1 Introduction

In 1899, A. Sommerfeld [1] published a paper about wave propagation along cylindrical wire of finite conductivity. The type of wave he investigated is a non-radiating mode, in which the attenuation is theoretically much smaller than that of waves in coaxial cables. Sommerfeld surface wave or surface electromagnetic waves (SEMWs) can be excited at the interface between two or more media. These two-dimensional electromagnetic waves damp exponentially along the third coordinate. They are of practical interest because their energy decreases in inverse proportion to the distance from the point-like source, while the energy of the bulk electromagnetic waves (BEMWs) decreases in inverse proportion to the squared distance to the source [2]. Sommerfeld’s wave can exist at the interface between two media only if the permittivity of one of them is negative or has a non zero imaginary part. In the former case, these waves are known as Fano waves and their phase velocity is less than the speed of light; in the latter case, they are called Zenneck waves and their phase velocity exceeds the speed of light [3]. If the conductivity of the conductor increases without limit, the extension of the field increases in such a manner that the power carried by the wave of finite amplitude would become infinite. Single metal wires have been theoretically and experimentally studied for guided propagation of electromagnetic waves from tens of meter to millimeter waves [4, 5].

There may have been some doubt about the possibility of generating this wave type, as there is in the case of Zenneck’s ground wave in wave propagation along the earth. The latter wave is guided by a plane interface separating a non-conducting medium from a conducting one, while in Sommerfeld’s case the interface is cylindrical. Both waves are possible solutions of Maxwell’s equations satisfying given boundary conditions. However, the solutions are special ones insofar as they refer to plane electro-magnetic waves and thus provide that the power sources are infinitely far removed. There has been much discussion about the physical reality of Zenneck’s ground wave and there is no convincing answer as yet. However, recent experiments [6] on wave propagation along cylindrical conductors prove that cylindrical surface waves can be generated with high efficiency, and thus they can be employed for guided power transmission without special conditioning and without reducing the wave velocity, while still using launchers of practical size. Of particular practical value, common un-insulated single or multi-strand overhead power conductors may be used to support very low attenuation propagation over the entire frequency range from below 50 MHz to above 20 GHz while employing a launch device of only 15–20 cm in diameter [7].

Such cylindrical wire of finite conductivity became easy to obtain after the discovery of the laser filamentation [8]. When an ultra-short laser pulse propagates in gases at atmospheric pressure, a trace in the form of a thin plasma filament with the diameter d pl ≈ 50–150 μm [9], electron density N e ≈ 1015–1017 cm−3 [10], and a length up to several hundred meters can be formed [11, 12]. The generated plasma through filamentation is cold plasma, and thus it can be suspended in free space to serve as a waveguide [13]. Laser plasma filaments can propagate through turbulent regions [14], low atmospheric pressure [15], a dense cloud [16, 17], or rain [16] and the filament is robust against obstacles [18]. Based on all these properties, one can say that the laser plasma filaments channel is an efficient robust transmission line.

Propagation of SEMWs along a slab of laser-induced plasma channels produced by ultrashort pulse lasers have been studied in [19]. Surface waves have also been studied in the context of discharge plasmas [20]. The single line of plasma filaments used to channel microwave radiation considerably alleviates requirements to the power of femtosecond laser pulses compared to that needed to form a circular waveguide formed by a considerable number of plasma filaments [21, 22].

In this paper, a cylindrical wire contains a bundle of laser plasma filaments used as a single transmission line carrying a Sommerfeld surface wave is considered. A proposed model and numerical analysis to calculate attenuation and phase velocity of the propagated wave in the microwave band for different configurations are introduced. Taking into consideration the plasma lifetime, the possibility for such a transmission line to carry electromagnetic surface waves and factors affecting the attaching distance are discussed. The attenuation of the propagating wave, its dependence on the geometry of the plasma wire, the operating frequency and the plasma effective electron density are investigated, where a thorough comparison between the propagation along a single wire of plasma filaments and free space is performed.

2 Single wire of plasma filaments

On a cylindrical conductor (wire), electromagnetic waves propagate as weakly guided surface waves, due to the finite conductivity of the metal. This type of guided propagation does not occur with a perfect conductor [23]. Since the plasma filament has a finite conductivity, it can be used as a guided structure to support this kind of surface waves. The principle mode of such a structure has a remarkable low loss and low group velocity dispersion, while all other modes vanish almost immediately upon excitation due to their high attenuation [24]. The propagation distance of the plasma channel mainly depends on the ratio of the input peak power over the critical power for self-focusing engaged in the input laser beam [25]. The instantaneous length of the plasma channel and, consequently, the length of the guiding line are determined by the time of electron–ion recombination (τ pl = 3–10 ns) [26]. Thus, to attach to the plasma wire, the microwave pulses should travel close to the speed of light. Also, the wavelength λ mw of microwave radiation is substantially less than the filament length; the propagation of a microwave pulse along a finite plasma wire is similar to its propagation along an infinitely long transmission line.

Sommerfeld showed theoretically that a cylindrical wire, of large but finite conductivity, immersed in a dielectric, could be used as a waveguide structure. Sommerfeld’s considerations are restricted to conductor wire dimensions such that the wire diameter is much bigger than the electromagnetic field penetration depth in the conductor, which is determined by the skin depth δ. The skin depth is inversely proportional to both the plasma conductivity and the microwave frequency. Skin depth imposes certain requirements on the structure of the plasma channel. In order to obtain a low-loss line, it is necessary to form an extended plasma beam with a diameter substantially exceeding the skin depth (a pl ≫ δ) and a sufficiently high concentration of free electrons (higher conductivity):

$$ a_{\text{pl}} \gg \delta = (2/\omega_{\text{mw}} \mu \sigma_{\text{pl}} )^{1/2} $$
(1)

where ω mw is the angular frequency of the microwave radiation, μ is the permeability and σ pl is the plasma conductivity defined as [27]:

$$ \sigma_{\text{pl}} = \omega_{\text{pl}}^{2} v_{\text{c}} \varepsilon_{0} /(\omega_{\text{mw}}^{2} + v_{\text{c}}^{2} ) $$
(2)

where v c = N a v e σ c is the effective rate of elastic collisions of the electrons and neutral particles of the medium, N a is the concentration of neutral particles, v e is the mean electron velocity, σ c is the effective collision cross-section for electrons and neutral particles, and ω pl is the plasma frequency defined as [28]:

$$ \omega_{\text{pl}} = \left( {N_{\text{e}} e^{2} /m_{\text{e}} \varepsilon_{0} } \right)^{1/2} $$
(3)

where N e is the electron density in the plasma filament, e and m e are the charge and mass of the electron, respectively. For plasma filaments in the atmosphere, where the air is weakly ionized, the collision frequency is about 1012 s−1 [29]. Both plasma frequency and collision frequency determine the conductivity of the plasma channel [28]. The values of conductivity and skin depth of plasma channels with different values of electron concentrations and different wavelengths of microwave radiation are listed in Table 1.

Table 1 Conductivity and skin depth of plasma filament for different values of wavelengths of microwave radiation and different concentrations of free electrons
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For an electron concentration of 1015 cm−3, the skin depth δ for microwaves in laser-induced filaments is about 1 mm (Table 1). Thus to meet the requirements of Sommerfeld, the wire diameter must be no less than several millimeters. Such wire diameter is much bigger than the conventional diameter of a single filament (≈100 μm). The plasma wire will be substantially smaller than skin depth if the wire is consisted of single filament. In order to increase the wire thickness to be suitable for carrying Sommerfeld surface wave, a wire formed by a set of closely spaced plasma filaments is required.

To form such plasma wire, one consider a femtosecond laser pulse with regular initial modulation of its intensity cross-section, each perturbation has a Gaussian profile and spacing by a distance b pl from the center of each others as shown in Fig. 1a. This initial intensity modulation will initiate the origin of the filaments in the plane of the pulse cross-section despite of random fluctuation which achieve the stabilization of the parameters of the plasma channel [30]. Formation of such a wire with a length of several tens of meters requires laser radiation with a peak power of about 1 TW, the total pulse energy E p = 100 mJ and the initial pulse duration τ0 = 100 fs [31]. During propagation of the laser pulse, each perturbation causes formation of a plasma filament (Fig. 1b). The effective concentration of free electrons in this plasma wire \( \tilde{N}_{\text{e}} \), as proposed by Valuev et al. [29], will depend on the spacing b pl between the centers of adjacent plasma filaments, the electron concentration of each filament N e and the filament diameter d pl [29, 32]. For example, if the filament has 1017 cm−3 electron density, 150 μm diameter, and the spacing between the centers of the filaments is about 1.5 mm [33], the effective (average) electron density will be about 1015 cm−3. Changing spacing between perturbations in the initial intensity distribution, it is possible to control the mean value of effective electron concentration \( \tilde{N}_{\text{e}} \) in the plasma beam.

Fig. 1
figure 1

a Intensity distribution in the cross section of a femtosecond beam with regular modulation for formation of a single wire of laser plasma filaments and, b distribution of the filaments in the cross section after propagation

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Thus, the bunch of laser filaments will present a guiding wire for microwave radiation if the distance b pl is much smaller than the wavelength λ mw. In this case the microwave radiation will see an over-dense plasma wire (ω mw < ω pl) has a diameter bigger than the field penetration depth and can guided by attaching to the interface between the plasma and air surrounding it.

3 The proposed mathematical model

An infinitely long cylindrical plasma wire of circular cross section is created in an infinite homogenous dielectric (air). The particular solution describes a radially symmetric transverse magnetic wave travelling along the wire (TM mode). There are also solutions describing higher order modes around the cylinder [34]. Because most of the energy of these unsymmetrical modes is inside the wire their attenuation is extremely large. Thus, they vanish almost immediately upon excitation due to their high attenuation.

For our proposed model, if the coordinate system shown in Fig. 2 is used, and applying the boundary conditions that the field components E z and \( H_{\phi } \) are continuous at the surface of the plasma wire (r = a pl), one can obtain the Eigen-value equation [35]:

$$ \frac{{\gamma_{0} H_{0}^{(1)} (\gamma_{0} a_{\text{pl}} )}}{{k_{0}^{2} H_{1}^{(1)} (\gamma_{0} a_{\text{pl}} )}} = \frac{{\gamma_{\text{pl}} J_{0} (\gamma_{\text{pl}} a_{\text{pl}} )}}{{k_{\text{pl}}^{2} J_{1} (\gamma_{\text{pl}} a_{\text{pl}} )}} $$
(4)

Where J 0, J 1 are the Bessel functions of the first kind, \( H_{0}^{(1)} ,\,H_{1}^{(1)} \) are the Hankel functions of the first kind. The parameter γ 0 and γ pl are defined by:

$$ \gamma_{0}^{2} = k_{0}^{2} - h^{2} ;\quad \gamma_{\text{pl}}^{2} = k_{\text{pl}}^{2} - h^{2} $$
(5)

where h is the propagation constant of the guided wave for both parts of the field. The wave-number k defined as [23]:

$$ k = \left\{ \begin{aligned} k_{\text{pl}} & = (\omega_{\text{mw}} \mu \sigma_{\text{pl}} )^{1/2} \exp ( - j\pi /4) \\ k_{0} & = \omega_{\text{mw}} (\varepsilon_{0} \mu_{0} )^{1/2} \\ \end{aligned} \right. $$
(6)

Where k pl is the wave-number inside the plasma wire, k 0 is the wave-number outside the plasma wire, and ɛ 0 is the free space permittivity.

Fig. 2
figure 2

Plasma wire in cylindrical coordinate system

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For plasma wires with large radii compared with the skin depth \( (\gamma_{\text{pl}} a_{\text{pl}} \gg \left| {\alpha^{2} - 1/4} \right|) \), where α is the order of Bessel function, J 0 and J 1 can be replaced by their asymptotic representations and the Eigen-value equation can be written as [36]:

$$ \frac{{H_{0}^{(1)} (\gamma_{0} a_{\text{pl}} )}}{{H_{1}^{(1)} (\gamma_{0} a_{\text{pl}} )}} = j\frac{{k_{0}^{2} }}{{\gamma_{0} k_{\text{pl}} }} $$
(7)

If furthermore the plasma wire radius is not too large, so that \( [0 < \gamma_{0} a_{\text{pl}} < (\alpha^{2} + 1)^{1/2} ] \), we can use in (7) the approximations:

$$ H_{0}^{(1)} (x) \approx \frac{2j}{\pi }\ln [ - j\frac{x}{2}\exp (\gamma )];\quad H_{1}^{(1)} (x) \approx - \frac{2j}{\pi x} $$
(8)

where exp(γ) = 1.781. Equation (8) is then simplified to:

$$ \gamma_{0} a_{\text{pl}} \ln ( - j0.89\gamma_{0} a_{\text{pl}} ) = j\frac{{k_{0}^{2} }}{{\gamma_{0} k_{\text{pl}} }} $$
(9)

Now, Eq. (9) can be written in the form:

$$ \zeta \ln \zeta = \eta $$
(10)

with:

$$ \zeta = ( - j0.89\gamma_{0} a_{\text{pl}} )^{2} $$
(11)

and:

$$ \eta = 2(0.89)^{2} \frac{{k_{0}^{2} a_{\text{pl}} }}{{\left| {k_{\text{pl}} } \right|}}\exp (j3\pi /4) = \left| \eta \right|\exp (j3\pi /4) $$
(12)

Thus, for plasma wire in air:

$$ \left| \eta \right| = 1.04 \times 10^{ - 8} a_{\text{pl}} f^{3/2} N_{\text{e}}^{ - 1/2} $$
(13)

where f is the frequency and N e is the electron density. a pl and N e are measured in cm and cm−3 respectively.

Equation (10) can be split into two real equations which are more convenient for numerical evolution. The absolute value ζ and the argument α of \( \zeta = \left| \zeta \right|\exp (j\alpha ) \) can be as:

$$ \left| \zeta \right|\ln \left| \zeta \right| = - \left| \eta \right|;\quad \alpha = - \frac{\pi }{4}\left(1 - \frac{1}{\ln \left| \zeta \right| + 1}\right) $$
(14)

The propagation constant consists of a real part \( h\prime \), which determines the phase velocity, and an imaginary part \( h^{\prime\prime} \), which describes the attenuation:

$$ h = h^{\prime} - jh^{\prime\prime} $$
(15)

This propagation constant of the surface wave is determined by k 0 and γ 0 (Eq. 7). As γ 0/k 0 is very small we can write \( h = k_{0} - \gamma_{0}^{2} /2k_{0} \) or, if \( \gamma_{0}^{2} \) is expressed in terms of ζ with Eq. (12):

$$ h = \left( {k_{0} + 0.63\frac{\left| \zeta \right|\cos (\alpha )}{{k_{0} a_{\text{pl}}^{2} }}} \right) + j\left( {0.63\frac{\left| \zeta \right|\sin (\alpha )}{{k_{0} a_{\text{pl}}^{2} }}} \right) $$
(16)

Comparing Eq. (16) with Eq. (15), we obtain the attenuation as:

$$ {\text{Attenuation }}({\text{dB/m}}) = - 5.47\frac{\left| \zeta \right|\sin (\alpha )}{{k_{0} a_{\text{pl}}^{2} }} $$
(17)

where k 0 is in m −1 and a pl in m. The real part of h yields the phase velocity of the wave:

$$ V_{\text{ph}} (m/s) = \frac{{\omega_{\text{mw}} }}{{h^{\prime } }} = c\left( {1 - 0.63\frac{\left| \zeta \right|\cos (\alpha )}{{k_{0} a_{\text{pl}}^{2} }}} \right) $$
(18)

And the attaching distance d attch, which represents the co-operating distance between the plasma filament and microwaves, can be calculated as:

$$ d_{\text{attch}} (m) \approx \frac{{4.76k_{0} a_{\text{pl}}^{2} }}{\left| \zeta \right|\cos (\alpha )} $$
(19)

Equation (19) has been deduced by imposing that the speed of laser pulses in air is equal to the speed of light in vacuum c, and that the time of electron–ion recombination (plasma life-time) is about 10 ns [37].

4 Results and discussion

Equation (14) was solved numerically within a wide range of operating frequency f, wire radius a pl and electron density N e. We have considered frequencies from 0.1 to 10 GHz, radii from 0.1 to 1 cm, and electron density from 1015 to 1017cm−3. Then, in case of a plasma wire, |η| ranges between 4 × 10−6 to 0.4. |ζ| and α are plotted in Fig. 3a and b as functions of |η|.

Fig. 3
figure 3

a Graph for determining the absolute value of |ξ|, b graph for determining the argument of |ξ| (see Eq. 14)

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Consider, for instance, a wave with a frequency of 5 GHz, propagating on a plasma wire of an effective electron density of 1015 cm−3 and 5 mm radius (>δ ≈ 1 mm) created in air, from Eq. (13) we obtain |η| = 0.0591 and from Fig. 3 the corresponding |ζ| = 1.2 × 10−2 and α = − 58°22′. Inserting in Eq. (19) we find that the reduction in phase velocity is less than 1.6 %. Thus, the microwave pulse can keep attaching to the plasma filament, and the plasma recombination at the wire trailing end does not affect propagation of microwave radiation to about 200 m. This distance is on the order of the typical propagation distance of ultra-short pulse laser plasma channels in air [38].

For a single transmission line based on Sommerfeld’s surface wave is favorable with regard to the losses effected by the conductor itself, the practical application is restricted to very high frequencies. Below 3 GHz, the size of low loss launching devices and the clearance area around the wire becomes too large [39]. Though, we consider frequencies from 3 to 30 GHz, radii from 0.1 to 1 cm, and an effective electron density from 5 × 1014 to 1016 cm−3.

Figure 4 shows the dependence of the attaching distance d attch on the wire effective electron density \( \tilde{N}_{e} \), wire radius a pl, and operating frequency f. As seen in the figure, by increasing the effective electron density, which is a direct consequence of the increased conductivity of the plasma channels, the reduction of the phase velocity of microwave radiation will decrease, and the co-operated attaching distance between the plasma filament and the microwaves will be longer. For example, the reduction of phase velocity reaches about 0.003 % in the case of a copper wire (N e = 1022 cm−3). Figure 4 shows that the larger the plasma wire diameter, the longer the attaching distance, which is clearly expected from Eq. (19). Also by increasing the operating frequency, the attaching distance will increase. In fact, increasing the wire effective electron density makes the skin depth lower (Table 1) and the wire diameter necessary to satisfy Sommerfeld requirements (i.e., a pl ≫ δ) becomes smaller; that means not much difference in the attaching distance d attch can be achieved. Moreover, increasing the operating frequency reduces the skin depth, hence making the required plasma wire diameter smaller and a constant attaching distance may be maintained.

Fig. 4
figure 4

Attaching distance as a function of effective electron density and plasma wire radius for a f = 3 GHz, and b f = 10 GHz

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Figure 5 represents the attenuation loss (dB/m) with respect to the operating frequency for various plasma wire diameters, using two different effective electron densities. Where it’s noticed from Fig. 5 and Eq. (17) that: the bigger the diameter of the plasma wire, the lower the loss of the microwave radiation. This can be explained by the loss caused by the plasma surface resistance (AC resistance), which is inversely proportional to the wire diameter [40]:

$$ R_{\text{ac}} \propto 1/(a_{\text{pl}} \sigma_{\text{pl}} \delta ) $$
(20)

Comparing Fig. 5a and b, one can find that: the higher the effective electron density of the wire, the lower the loss of the microwave radiation. Higher electron density translates to higher number of electrons per volume, causing an increase in the conduction of electric field through the plasma wire; higher conductivity leads to lower wire resistance and consequently lower loss.

Fig. 5
figure 5

Attenuation in (dB/m) of single plasma wire of 0.5, 1, 2 and 4 cm radius at different effective electron densities a \( \tilde{N}_{\text{e}} = 5 \times 10^{14} {\text{cm}}^{ - 3} \) b \( \tilde{N}_{\text{e}} = 5 \times 10^{15} {\text{cm}}^{ - 3} \)

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Also Fig. 5 shows (as the general trend) that the higher the frequency, the higher the loss of microwave radiation propagating along the plasma wire. This can also be explained through Eqs. (17) and (20), as the frequency increases the skin depth δ and plasma conductivity σ pl will decrease leading to higher plasma surface resistance and higher attenuation of the microwave radiation. Based upon the results shown in both Figs. 4 and 5, one could study parameters that are feasible in the laboratory and provide optimal attaching distances, in addition to obtain a lowest propagation loss.

To demonstrate the advantage of surface waves propagating along a single wire of plasma filaments, a comparison is performed between the power of a microwave radiation transmitted as a Sommerfeld surface wave on a single wire and a microwave radiation freely propagating through the atmosphere as functions of the propagation length. Let’s Introduce the parameter \( \xi ({\text{dB}}) = P_{\text{WG}} - P_{\text{FS}} \) of the powers \( P_{\text{WG}} \) and \( P_{\text{FS}} \) which represent the difference in decibel between the power transmitted through the guiding line and that propagate in free space, respectively [22, 41]. Thus, the guiding microwave radiation was normalized by subtracting out the free space attenuation of the radiation to examine the improvement of the plasma wire over free space propagation. Attenuation in (dB/m) and attenuation length in (m) of microwave radiation propagating along single wire of plasma filaments as a surface wave for different configurations is summarized in Table 2.

Table 2 Attenuation (Att.) in dB/m and attenuation length (AL) in m of microwave radiation propagating along a laser plasma filament for different possible configurations
Full size table

As shown in Fig. 6, the parameter ξ displays a clearly pronounced maximum as a function of the propagation length. The maximum difference in the transmission of a microwave radiation for these guiding line ranges from 43 dB (2.35 dB/m) to 49 dB (1.39 dB/m), with the propagation lengths providing the maximum difference of the transmitted microwave radiation ranging from 4 to7 m.

Fig. 6
figure 6

Transmission improvement factor ξ (dB) as a function of the propagation distance for single plasma wire with \( \lambda_{\text{mw}} = 10\,{\text{cm}},\;\tilde{N}_{\text{e}} = 5 \times 10^{15} {\text{cm}}^{ - 3} \) for a pl = 1 cm (dash line) and a pl = 2 cm (solid line)

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The analysis also shows that, in spite of the attenuation of microwave radiation induced by plasma absorption, propagation along a single wire of plasma filaments can provide a considerable enhancement of microwave radiation transmission relative to freely propagating microwave radiation over some limited distances (31, 54 m), these distances is in the order of the typical propagation distance of ultra-short pulse laser plasma channels in air [32]. Beyond these ranges of distances, the attenuation related with the free space propagation is lower than that for the single wire of the plasma filaments.

5 Conclusion

A single wire of laser plasma filaments, formed by the propagation of femtosecond laser pulses in air, can work as a single wire transmission line to guide the electromagnetic radiation in the form of Sommerfeld surface waves. A proposed model for electromagnetic waves propagating along this virtual wire of plasma filaments was introduced. The attenuation and the phase velocity of the electromagnetic radiation propagating along this single thread filament at different operating conditions were numerically calculated as well as the possible attaching distance that represents the co-operating distance between the plasma wire and the electromagnetic wave. The phase velocity of the propagating wave proved to be close to the speed of laser pulses, which makes attaching to such instantaneous plasma channel feasible over distances in the order of the filament length. The attaching distance and the loss of the microwave radiation that propagates along this single wire were found to be dependent on the effective electron density, operating frequency and the wire diameter. Energy loss of the propagating wave along the plasma wire appears to be lower than that of free space over some distances. For this reason, a single wire of plasma filaments may be a potential channel for short distance point to point wireless transmission of pulsed-modulated microwaves.