}{r}, \tag{3} \]

where \(m=1,2,3,\ldots\).

Solving jointly two Maxwell equations\(^{1)}\)

\[ \operatorname{rot}\vec{H}=\frac{\varepsilon}{c}\frac{\partial \vec{E}}{\partial t} \]

\[ \operatorname{rot}\vec{E}=-\frac{\mu}{c}\frac{\partial \vec{H}}{\partial t}, \]

we obtain the equation for the magnetic-field intensity

\[ \frac{d^2H_m}{dr^2}+\left(\frac{1}{r}-\frac{\varepsilon'(r)}{\varepsilon(r)}\right)\frac{dH_m}{dr} +\left(\varkappa^2\varepsilon(r)-\frac{m^2}{r^2}\right)H_m=0, \tag{4} \]

\(^{1)}\) In the following exposition of the theory of Kaiser and Kloss, the Gaussian system of units is used; the time factor is chosen in the form \(e^{j\omega t}\).

\[ H(r,\varphi)=\sum_{0}^{\infty} H_m(r)\cos m\varphi . \]

For the external region of the column, i.e., for \(\varepsilon=1\), the solution of equation (3) is known and has the form

\[ V_m(r)=A_m r^m+B_m r^{-m}, \]

where \(A_m\) and \(B_m\) are arbitrary constants.

The exact solution of equation (4) for the external field contains Bessel functions of the first and second kind.

Let us find an approximate solution for \(H_m(r)\). To do this, assuming in simplified fashion that the field, having \(\operatorname{rot}\vec H\), is potential with respect to the vector \(\vec E\), we find the relation between \(H_m\) and \(V_m\).

\[ \vec E=\operatorname{grad} V \,^{1)} \]

\[ \operatorname{rot}\vec H=j\kappa\varepsilon \vec E , \]

whence

\[ H_m=j\kappa\varepsilon r\,\frac{\dfrac{dV_m}{dr}}{m}, \]

i.e.,

\[ H_m(r)=j\kappa\varepsilon\left(A_m r^m-B_m r^{-m}\right)\ [2]. \tag{5} \]

To determine the constants \(A_m\) and \(B_m\), it is necessary to specify boundary conditions and, consequently, to find the field inside the cylinder. This problem is complicated by the fact that inside the cylinder \(\varepsilon\) depends on \(r\), and can be solved only by numerical integration of the initial equation (3).

Despite its approximate character, this method gives practically the same quantitative result as the method of direct solution of equation (4) [4].

To find the reflection coefficient one may use the expansion of the incident plane wave into cylindrical harmonics by means of Bessel functions, and express the reflected wave through a Hankel function:

\[ H_{\text{inc}}=1\cdot e^{j kx}=I_0(\kappa r)+2\sum_{1}^{\infty} j^m I_m(\kappa r)\cos m\varphi, \]

\[ H_{\text{refl}}=\sum_{0}^{\infty} t_m\left[I_m(\kappa r)-jN_m(kr)\right]\cos m\varphi, \]

\(t_m\) is the reflection coefficient for the cylindrical harmonic. The total magnetic-field intensity

\(^{1)}\) This assumption is made only for the region located near the cylinder.

\[ H_{\text{total}}=\sum_{0}^{\infty} H_m \cos m\varphi=H_{\text{inc}}+H_{\text{refl}}. \]

Under the condition \((kr)^2 \ll 1\), one may use approximate analytical expressions for \(I_m(kr)\) and \(N_m(kr)\). Comparing the expression obtained with (5), we find \(t_m\):

\[ t_m \cong -\frac{2j^m}{1+j\frac{m!(m-1)!4^m}{\pi \kappa^{2m}}\frac{A_m}{B_m}} \quad [2]. \tag{6} \]

The total reflection coefficient for transverse polarization is

\[ g_1=\sum_{0}^{\infty} j^m t_m. \tag{7} \]

The resonance condition is \(\operatorname{Re}(A_m)=0\). Determination of \(g_1\) for various cases of the distribution of electron density over the radius of the column, and comparison of it with \(g_{11}\) (the reflection coefficient for parallel polarization), carried out by Kaiser and Closs, showed that resonance occurs when the linear electron density \(a<10^{12}\ \dfrac{\text{el}}{\text{cm}}\)¹).

The ratio \(\dfrac{g_1}{g_{11}}\) has a resonance maximum that depends on the diffusion of the column. The authors of paper [2] studied in detail the case of a Gaussian density distribution

\[ n(r)=n_0 e^{-\left(\frac{r}{r_0}\right)^s} \quad \text{for } s=2. \]

Here \(n(r)\) is the volume electron density,

\(r_0\) is the radius at which the density is \(37\%\) of the density on the axis,

\[ n_0=\frac{a}{\pi r_0^2}. \]

Cases with \(s>2\) are considered only at the resonance point. Observations carried out with special equipment by Clegg, Closs, and Kaiser [3] confirmed the existence of resonance and gave satisfactory agreement between theoretical and experimental results.

The aim of the present work is to investigate the influence of the resonance properties of meteor trails on the shape of the reflected signal.

Frequency characteristics for various cases of the electron distribution in the trail

A meteor trail may be represented in the form of a four-terminal network, whose transmission coefficient (in the present case, the coefficient

¹) By linear electron density is meant the number of electrons per unit length of the column.

reflection) depends on the frequency of the harmonic signal arriving at the input. This means that the spectral components of the signal arriving at the input (the pulse sent by the transmitter) will be transmitted to the output (reflected) to different degrees. Thus, in considering distortions of a pulse signal upon reflection, we must be interested in the frequency dependence of the complex reflection coefficient

\[ g_1 = G e^{-j\psi}. \]

The resonant properties of a column of ionized gas are determined mainly by the distribution of the electron density over the cross section of the column¹, which depends on the degree of diffusion at its boundaries.

The most favorable for resonance is a column with sharply defined boundaries in the form of a homogeneous cylinder. Physically this case is unrealistic, but it can serve for comparison when considering the influence of diffusion on the resonant properties of trails.

Upon reflection from an ionized cylinder, the dipole type of oscillation is predominant; therefore in the following calculations we take \(m = 1\).

For a homogeneous cylinder [2]

\[ \frac{A_1}{B_1}=\frac{1+\varepsilon}{1-\varepsilon}\,a^{-2}, \]

where \(a\) is the radius of the cylinder.

\[ g_1 = j t_1 = \frac{2}{1 + j\,\dfrac{4}{\pi(\kappa a)^2}\,\dfrac{1+\varepsilon}{1-\varepsilon}} . \tag{8} \]

The resonance condition is \(\varepsilon = -1\).

In expression (8), the frequency-dependent terms are

\[ \kappa=\frac{\omega}{c} \quad \text{and} \quad \varepsilon=1-\frac{4\pi e^2 n}{m\omega^2}. \]

\[ g_1=\frac{2}{b^2+1}(1-jb), \]

where

\[ b=\frac{4c^2m}{\pi^2 a^2 e^2 n}-\frac{4c^2}{\pi a^2\omega^2}. \]

The modulus of the reflection coefficient is

\[ G=\frac{2}{\sqrt{b^2+1}}, \]

the phase

\[ \psi=\operatorname{arctg} b. \]

Figure 2 gives the frequency characteristics for columns with different radii.

For cylinders with one and the same volume electron density, the resonant frequency does not depend on the radius; the passband increases with increasing \(a\).

For the same linear electron density, the resonant frequency decreases with increasing radius, while the passband narrows.

Fig. 2. Reduced resonance characteristics for a homogeneous cylinder at \(N=2.2\cdot 10^{7}\ \text{electrons}/\text{cm}^{3}\), \(\omega_{\mathrm{res}}=1.88\cdot 10^{8}\ \text{s}^{-1}\)

A real meteor trail cannot be represented in the form of a homogeneous cylinder. Owing to diffusion, the boundaries of the column will be blurred. The degree of blurring can be characterized by means of a function showing the distribution of the volume electron density along the radius. In the literature several models of the meteor trail are considered, differing in the distribution of the electron concentration along the radius.

Gerlovson represented the meteor trail in the form of a cylinder in which the electron density is constant from the axis to some radius, and then falls linearly to zero. Feinstein assumed that the density decreases linearly, starting from the axis and ending at a certain radius where the density is equal to zero.

According to Kaiser and Closs, the volume density depends on the radius in the following way:

\[ n(r)=n_{0}e^{-\left(\frac{r}{r_{0}}\right)^{s}}, \]

\[ r_{0}^{2}=4Dt, \]

where \(D\) is the diffusion coefficient, and \(s\) determines the degree of blurring of the column boundaries. For \(s \gg 1\), this distribution is close to Gerlovson’s distribution. Various types of distributions are shown in Fig. 3.

The Kaiser and Closs model may be regarded as the most acceptable. The distribution function introduced by them covers both trails with an almost linearly decreasing density and trails close to those considered by Herlofson. In this case the determination of the reflection coefficients for different cases of the distribution is carried out on a common basis. The special case with \(s=2\) (Gaussian distribution) is used in some subsequent works [4].

Fig. 3. Various types of electron-density distribution along the radius:
a) Herlofson model,

\[ \varepsilon = 1 - \frac{4\pi e^2 n(r)}{k^2 m c^2}, \]

b) Kaiser and Closs model,

\[ n(r)=n_0 e^{-\left(\frac{r}{r_0}\right)^s}. \]

But even the Kaiser–Closs model does not encompass the full range of physical phenomena occurring during the formation of a meteor trail. In particular, collisions are not taken into account, nor is the dependence of the diffusion coefficient on radius and time under the influence of changes in the temperature gradient at the boundary of the column included in the calculation.

The question of the conditions under which one or another distribution is possible has not at present been sufficiently studied. The solution of this question presents considerable difficulty and is not the aim of the present work.

The resonant properties of trails and the degree of pulse distortion are considered for several values of \(s\) and for the Herlofson model.

Let us determine the frequency dependence of the reflection coefficients for various distributions.

In the general case, for \(m=1\),

\[ g_1 = j t_1 = \frac{2}{1 + j \dfrac{4}{\pi (k r_0)^2} - \dfrac{A'}{B'}} , \]

\[ A_1 = A' r_0^{-1}, \qquad B_1 = B' r_0 . \]

The ratio \(\dfrac{A'}{B'}\) depends on the electron-density distribution.

Let us first consider the Herlofson model (Fig. 3a). In this case the column is inhomogeneous in the region \(r=a\).

\[ \frac{A'}{B'} = \frac{ (1+\varepsilon) + \dfrac{\varepsilon}{a\varepsilon'(a)}\,(j\pi-\ln|\varepsilon|) }{ (1-\varepsilon) + \dfrac{\varepsilon}{a\varepsilon'(a)}\,(j\pi-\ln|\varepsilon|) } \ [2]. \]

At resonance \(\varepsilon=-1\).

\[ \varepsilon = 1 - f e^{-\left(\frac{r}{r_0}\right)^s}, \qquad f = \frac{4\alpha}{(k r_0)^2}\,\frac{e^2}{mc^2}, \tag{9} \]

\[ a\varepsilon'(a)=s\ln f . \]

When considering the Kaiser and Closs model,

\[ \varepsilon(\rho)=1-f e^{-\rho^s}, \]

where

\[ \rho=\frac{r}{r_0}, \]

\(\dfrac{A'}{B'}\) is found by numerical integration of equation (3), represented in the form

\[ \frac{d}{d\rho}\left(\rho \varepsilon \frac{dV}{d\rho}\right) = \frac{\varepsilon V}{\rho}. \tag{10} \]

At the boundary of the column \(\varepsilon=1\) and

\[ V=A'\rho=B'\rho^{-1}. \tag{11} \]

The conditions on the axis of the column are as follows:

\[ \text{for } \rho \ll 1 \qquad \varepsilon \simeq \mathrm{const} \quad \text{and} \quad V \simeq \rho, \]

\[ \text{on the axis } V=0, \qquad \frac{\varepsilon V}{\rho}=\varepsilon(0). \]

Numerically integrating (10) point by point, starting from the axis, we find \(V\) inside the cylinder near its boundary.

Imposing the boundary conditions and using (11), we find \(A'\) and \(B'\). In this case

\[ A' = A'_R + jA'_y, \qquad B' = B'_R + jB'_y . \]

In calculation it is convenient to use the relations

\[ A=\frac{A'_R}{B'_R},\qquad B=\frac{A'_j}{B'_R}\quad \text{and}\quad C=\frac{B'_j}{B'_R}. \]

\[ g_1= \frac{2}{ 1+\dfrac{4}{\pi(kr_0)^2}\,\dfrac{-B+AC}{1+C^2} +j\,\dfrac{4}{\pi(kr_0)^2}\,\dfrac{A+BC}{1+C^2} }. \tag{12} \]

\[ \frac{A'_R}{B'_R},\qquad \frac{A'_y}{B'_R} \quad \text{and}\quad \frac{B'_y}{B'_R} \]
are specified graphically as functions of \(f\).

In [2] numerical integration was carried out only for the Gaussian distribution. Since the pulse shape is substantially affected only by trails with a distribution for \(s>2\), we carried out numerical integration for \(s=12\), as a result of which the curves in Fig. 4 were obtained.

Fig. 4. Curves for determining the ratio of arbitrary constants \(A_m/B_m\) for the case \(s=12\), obtained by numerical integration of equation (3).

Of interest is the dependence of the resonant frequency, the resonant reflection coefficient, and the pass band on the linear electron density, the radius of the column, and \(s\).

The curves in Fig. 5 were obtained by Kaiser and Closs. On the basis of curve (c), the dependence of the resonant frequency on \(a\) and \(r_0\) was constructed (Fig. 6).

For columns with one and the same linear electron density, the resonant frequency decreases as \(r_0\) increases. For trails of identical dimensions, the resonant frequency is the higher, the larger \(a\) is.

The resonance characteristic is determined by the distribution function (in this case \(s\)), by the linear electron density, and by the radius of the column. When an ionized trail is formed, its initial diameter is of the order of the mean free path of the molecules. With transverse polarization and for a certain \(a\), conditions exist for resonance at a definite frequency. With the gradual spreading of the column boundaries, the law of which is determined by the value of \(s\), there occurs a gradual increase in \(r_0\) (the radius at which the volume electron density is 37% of the density on the axis). In this process the density on the axis remains constant for some time, resonance is still observed, but the resonant peak shifts in frequency and, at certain moments, may pass through the frequency band occupied by the radar pulse. Then gradual destruction of the trail takes place. It is evident that, without taking collisions into account, the linear electron density must retain its value as the column expands. For one and the same linear density, the resonant frequency decreases as the radius increases (Fig. 6), i.e., the resonant peak moves to the left with time (Fig. 7). In this case the reflected signal should change its amplitude and shape. Changes in the signal amplitude in this case were observed by Clegg, Closs, and Kaiser. The amplitude for transverse polarization first increases, reaches a maximum, and then decreases.

The shape of the reflected signal can be found from consideration of the frequency characteristics. The change in shape can be traced by establishing the dependence of the form of the frequency characteristic on \(r_0\) for one and the same value of \(s\) and density \(\alpha\).

Fig. 5. a) \(\left(\dfrac{g_1}{g_{11}}\right)_{\mathrm{res}}\), calculated by numerical integration of equation (3); b) the same for the Terlofon model; c) \(f_{\mathrm{res}} = 1-\varepsilon_{\mathrm{res}}\) as a function of \(s\).

Fig. 6. Dependence of the resonance frequency on \(\alpha\) and \(r_0\)

— for distributions with \(s \gg 1\).

– – – for the case \(s = 2\).

The behavior of the reflection coefficient during expansion of the trail is seen from Fig. 7. Within the frequency band of the sounding pulse, the frequency characteristic substantially changes its shape. These changes occur over a time of the order of \(0.1\) sec.

The change in the shape of the frequency characteristic within the band is the more significant, the greater the gradient of the electron density at the boundary of the column, i.e., the larger \(s\) is.

Fig. 7. Resonance curves of an expanding trail, \(a = 10^{10}\) el/cm, \(s = 12\)

Fig. 8. Dependence of the passband on \(s\) (for the Gerdjoffson model)

The influence of the resonance properties of the trail on the shape of the reflected signal can be characterized by the bandwidth of the amplitude-frequency-

characteristic. The bandwidth depends strongly on \(s\) (Fig. 8). (The curve was constructed for the Gerloffson model under the assumption that all other trail parameters are constant.)

The signal can be distorted most strongly upon reflection from a homogeneous ionized column.

In order to determine the possible types of distortion, the transient process was considered for reflection from a homogeneous cylinder (with radius \(a = 30\ \text{cm}\) and \(\alpha = 6 \cdot 10^{10}\ \frac{\text{el}}{\text{cm}}\)) of a unit rectangular step of the envelope of a sinusoidal voltage.

Since in the present case obtaining exact quantitative results is not necessary, the resonance characteristic of the trail was approximated by the resonance curve of an oscillatory circuit with quality factor \(Q = 30\).

Fig. 9. Transient processes in the reflection of the rectangular front of a pulse from a homogeneous column of ionized gas

The agreement of both the amplitude-frequency and the phase-frequency characteristics is satisfactory. Figure 9 shows the transient process for the case in which the carrier frequency of the signal coincides with the resonance frequency, and also for the case \(\omega_0 \ne \omega_{\text{res}}\). It may be said that for other cases of distribution as well, the expected types of distortion are lengthening of the pulse fronts and oscillations on the flat portion.

Determining the shapes of the envelopes of the reflected signal in the general case cannot be carried out with sufficient accuracy by means of such an approximation. This problem can be solved by applying approximate methods based on operational calculus.

Conclusions

1. In the case of resonant reflection, the amplitude and shape of the reflected signal must change as the trail expands.

2. These changes are the more pronounced, the greater the gradient of electron density at the boundary of the column.

1. The effect of resonance upon reflection may manifest itself in a prolongation of the pulse fronts and in oscillations at its peak.

2. Changes in amplitude and form may introduce an additional error into the measurement of range and meteor velocity, and into the measurement of the drift velocity of a meteor trail by the coherent-pulse method; they may also cause errors in certain special measurements.

The determination of transient processes for different cases of distribution and for different trail parameters is of considerable interest and is a task for further work.

REFERENCES

1. Herlofson N., Plasma resonance in ionospheric irregularities, Arkiv för Fysik, Bd 3, N 15, 1951.

2. Kaiser T. R. a. Closs R. L., Theory of radio reflections from meteor trails, Phil. Mag., V. 43, N 336, 1952.

3. Closs R. L., Clegg J. A. a. Kaiser T. R.,—An experimental study of radio reflections from meteor trails, Phil. Mag., V. 44, N 350, 1953.

4. Kettel G. H. Certain mode solutions of forward scattering by meteor trails, Proc. JRE, October, 1955, special issue.

1. In the analysis it is assumed that along the axis of the cylinder the density is distributed uniformly. ↩