1 Introduction

It is well known that a time-varying current flowing through an antenna creates radiation [1]. The alternating current is concentrated near the surface of a conductor [2] and it is assumed that its interior seldom influences the antenna radiation characteristics. It is true for the metal antenna, where the metal conductance is very high and its skin depth is very thin, and there are very few situations when the antenna metal is thinner than its skin depth.

Contrastively, the plasma antennas use a plasma generated in a low-pressure dielectric tube by microwave excitation, or in atmospheric air by femtosecond laser irradiation as their conducting element, instead of metal. Their importance arises from many advantages, such that they are reconfigurable, stealthy and can be turned on or off rapidly. In particular, the femtosecond laser-induced plasma-channel antenna has a very high flexibility and reconfigurability, and its Vee arrangement made of two plasma channels can have a good directivity and large total gain (more than 8 dB) [3].

However, the conductivity of plasma channel is much smaller, its skin layer is much thicker than that of a metal, and for the same antenna geometry, the plasma antenna is not up to a metal counterpart in the radiation efficiency and gain etc. The plasma used as an antenna leg has a certain radial inhomogeneity at any rate: the plasma density and conductivity has a gradient decreasing in radial direction, especially in the vicinity of plasma-air or plasma-wall boundary. The plasma with the lowest conductivity would take the outermost place, making a main contribution to the antenna radiation characteristics. Nevertheless, many theoretical research and calculations approximate the plasma channel as a column with a homogeneous plasma density and conductivity in its cross section or volume, as a rough approximation [3,4,5,6,7,8], and even when considering the inhomogeneity, there is no immediate comparison with the homogeneous one [9, 10], so it is difficult to estimate the relative accuracy and applicability of this particular approximation. The comparison is impossible experimentally, as all examples of the real plasma have a radial conductivity gradient. In this paper, using high-frequency structure simulator (HFSS), the effect of radial plasma inhomogeneity on the radiation characteristics of the plasma-channel antenna is considered quantitatively in comparison with the homogeneous counterpart. From the viewpoint of the practical application, the total gain and the radiation efficiency are taken as the main far-field radiation characteristics to be considered, as they can be regarded as the ones of the universal antenna characteristics concerning the antenna impedance, accepted and radiated power, and the spatial concentration degree of radiation intensity. Along with the reflection (mismatch) efficiency, the radiation efficiency determines the overall efficiency [1]. In our calculation below, the reflection efficiency is regulated to be close to one (voltage reflection coefficient to be zero at the input terminals) through the impedance matching procedure and hence the radiation efficiency is close to overall efficiency.

2 Plasma characteristics for antenna calculation

The plasma conductivity and its dielectric permittivity can be written as follows, respectively [11]

$$\sigma {\text{=}}\frac{{{\varepsilon _{\text{0}}}{\nu _{\text{m}}}\omega _{{\text{p}}}^{2}}}{{{\omega ^2}+\nu _{{\text{m}}}^{2}}}$$
(1)
$${\varepsilon _{\text{r}}}=1 - \frac{{\omega _{{\text{p}}}^{2}}}{{{\omega ^2}+\nu _{{\text{m}}}^{2}}},$$
(2)

where \({\omega _{\text{p}}}=\sqrt {{n_{\text{e}}}{e^2}/{m_{\text{e}}}{\varepsilon _0}}\) is plasma frequency and is dependent on plasma density. The effective collision frequency \({\nu _{\text{m}}}\) is estimated as \({\nu _{\text{m}}}=3 \times {10^8}(\rho /{\rho _0})T \approx 3 \times {10^9}{P_0}\), where \({P_0}\) is the gas pressure in torrs [10, 12]. At atmospheric pressure, \({\nu _{\text{m}}} \approx 2.3 \times {10^{12}}{{\text{s}}^{ - 1}}\) is much larger than most antenna signal frequencies. The larger the plasma conductivity, the smaller the (negative) plasma permittivity. For \(f=600\;{\text{MHz}}\), \({n_{\text{e}}} \geq {\text{1.66}} \times {10^{15}}{\text{c}}{{\text{m}}^{-{\text{3}}}}\) corresponds to \({\omega _{\text{p}}} \geq 2.99 \times {10^{12}}{{\text{s}}^{-1}}\), \(\sigma \geq {\text{20.36}}\) S/m, and the plasma permittivity starts to be negative. In general, at atmospheric pressure, \(\omega <<{\nu _{\text{m}}}\) holds, and therefore, the plasma conductivity and permittivity is weakly dependent on signal frequency and strongly depends on plasma density.

The skin depth of plasma is defined as follows. The spatial decay constant α within a plasma for an electromagnetic wave normally incident on the boundary of a uniform density plasma can be calculated as discussed in [13]

$$\alpha = - \frac{\omega }{c}\operatorname{Im} \varepsilon _{{\text{r}}}^{{1/2}} \equiv {\delta ^{ - 1}}.$$

For \(\omega <<{\nu _{\text{m}}}\), we can get

$$\delta ={\left( {\frac{2}{{\omega {\mu _0}\sigma }}} \right)^{1/2}}.$$
(3)

Figure 1 shows the dependence of plasma conductivity and permittivity on plasma density for \(\omega <<{\nu _{\text{m}}}\) according to (1) and (2).

Fig. 1
figure 1

Dependence of plasma conductivity and permittivity on plasma density for \(\omega <<{\nu _{\text{m}}}\)

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From (1) to (3), it can be seen that the skin depth of the plasma can be larger than the plasma radius, depending on signal frequency and plasma conductivity, and to decrease the channel resistance and increase the radiation efficiency, it is necessary to make the radius of the plasma channel large enough [3].

3 Effect of radial plasma inhomogeneity

The above analytical relation can be effective to study the plasma with a homogeneous conductivity, but goes out of force when applying direct to the problem dealing with plasma inhomogeneity near the plasma surface. As can be seen later, the consideration of the radial plasma inhomogeneity for an antenna calculation not only is an approach to a more accurate simulation, but also is useful to estimate the skin depth of an inhomogeneous plasma properly. The fact that a single conductivity characterizes a whole plasma could be accepted, as it is averaged across the plasma cross section. However, in fact, various radial distributions of conductivity can correspond to a given cross-sectional averaged conductivity and the radiation characteristics of a real plasma antenna might deviate from that of a single or averaged conductivity.

The geometry of the simulated femtosecond laser-induced plasma-channel Vee antenna and its general 3D radiation pattern are illustrated in Fig. 2. It consists of two plasma columns forming the Vee shape, and it is with the open area of the Vee pointing in the desired direction of transmission or reception. The angle between the legs (apex angle) can vary with length of the legs to achieve maximum performance. By reasonably choosing the angle \({\theta _0}\), so that inner main lobes could approximately overlap, the two inner main lobes align with each other along the middle direction (Y-axis) and produce a stronger main lobe, thus increasing the directivity of the antenna. In Ref. [8], the radiation characteristics of the antenna have been carefully analyzed for the antenna frequency of 600 MHz on the assumption that the plasma channels have a homogeneous plasma conductivity in radial and longitudinal directions of the channels. In addition, now, the apex angle 60°, radius, and conductivity of the plasma channel 10 mm and 100 S/m, respectively, and the leg length in third resonance 615 mm with which the antenna gives a moderate gain and directivity are taken as the reference values and the radiation characteristics of the antenna are compared to some cases of the radially inhomogeneous conductivity distributions with

Fig. 2
figure 2

Schematic diagram for the simulated Vee antenna (a) and its radiation pattern (b). Color variation from blue to red in (b) represents the increase in gain

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$$\int_{0}^{{{r_{{\text{calc}}}}}} {\sigma \cdot 2\pi r{\text{d}}r} ={\sigma _{{\text{ref}}}} \cdot \pi r_{0}^{2}=31,400\;{\text{S/m}}\;{\text{m}}{{\text{m}}^{\text{2}}},$$

where \({\sigma _{{\text{ref}}}}=100\) S/m and \({r_0}=10\) mm. In Fig. 3, several plasma conductivity distributions for calculation and comparison are shown. Every inhomogeneous distribution has a radially layered structure for simulation. The radial gradient of the first three cases (\({r_{{\text{calc}}}}={r_0}=10\) mm) steepens down in order from Fig. 3a—flat distribution, through Fig. 3b—Gauss distribution with the cut periphery outside radius of 10 mm, to Fig. 3c—Bessel function-shaped distribution. In most cases of calculation, the antenna leg length is slightly varied from the reference value with the difference of average plasma conductivity and its distribution to achieve the antenna resonance (impedance matching procedure), although the detailed variation is not included in the description. The calculations for these three distributions show that their 3D far-field radiation patterns remain very similar, but the total gains are 6.3748 dB, 6.1149 dB, and 5.8148 dB, respectively, lowering down by about 0.3 dB, and their radiation efficiencies are 0.6593, 0.6063, and 0.5556, respectively, decreasing by more than 5%. For the worst case—Bessel distribution, to achieve the radiation characteristics similar to the case of the flat distribution, Fig. 3a, cross-sectional averaged conductivity of 300 S/m is required instead of 100 S/m, the total gain and radiation efficiency being then 6.5042 dB and 0.6791, respectively. This shows that, although the conductivity is sufficiently high in the center of the cross section, the lower the conductivity in the vicinity of the channel boundary, the lower the total gain and radiation efficiency. To confirm this, the calculation has been performed for the distribution \(\sigma \sim {r^2}\) (\(0 \leq r \leq {r_{{\text{calc}}}}={r_0}\)), provided that averaged conductivity is the same, 100 S/m, although it seems to be an almost unrealistic case now. Its total gain and radiation efficiency are 6.7146 dB and 0.7211, respectively, being much better than other decreasing distributions. For the distribution of Fig. 3e, the radiation characteristics corresponding to flat distribution (Fig. 3a) are obtained only with cross-sectional averaged conductivity 70 S/m. It is more desired to modulate the laser beam intensity to have radially increasing or annular distribution, rather than the regular initial modulation [14] if it is possible.

Fig. 3
figure 3

Plasma conductivity distributions used for calculation

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All the above show that the outer layer with a certain thickness (related to the signal frequency) is more important for the antenna radiation, and at the same time, point out the error range of the approximation. Although the femtosecond laser-induced plasma channel in the raw can have a more complex radial conductivity distribution rather than Gaussian or Bessel distribution, the calculation for the flat distribution can give an aroused radiation characteristics corresponding to the inhomogeneous distribution with more than double reference average conductivity − 200 S/m.

The (full) Gaussian conductivity distribution (Fig. 3d) deserves special emphasis for the approximate calculation of antenna. Unlike the distributions above, it has more than one-third of plasma particles beyond its reference effective radius \({r_0}=10\) mm. If the above calculations are for the various distributions within a given effective radius, the Gaussian distribution is for a plasma diffused to, actually, a limitless radius. In general, for a given plasma conductivity, increasing the antenna radius results in an improvement in the radiation characteristics [8]. If, however, the radius increase is accompanied by a gradual decrease of conductivity to null, the outcome might not be affirmative. The simulation of the Gaussian distribution could demonstrate a simultaneous effect of a sufficient decrease of the boundary plasma conductivity related to the spatial plasma diffusion and a sufficient increase of the calculation radius for a given effective radius.

Figure 4 shows the dependence of the total gain and radiation efficiency on the calculation (cut) radius \({r_{{\text{calc}}}}\) for the Gaussian distribution. Along with the increase of the calculation radius, the modeled distribution approaches to the full Gaussian. As the calculation radius increases, the radiation characteristics become worse, and if we take the calculation radius 21.5 mm within which more than 99% of plasma medium is included, the relevant distribution could be said to be close enough to full Gaussian (\({r_{{\text{calc}}}}=\infty\)). Its total gain and radiation efficiency are 5.2622 dB and 0.4115, respectively. To get radiation characteristics similar to that of the referenced flat distribution, the effective cross-sectional averaged conductivity must be raised by a factor of 8, i.e., the integral of conductivity and calculation cross-sectional area product must be \(\int_{0}^{{21.5{\text{mm}}}} {\sigma \cdot 2\pi r{\text{d}}r} =8 \cdot {\sigma _{{\text{ref}}}} \cdot \pi r_{0}^{2}=8 \times 31400\;{\text{S/m}}\;{\text{m}}{{\text{m}}^{\text{2}}}.\) Then, they are 6.7865 dB and 0.6397, respectively. This shows that the calculation results for the full Gaussian distribution and the relevant referenced flat one with the limited radius 10 mm diverge largely and in reality, we cannot approximate the Gaussian distribution (Fig. 3d) reasonably by the reference flat distribution (Fig. 3a). It is caused by excessively low plasma conductivity in the boundary layer due to an excess of plasma diffusion, compared to the reference flat distribution.

Fig. 4
figure 4

Dependence of the total gain and radiation efficiency on the calculation radius \({r_{{\text{calc}}}}\) of Gaussian distribution

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On the other hand, in general, with the decrease of an antenna channel radius, the antenna gain and radiation efficiency decrease when the channel conductivity remains unchanged [3]. However, in the present calculation model for Gaussian distribution, the decrease of calculation radius is accompanied with the conductivity increase within it which causes the enhancement of antenna performance. As a consequence of these two effects, the radiation characteristics begin to diminish at a sufficiently small antenna radius. However, the calculation below 10 mm in Fig. 4 is meaningless, because there is no need to take the calculation radius smaller than the referenced effective radius of Gaussian distribution 10 mm to include the plasma outside it in consideration.

Stressing the significance of outer plasma layer in an antenna radiation is equivalent to emphasizing the inferiority of the inner layer. The seeming effects of the outer layer on the radiation result from the current flow there by skin effect and the radiation absorption by itself. To see which mechanism is predominant, how thick inner/outer layer contributes to the radiation and how much its does, let us imagine the hollow plasma channel with homogeneous conductivity in the annular cross-sectional plasma tube. The radiation characteristics of the plasma-tube-channel antenna with the fixed outer radius 10 mm as function of different inner radii for several constant conductivities are shown in Figs. 5 and 6. Along with the increase of the inner radius, the tube wall is getting thinner.

Fig. 5
figure 5

Total gain as a function of the inner radius of plasma-tube-channel antenna

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Fig. 6
figure 6

Radiation efficiency as a function of the inner radius of plasma-tube-channel antenna

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As can be seen from the figures, as long as the channel tube wall is not thinner than its skin depth, the radiation characteristics do not deteriorate, highlighting the role of skin effect. As the plasma-tube wall thickness from which the radiation characteristics start to deteriorate coincides with the skin depth for a given conductivity, the same calculation scheme can be used to estimate the skin depth of the plasma with inhomogeneous conductivity in cross section. For instance, the skin depth of the plasma channel with the Gaussian radial conductivity distribution discussed above is estimated to be about 11 mm.

Diffraction of laser beams always occurs, even if they propagate in vacuum, causing its axial or longitudinal inhomogeneity [15, 16]. For the femtosecond laser, the natural linear diffraction is largely suppressed by nonlinear self-focusing. Its intensity is always maintained at the verge or below the value corresponding to ionization threshold via the clamping mechanism [15]. The femtosecond laser beam, therefore, remains focused over an extended distance and it can be deduced that the axial inhomogeneity is not serious compared to the radial inhomogeneity.

Let us consider the case of conductivity linearly decreasing in the axial direction of the plasma channel and constant across its cross sections. For an approximate calculation, we divide the leg length of the reference antenna into ten equal parts having the gradually reduced conductivities down to zero at its ends, so that the average conductivity over the whole leg length could be the reference conductivity 100 S/m. Then, the total gain and the radiation efficiency are obtained to be 6.2563 dB and 0.6028, respectively. To achieve the radiation characteristics similar to or better than the case of axially homogeneous laser propagation (with no consideration of axial inhomogeneity), the plasma-channel length-averaged conductivity of 143 S/m is required instead of 100 S/m, the total gain and radiation efficiency being then 6.5912 dB and 0.6597, respectively. More realistically, the more steeply (nonlinearly) decreasing conductivity, \(\sigma \sim {(l - x)^{1/\alpha }}\) (where \(x\) is axial coordinate, \(l\) is the antenna leg length and \(\alpha\) is an integer more than two), gives the radiation characteristics very close to that for the constant conductivity distribution. For instance, \(\alpha =2\) and the average conductivity 100 S/m give the total gain 6.4313 dB and the radiation efficiency 0.6397. In addition, the average conductivity of only 116 S/m is needed to achieve the radiation characteristics close or superior to the case of axially constant conductivity 100 S/m, giving the total gain 6.6658 dB and radiation efficiency 0.6598. That is, the consideration of axial plasma inhomogeneity only results in disagreement with the case of no consideration for it by a factor of less than 1.5 in the average conductivity for the similar radiation characteristics. (When considering the radial plasma inhomogeneity, the average conductivity more than 200 S/m is required to obtain the radiation characteristics close to the case of the flat radial distribution of conductivity 100 S/m—see above.) These all confirm the more significant effect of the radial plasma inhomogeneity on the antenna radiation compared to its axial inhomogeneity.

4 Outer plasma layer around metal antenna in fast-flow field and blackout phenomenon

From the above, it is obvious that the inner part of an antenna channel beyond the skin depth does not have an essential effect on its radiation characteristics, whatever the inner conductivity is very high or not, even though the inner is a metal, as the current can hardly flow through it. The outer plasma layer not only can give an important effect on the plasma antenna performance, but also can be formed around the solid metal antenna and produce a complicated phenomenon when it is immersed in a sufficiently fast-flow field. The plasma layer with finite thickness can be generated around a reentry vehicle by high-speed flow and can attenuate the intensity of metal antenna radiation or even interrupt communication (i.e., the blackout phenomenon) [17].

However, another phenomenon, the enhancing of the intensity of antenna radiation when parameters of the plasma layer are chosen properly, has been demonstrated in numerical and experimental studies. This phenomenon suggests a way of overcoming the blackout problem [18, 19].

Now, for the solid copper Vee antenna with the same geometry and excitation as the above reference antenna, the effect of different plasma cover configurations and plasma parameters on the radiation characteristics is considered when it is immersed in a high-speed flow field or it travels at a hypersonic speed as a part of vehicle, similar to reentry condition. In general, an antenna consists of two poles (legs) and is excited by the driving signal launched between them. The linear leg part of the Vee antenna is long and the exciting part is relatively short. The calculation shows that if the plasma layer directly covers the whole antenna, including the linear and exciting parts, even 1 mm thick plasma layer of 100 S/m can attenuate the radiation efficiency to less than 0.1. The direct plasma layer covering not only steeply reduces the radiation efficiency, but also largely changes the far-field radiation pattern, indicating the big change in antenna structure. It is due to nothing but the short circuit of the two antenna poles by the conductive plasma. Apparently, the antenna short circuit can be a cause of the blackout phenomenon.

The total radiated power \({P_{{\text{rad}}}}\) of an antenna is related to the total input power \({P_{{\text{in}}}}\) by \({P_{{\text{rad}}}}=\varepsilon {P_{{\text{in}}}}\), where \(\varepsilon\) is the antenna radiation efficiency [1]. Hence, the antenna radiation power depends only on the radiation efficiency in the case of constant input power, and the less the radiation efficiency, the lower the radiation power and the more severe the signal attenuation. In addition, the point of the lower radiation efficiency can be thought to represent a more conceivable communication disruption, although the communication characteristics depend on many factors, including the input power of the transmitting antenna and the performance of the receiving antenna, as well.

Figure 7 shows the dependence of the radiation efficiency of the copper Vee antenna on conductivity and thickness of the outer plasma layer for the direct plasma covering only on the antenna legs, assuming the avoidance of the short circuit. For the three conductivities, with the increase of plasma layer thickness, the radiation efficiency first decreases and then rises again to some extent. (The far-field radiation pattern remains almost unchanged there.) It results from the combined effect of the signal wave absorption/attenuation and skin effect, as we can see below. The refractive index and attenuation are expressed, respectively, as follows [11]:

Fig. 7
figure 7

Dependence of the radiation efficiency on the plasma layer conductivity and thickness of the short circuit-avoided copper Vee antenna

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$$n=\sqrt {\frac{{{\varepsilon _{\text{r}}}+\sqrt {\varepsilon _{{\text{r}}}^{2}+{{({\sigma \mathord{\left/ {\vphantom {\sigma {{\varepsilon _0}\omega }}} \right. \kern-0pt} {{\varepsilon _0}\omega }})}^2}} }}{2}} ,\;\chi =\sqrt {\frac{{ - {\varepsilon _{\text{r}}}+\sqrt {\varepsilon _{{\text{r}}}^{2}+{{({\sigma \mathord{\left/ {\vphantom {\sigma {{\varepsilon _0}\omega }}} \right. \kern-0pt} {{\varepsilon _0}\omega }})}^2}} }}{2}} .$$
(4)

The numbers \(n\) and \(\chi\) determine the relation between the amplitudes of the field and the phase shift between them: \(H=(n+i\chi )E=\sqrt {{n^2}+{\chi ^2}} {{\text{e}}^{{\text{i}}\psi }}E\), \(\psi =\arctan (\chi /n)\). As only the value of the energy flux density averaged over one period is of practical importance, let us estimate the signal wave absorption related to the wave energy flux in the plasma layer. In a homogeneous medium, the energy flux of a wave decays exponentially:

$$\bar {S}=\frac{1}{2}{\left( {\frac{{{\varepsilon _0}}}{{{\mu _0}}}} \right)^{1/2}}\operatorname{Re} (E{H^*})={\left( {\frac{{{\varepsilon _0}}}{{{\mu _0}}}} \right)^{1/2}}\frac{n}{4} \cdot {\left| {E(0)} \right|^2}{{\text{e}}^{ - {\mu _\omega }x}},$$

where \(x\) is the signal wave propagation axis normal to the copper surface, \(E(0)\) is the amplitude at the point \(x=0\), and

$${\mu _\omega }=2{({\varepsilon _0}{\mu _0})^{1/2}}\chi \omega ={({\mu _0}/{\varepsilon _0})^{1/2}}\sigma /n$$
(5)

is absorption coefficient. For the plasma conductivities \(\sigma =\)10 S/m, 100 S/m and 1000 S/m and signal frequency \(f=600\) MHz, the absorption coefficients are 307.55 m−1, 974.02 m−1, and 3080.6 m−1, respectively, and the lengths over which the energy flux decreases by a factor e are 3.252 mm, 1 mm, and 0.325 mm, respectively. In addition, their skin depths are 6.5 mm, 2 mm, and 0.65 mm, respectively. At first, when the plasma layer is very thin, the larger the conductivity, the more serious the absorption, and the more steep the radiation efficiency decreases. On the other hand, when the plasma covers the antenna legs, the current refinds its way on the outer plasma layer out of the inner copper layer. The larger the conductivity, the more the current can flow through the outer plasma layer for a given thickness. Therefore, the larger the conductivity, the thinner the thickness of the plasma layer from which the radiation efficiency start to increase. As there exists the competition between reduction of the radiation efficiency by absorption and its increase by loading alternating current in the plasma layer, the resultant increase begins at a thickness larger than the skin depths. It can also be seen that when the plasma layer directly covers the antenna structure but the exciting part, at a certain plasma thickness, the antenna could maintain its operation, unless the plasma conductivity is very low (~ 10 S/m), although not so good as no plasma cover case.

In general, the electron density and conductivity distribution are very nonuniform and can vary several orders of magnitude along a normal from the vehicle body (antenna surface), first steeply and then flatly, and hence, roughly, there exists a plasma sheath (with very large conductivity) between the metal antenna surface and the plasma layer, separating them, by a high-speed flow [19, 20]. The thickness of the plasma layer depends on vehicle shape, velocity, and altitude but as an order of magnitude estimate, e.g., for a blunt body is 5–10% of the body radius at the stagnation point and somewhat larger at positions off the stagnation region. The more slender the body and shallower the angle of attack will generally result in lower ionization rates, a thinner plasma sheath, and a much more benign flowfield for telemetry considerations. However, shallow angles of attack result in all of the stagnation flow passing over the vehicle, and a thinner plasma sheath may indicate a relative increase in electron concentration. Over a wide range of flight parameters, the electron density in the stagnation region can vary from a few cm−3 to 1018 cm−3 corresponding to a conductivity more than 10,000 S/m [21, 22].

Figure 8 shows the dependence of the radiation efficiency on the plasma layer conductivity and thickness for the short circuit-avoided copper Vee antenna in the case of existence of the plasma sheath. The sheath thicknesses are 3 mm for solid line and 1 mm for dashed line, respectively, and the plasma sheath conductivities are taken as high as 10 times higher than that of the attached plasma layer. It can be seen that the existence of the plasma sheath with high conductivity is positively reflected on the antenna radiation, and the larger its thickness, the better the radiation characteristics. In addition, with the increase of the plasma layer thickness, the action of the sheath is getting fainter. In Figs. 7 and 8, the overhigh conductivity of the plasma layer and sheath seems nonrealistic, but it is aimed to show the very high plasma frequency relative to the given antenna frequency, causing a radiation enhancement. It is compatible with the experimental results that the radiation of an antenna can be enhanced if the antenna frequency is much lower than the plasma frequency and the antenna dimension is properly chosen [18].

Fig. 8
figure 8

Dependence of the radiation efficiency on the plasma layer conductivity and thickness for the short circuit-avoided copper Vee antenna with the plasma sheath (plasma sheath thickness: solid line—3 mm, dashed line—1 mm, plasma sheath conductivities are taken as high as ten times higher than that of the attached plasma layer)

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The plasma sheath with a high degree of ionization can be replaced by a boundary layer with a low plasma density, according to the body region of a vehicle (down to its rear) [20, 23]. In Fig. 9, the radiation efficiency variation in the case of existence of non-conducting sheath between the copper surface and plasma layer is shown. With the increase of the plasma layer thickness, the radiation efficiency steeply decays first, and then gradually increases, but the thicker the sheath, the lower the final radiation efficiency achieved. This is because when the plasma layer covers the antenna legs, the current tends to resettle itself on the plasma layer, finding its path there, and, the thicker the non-conducting sheath, the larger its high-frequency resistivity, and therefore, the plasma layer cannot act as a current path, predominantly absorbing the radiation.

Fig. 9
figure 9

Dependence of the radiation efficiency on the plasma (\(\sigma =100\) S/m) layer thickness for several air sheath thickness

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Figures 7, 8, and 9 and calculations show that, for the given reference antenna, when the plasma layer conductivity is about 1000 S/m (or accordingly lowering the antenna frequency compared to the plasma frequency), an enhancement of the antenna radiation can be expected nearly up to the state with no plasma covering by a combination of the plasma layer/sheath parameters. In addition, if some measures to prevent the exciting part covering by the plasma layer are taken, the fatal consequence up to the blackout phenomenon could be avoided, although the signal power drop is inevitable. In other words, if we can make the plasma layers cover each antenna legs separately, maintaining their electric polarity distinction (cover the antenna system not as a whole), generally the antenna signal attenuation might not come up to a communication disruption. In this case, the plasma layer generation would mainly result in an increase of antenna radius, with a decreased conductivity; hence, the antenna radiation pattern might not be distorted seriously, and accordingly, the signal intensity could decrease only to some extent due to the decrease of the outer plasma layer conductivity, unless the plasma layers are too thick, and thus, a serious communication blackout could be prevented and the its threshold heightens. For example, in Fig. 7, the radiation efficiency decreases only to 99% for conductivity 1000 S/m and the plasma layer thickness 40 mm, 85% for conductivity 100 S/m and the plasma layer thickness 42 mm, and 57% for conductivity 10 S/m and the plasma layer thickness 45 mm, respectively. This indicates that when manipulating separated plasma covering for the antenna legs, the blackout threshold increases considerably, compared to whole plasma covering (the above mentioned short circuit case). This is true for Figs. 8 and 9 as well, but, the smaller the plasma layer conductivity, the lower the radiation efficiency. Here, another merit of the femtosecond laser-induced plasma-channel antenna emerges: the plasma layer is not generated around the plasma channel by a high-speed flow.

Finally, to see the effect of plasma layer with negative permittivity on the antenna far-field radiation pattern, the calculation giving Fig. 7 is repeated but with the permittivity \({\varepsilon _{\text{r}}}=1\) for all cases, not according to expression (2). The two results are very similar with divergence less than 1%. It can be explained by (4) and (5). The refractive and attenuation indices as well as absorption coefficient are determined mainly by the conductivity-containing term of (4) due to very littleness of \({\varepsilon _0}\) in its denominator, and is less dependent on \({\varepsilon _{\text{r}}}\). In addition, for the skin effect, the current flowing through the outer plasma layer is determined by conductivity (2), irrespective of the magnitude or sign of plasma permittivity.

5 Conclusion

We have investigated the effect of the radial inhomogeneity of conductivity and dielectric permittivity on the far-field radiation characteristics, i.e., the total gain and radiation efficiency of the femtosecond laser-induced plasma-channel Vee antenna and compared to that of the radially homogeneous conductivity distribution, frequently used as a rough approximation. The radiation characteristics of the plasma-channel antenna are affected mainly by the outer boundary plasma layer due to skin effect and are fairly different from that of the approximated flat distribution. To obtain the radiation characteristics corresponding to the flat distribution of conductivity 100 S/m and effective radius 10 mm, the effective cross-sectional averaged conductivity 300 S/m for the Bessel distribution, and 800 S/m for the Gaussian distribution are needed, and it is shown that the modulation of a laser beam intensity to have radially increasing or annular distribution is preferable. In addition, using the fact that the outer current-carrying layer makes a main contribution to the radiation, the skin depth of a plasma with radially inhomogeneous conductivity distribution, like the Gaussian distribution, can be estimated.

The boundary plasma layer around a metal antenna generated by high-speed flow can attenuate the intensity of antenna radiation and even interrupt communication. The simultaneous action of the radiation absorption and current flow in the plasma layer, benefiting to the radiation, can vary the antenna radiation characteristics in a complicated manner. The antenna short circuit by the plasma layer can bring about a fatal consequence in the radiation and communication. The higher the conductivity of a plasma sheath, separating the antenna metal and the plasma layer, the better its action can be reflected on the antenna radiation, provided that the short circuit is avoided. In addition, it is another merit of the femtosecond laser-induced plasma-channel antenna that the plasma layer is not generated around the plasma channel by a high-speed flow.

The effect of a plasma layer with negative permittivity on the antenna radiation characteristics is considered based on the role of plasma permittivity in the radiation absorption coefficient, and its insignificance being examined in view of the skin effect and antenna radiation characteristics.