Guiding microwave radiation using laser-induced filaments: the hollow conducting waveguide

concept

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IOP PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 45 (2012) 265401 (9pp) doi:10.1088/0022-3727/45/26/265401

Guiding microwave radiation usinglaser-induced filaments: the hollowconducting waveguide conceptMostafa Alshershby, Zuoqiang Hao and Jingquan Lin

School of Science, Changchun University of Science and Technology, Changchun 130022,People’s Republic of China

E-mail: linjingquan@cust.edu.cn

Received 25 February 2012, in final form 13 May 2012Published 13 June 2012Online at stacks.iop.org/JPhysD/45/265401

AbstractA microwave waveguide that consists of a set of laser plasma filaments produced in air by thepropagation of femtosecond laser pulses is investigated according to the hollow conductingwaveguide concept. The conductivity, skin depth of the electromagnetic waves in this plasmawaveguide and the energy required to excite such a waveguide are calculated for differentpossible configurations. A hollow conducting plasma waveguide is shown to support guidedmodes of electromagnetic radiation from millimetre to centimetre wavelength range. Ourcalculations show that, under the concept of conducting waveguide, it is more suitable to usethe TE01 mode rather than TE11 to achieve an extended attenuation length. The attenuationlength of the low-loss mode TE01 is shown to be dependent on the geometry of the plasmawaveguides, the operating frequency and the plasma effective electron density. The effect ofthe plasma wall density spread on TE01 propagation is evaluated. Using the hollow conductingplasma waveguide operating in TE01 mode, an enhancement of microwave transmission overboth free space propagation and dielectric plasma waveguide is obtained.

(Some figures may appear in colour only in the online journal)

1. Introduction

The formation of a long plasma conducting channel becamepossible after the discovery of the filamentation of intenseultra-short laser pulses [1–3]. When a laser pulse propagatesin gases at atmospheric pressure, a trace in the form of athin plasma filament with diameter dpl ≈ 50–120 µm [4],electron density Ne ≈ 1015–1017 cm−3 [5] and a length ofseveral hundred metres is formed. Plasma filaments are usedin terahertz radiation sources [6] for triggering high-voltagedischarges in megavolt switches, controlling lightning in stormclouds [7, 8], and to guide energy in the form of electriccurrent and electromagnetic radiation [9]. The latter is ofgreat interest since it offers a solution to the high naturaldivergence of the EM radiation by confining the radiation andmaintaining a high energy density over long distances equalto the filamentation length [10]. More recently, the plasmawaveguide properties were experimentally demonstrated toform a circular waveguide and two parallel lines [9, 11].

The virtual plasma waveguide consists of a bundle oflaser filaments separated by a short distance and arranged inthe form of a hollow cylinder. Inside this array of filaments,microwave radiation will see an external uniform plasma walland, thus, can be confined and guided by the reflection on theplasma wall and can propagate over long distances avoidingtheir natural divergence. The efficiency of the microwaveradiation confinement depends on the number and distributionof the filaments. There are two main regimes for modellinga set of filaments serving as a circular guiding structure formicrowaves depending on the plasma parameters (electrondensity and collision frequency) and operating wavelength.The first regime is the same as the dielectric waveguidewith the waves confined to the dielectric by total internalreflection at its surface of optically less dense plasma walls.This regime is applied to some specific value of both thefilaments’ electron densities (�1015 cm−3) and the operatingwavelengths (1 mm � λ � 6 mm) to meet the refractive indexrequirement of the cladding layer of the dielectric waveguide

0022-3727/12/265401+09$33.00 1 © 2012 IOP Publishing Ltd Printed in the UK & the USA

J. Phys. D: Appl. Phys. 45 (2012) 265401 M Alshershby et al

[12–16]. An excessive concentration of free electrons oroperating at different wavelengths makes the condition of totalinternal reflection invalid, i.e. the plasma–cladding refractiveindex becomes higher than the air–core refractive index. Thesecond regime is that used in an ordinary hollow conductingwaveguide with the waves confined to the conducting plasmaby being repeatedly reflected between opposite overdenseplasma walls. When the plasma electron density is sufficientlyhigh (>1015 cm−3), the plasma wall refractive index will bemuch higher than the air it surrounds, and the plasma behavesmore like a metal than a dielectric and the second regime willbe present. Since the typical measured electron density in alaser-induced filament in air is between 1015 and 1017 cm−3 [5],in addition to extending the waveguide bandwidth to includemore operating wavelengths, it is more realistic to examine thisvirtual structure from the viewpoint of the hollow conductingwaveguide regime side-by-side to the limited-band dielectricplasma waveguide regime [12–16].

In this paper we examine the propagation of microwaveradiation in such a hollow conducting waveguide consistingof filament array. The conductivity, skin depth of theelectromagnetic waves in these plasma filaments and theenergy required to excite such a waveguide are calculatedfor different possible configurations. The attenuation lengthof the radiation and its dependence on the geometry ofthe plasma waveguides, the operating frequency and theplasma effective electron density are investigated for the low-loss TE01 mode. The effect of the plasma wall densityspread on TE01 propagation is evaluated. Comparisonsof microwave transmission along the conducting plasmawaveguide, dielectric plasma waveguide and the wavepropagation in free space are performed.

2. Plasma filaments as a hollow conductingwaveguide

A general rule is that when an incident microwave upon aplasma surface is such that the frequency of the incidentmicrowave is greater than the plasma frequency, the incidentmicrowave passes through the plasma with or withoutattenuation depending on the relative magnitude of the plasmafrequency and incident frequency. If the incident microwavehas a frequency much less than the plasma frequency, theplasma behaves similarly to a metal [17]. The main ideabehind the proposal of using laser-induced filaments to forma waveguide is based on the property that wave propagationthrough plasma is only possible for frequencies higher thanthe plasma frequency defined by [18]

ωpl = (Nee2/meε0)

1/2 (1)

where Ne is the electron density of the free electron, e andme are the charge and mass of an electron, respectively, andε0 is the vacuum permittivity. Thus, the microwave radiationcannot penetrate the plasma wall [9, 17], and the reflectioncoefficient in this case depends mainly on the ratio betweenthe plasma frequency and the microwave frequency [17]. Atypical measured electron density in a laser-induced filament

in air is between 1015 and 1017 cm−3 [5], resulting in a plasmafrequency fpl = 300–900 GHz. This equation gives the upperlimit of the microwave frequency that can be propagated alongthis virtual waveguide. Thus, a rough estimation for thepropagated microwave wavelength is as follows:

2πc/ωpl = λpl < λmw (2)

where λpl and λmw are the plasma and the microwavewavelengths, respectively. The propagation distance of theplasma channel depends mainly on the ratio of the input peakpower over the critical power for self-focusing engaged in theinput laser beam [19]. The instantaneous length of the plasmachannel and, consequently, the length of the guiding structureare determined by the time of electron–ion recombinationτpl = 3–10 ns and achieve several metres [20]. Since thewavelength λmw of the microwave radiation is much less thanthe filament length, propagation of a microwave pulse alonga finite flying plasma waveguide is similar to the propagationof this pulse in an infinitely long waveguide, and to attach themicrowave pulses to the plasma waveguide, the microwavepulses should travel close to the speed of the laser pulses [12].

The diameter of the plasma waveguide should be largeenough to guide at least one microwave mode, and the gapbetween the filaments on the periphery of the waveguideshould be much less than the wavelength to be transmitted.Moreover, the plasma wall’s thickness must exceed theeffective penetration depth of the microwave field in theplasma, which is determined by the skin depth. The skindepth is inversely proportional to both the plasma conductivityand the microwave frequency. The skin depth imposes certainrequirements on the structure of the plasma waveguide. Inorder to obtain a low-loss waveguide, it is necessary toform a plasma waveguide with a wall thickness substantiallyexceeding the skin depth (hwg � δ) and a sufficiently highconcentration of free electrons (higher conductivity):

hwg > δ = (2/σplωmwµ)1/2 (3)

where ωmw is the angular frequency of microwave radiation, µis the plasma permeability and σpl is the plasma conductivitydefined as [18]

σpl = (ω2plv)/(ω2

mw + v2) (4)

where v is the collision frequency. For plasma filaments inthe atmosphere, where the air is weakly ionized, the collisionfrequency is given by [13, 21]

v = vc + vm (5)

where vc ≈ 2.91 × 10−6(ln �)Ne/T3/2

e is the electron–ionCoulomb collision frequency (in s−1), Te being the electrontemperature (in eV), ln � being the Coulomb logarithm, withNe measured in cm−3, and vm ≈ 3.91 × 10−8NaT

1/2e is the

effective electron neutral momentum transfer frequency (ins−1), where Na is the density of neutral atoms (in cm−3).For a weakly ionized gas at pressure 1 atm, the energyof the free electrons of the plasma is about 1 eV [13, 22],and Na = 2.7 × 1019 cm−3, the collision frequency is

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J. Phys. D: Appl. Phys. 45 (2012) 265401 M Alshershby et al

Table 1. Conductivity and skin depth of plasma filament for different values of wavelengths of microwave radiation and differentconcentrations of free electrons.

λmw = 3 mm λmw = 1 cm λmw = 6 cm

Ne (cm−3) σ (−1 m−1) δ (µm) σ (−1 m−1) δ (µm) σ (−1 m−1) δ (µm)

1015 19.61 359 25.65 574 26.33 1.38 × 103

1016 193.10 115 249.12 184 256.26 4451017 1.90 × 103 36.50 2.42 × 103 59 2.48 × 103 142Copper 1022 5.57 × 107 0.2 5.61 × 107 0.4 5.63 × 107 1

Figure 1. (a) Beam pattern and filament distribution after propagation and (b) schematic diagram of a cylindrical microwave waveguideformed by plasma filaments.

v ≈ 1.06 × 1012 s−1 at 1015 cm−3 and v ≈ 1.13 × 1012 s−1

at 1017 cm−3, respectively. At a gas pressure of 1 atm, agas temperature of 300 K, Ne = 1017 cm−3, Te = 1 eV andNa = 2.7×1019 cm−3, the collision frequencies in equation (5)are estimated as vc ≈ 7.61×1010 s−1 < vm ≈ 1.05×1012 s−1.Both plasma frequency and collision frequency determinethe conductivity of the plasma channel [18]. The values ofconductivity and skin depth of plasma channels with differentvalues of electron concentrations and different wavelengths ofmicrowave radiation are listed in table 1.

For an electron concentration of 1015 cm−3, the skin depthδ is about 1 mm (table 1) and the plasma wall thickness must beno less than several millimetres (hwg � 5δ). The wall thicknesshwg of the plasma waveguide will be substantially smaller thanthe skin depth δ if the plasma wall consists of a single ringof filaments. Obviously, a plasma wall consisting of a singlefilament is not suitable for guiding microwave energy. In orderto increase the thickness hwg of the wall used for microwaveenergy guiding, a plasma wall formed by a set of closely spacedplasma filaments and containing several rings is required, asillustrated in figure 1(a). This virtual plasma waveguide isexperimentally demonstrated in [9].

We assume a plasma waveguide to be an air cylinder ofradius Rwg, bounded by a conducting plasma layer whosethickness hwg significantly exceeds the skin depth δ. Thedensity of the plasma is assumed to be homogeneous in thecross-section as well as along the propagation length; a stepdensity profile for the air–plasma interface is proposed forsimplification.

3. Plasma waveguide modelling

When the plasma conduction current Jc = (σpl +ωmw|ε′′|)E is sufficiently larger than the displacement current

Jd = ωmw|ε′|E, i.e. Jc/Jd � 1, the plasma filaments tend tobehave more like a perfect conductor rather than a dielectric,and as a consequence, we can deal with this configurationas a hollow conducting waveguide rather than as a dielectricwaveguide. The complex refractive index n̄pl of the plasmafilaments is given by [23]

n̄pl = npl + iKpl =(

1 +ω2

pl

iωmwv − ω2mw

)1/2

(6)

where npl is the real part of n̄pl, simply referred to in whatfollows as the refractive index, Kpl = αplλmw/2π is theimaginary part of n̄pl, with αpl and λmw being the absorptioncoefficient and the radiation wavelength, respectively, and ωmw

is the microwave angular frequency. The effective electrondensity of the filaments in the waveguide wall region iscalculated in [12] as

N(e-eff) ≈ PFF · Ne (7)

where PFF is the plasma filling factor, the ratio of the plasmachannel volume to the total wall thickness volume. For thecircular geometry shown in figure 1(a) it is calculated by astraightforward geometry as

PFF = Npld2pl

4hwg(2Rwg + hwg)(8)

where dpl is the filament diameter, Rwg is the waveguideradius and Npl is the total number of plasma filaments in thewall region. The total number of plasma filaments Npl isdistinguished and taken into account in the experimental work[9], and is found to be approximately 1000 distinguishablefilaments chaotically distributed along the entire circumference

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J. Phys. D: Appl. Phys. 45 (2012) 265401 M Alshershby et al

of the ring, separated by an average distance of 585 µm,with the thickness of the wall of filaments varying between1 and 3 mm along the ring. From equations (7) and (8), theeffective electron density can be calculated. Substituting from(7) and (8) into (6), the plasma effective refractive index canbe obtained:

npl =(

1 +N(e-eff)e2/ε0me

jωmwv − ω2mw

)1/2

. (9)

As seen from (9), the calculated plasma refractive index isa function of the plasma filament diameter dpl, the plasmawaveguide core radius Rwg, the wall thickness hwg and themicrowave wavelength λmw.

The plasma wall reflectivity depends on both the plasmarefractive index and the microwave diffraction angle at thewaveguide aperture, thus, the reflection coefficient at the air–plasma interface can be determined as [24]

R =∣∣∣∣cos θi − npl cos θt

cos θi + npl cos θt

∣∣∣∣2

(10)

where θi is the incident angle which is related to the transmittedangle θt through Snell’s law. The boundary conditionsdetermine the dispersion equation given [25]:[

1

u

J ′n(u)

Jn(u)− 1

w

H(1)′n (w)

H(1)n (w)

] [k2

1

u

J ′n(u)

Jn(u)− k2

pl

w

H(1)′n (w)

H(1)n (w)

]

= n2h2

(1

w2− 1

u2

)2

(11)

where u = Rwg(k20 − h2)1/2 and w = Rwg(k

2pl − h2)1/2

are the normalized transverse wavenumbers, h with themode propagation constant, and n refers to the numberof circumferential (φ) variations (mode number), k0 =ωmw(µ0ε0)

1/2 is the wavenumber inside the waveguide, kpl =(ωmwµplσpl)

1/2ejπ/4 is the wavenumber inside the plasma layer,µ and σpl are the permeability and plasma conductivity,respectively. The Bessel functions are transcendental; thus,for each value of n there is a denumerable infinity of roots,any one of which can be denoted by the subscript m. Anyroot of (11) can then be designated by hnm. If the conductivityis finite, such as in a plasma, a superposition of electric andmagnetic types of waves is necessary to satisfy the boundaryconditions, except in the symmetrical case n = 0. Thus, it ismore convenient to study the propagation of the lowest lossmodes TE0m than to study the propagation of the fundamentalmode TE11 (lowest cut-off frequency). For TE0m modes, thedispersion equation (11) becomes

1

u

J1(u)

J0(u)= 1

w

H(1)1 (w)

H(1)0 (w)

. (12)

Equation (12) generally has a number of roots correspondingto various transverse axisymmetric modes. The maximumpropagation distance will be corresponding to the minimumnormalized transverse wavenumber, i.e. to the lowesttransverse mode TE01. An important parameter for each modeis its cut-off wavelength. This wavelength is determined by

the condition w = 0. The value of u when w = 0 determinesthe cut-off wavelength which is referred to as the cut-offnormalized wavenumber uc, where uc is the smallest solutionof J ′

0(uc) or uc ≈ 3.832. Thus, the cut-off wavenumberdetermines the upper boundary of the wavelength of themicrowave radiation:

kcut = 2π/λcut = uc/Rwg (13)

where λcut is the cut-off wavelength. Thus, for geometricparameters of a hollow cylindrical plasma filament waveguideworking in the TE01 mode, the following relations are valid:

λpl < λmw � λcut ≈ 1.64Rwg � Lpl and hwg > δ.

(14)

Equation (12) can be solved numerically for the propagationconstant h = α + jβ, where the real part α describes the lossand the imaginary part β determines the phase velocity.

4. Results and discussion

A rule of thumb is that the plasma frequency should be abouttwice or greater than the microwave operating frequency toconsider that the plasma behaves as an effective metal [26, 27].Since the typical measured electron density in a laser-inducedfilament in air is between 1015 and 1017 cm−3 [5], the resultingplasma frequency fpl = 300–900 GHz. Thus, it is betterto limit the operating frequency of the microwave to be lessthan 100 GHz. According to the results of the experimentalwork [9], we confine our study to an effective electron densityin the range 1014–1016 cm−3 and microwave frequency inthe range 5 –100 GHz. Lower operating frequencies needbigger waveguide radii and thicker plasma walls (bigger skindepths), which need a higher energy femtosecond laser sourceto achieve the required operating conditions (high effectiveelectron density and long filament waveguide).

The calculated plasma reflectivity as a function of botheffective electron density and operating frequency is shownin figure 2. This figure yields the conditions for the requiredeffective electron density and operating frequency to maintainthe plasma reflectivity much higher so as to ensure thatthe plasma tends to behave more like a perfect conductor.Thus, the transmission mechanism of the microwave pulse insuch a waveguide is caused by the higher reflections on theoverdense plasma walls. As shown in figure 2 the reflectivitybecomes higher with the increase in both the effective electrondensity and the operating frequency, leading to more effectivemicrowave guiding.

It is interesting to note that figure 2(b) shows a somewhatunexpected result that the plasma reflectivity increases byincreasing the frequency, in spite of the decrease in both plasmarefractive index and plasma conductivity (higher surfaceresistance). For a waveguide of diameter D, the microwavediffraction angle at its aperture will be φ ≈ sin−1(λmw/D).Thus, the incident angle of the microwave on the plasma wallis θi = π/2 − φ = π/2 − sin−1(λmw/D). As seen fromequation (10), the plasma reflection coefficient is a function ofboth the plasma refractive index and the incident angle which

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J. Phys. D: Appl. Phys. 45 (2012) 265401 M Alshershby et al

Figure 2. Plasma reflectivity as a function of (a) effective electron density and (b) frequency.

Figure 3. Attenuation length of TE01, TE11 and TM01 modes versus wavelength/radius for Ne-eff = 1015 cm−3, Rwg = 5 cm.

is a function of the frequency. When the frequency increases,the microwave incident angle approaches π/2 and the plasmareflectivity reaches a high value regardless of the difference inrefractive index between the two media (plasma and air) and inthis case the mode of operation will resemble the propagationof sliding mode, which has been investigated in [14–16].

Equation (11) was solved numerically for TE11, TM01 andTE01 modes of microwave propagation in the proposed hollowconducting plasma waveguide, taking into account the plasmaconductivity, plasma permittivity and the effective electrondensity.

Figure 3 shows the dependence of attenuation length onthe ratio of radiation wavelength to the waveguide radius(λmw/Rwg) for the above-mentioned three operating modes.While working in the range 0.7 � λmw/Rwg � 2 as thecase of a conventional circular waveguide, the TE11 modewill dominate over the modes TE01 and TM01. The maximumattenuation length (minimum loss) occurs at λmw ≈ 1.1Rwg

[22] and λmw ≈ 1.51Rwg for both TE11 and TM01, respectively(black squares in figure 3) regardless of the value of thewaveguide radius. It should also be noted that, for TE11 andTM01 modes, the attenuation lengths will be decreasing if itis moved away from the optimum condition. The behaviourof TE01 mode is quite different from that of modes TE11 and

TM01. The attenuation length will drastically increase with thedecrease in the ratio λmw/Rwg. As can be seen from the figure,the attenuation length of the TE01 mode will be higher thanthat of TE11 when the ratio λmw/Rwg is less than 0.7, whichmakes TE01 dominating over TE11. It can also be seen fromfigure 3 that, when working at even lower values of λmw/Rwg,the attenuation length of the TE01 mode can increase by abouttwo orders of magnitude over that of TE11. By decreasingλmw/Rwg, the energy propagates not in the plasma layer butessentially within the hollow guiding structure, bouncing atgrazing angle against the plasma wall. Consequently, there isa small energy loss due to refraction, and the microwave energywill propagate to longer distances. Thus, a larger attenuationlength can be achieved via increasing the operating frequencyor more accurately, the ratio of the waveguide radius to theradiation wavelength. Thus, one can control the geometry ofthis plasma waveguide by damping the TE11 mode, letting theTE01 mode survive, to make use of this mode in the propagationto longer distances.

It is important that, as shown in figure 3, operating inthe TE01 mode at a higher frequency or a smaller ratio ofthe microwave wavelength to the waveguide radius breaks thelimitation of the TE11 mode that has a maximum attenuationlength of around 10 m. Moreover, the required plasma wall

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J. Phys. D: Appl. Phys. 45 (2012) 265401 M Alshershby et al

Figure 4. Calculated dependence of the attenuation length ofmicrowaves on the plasma effective density at various waveguideradii Rwg, for the low-loss mode TE01

(λmw = 3 cm, Ne-eff = 1015 cm−3).

Figure 5. Calculated dependence of the attenuation length ofmicrowaves on the wavelength for the low-loss mode TE01

(Ne−eff = 1015 cm−3).

thickness becomes less (small skin depth) which helps theformation of the plasma waveguide for longer distances usingless pulse energy. Thus, we can use laser sources with higherpulse repetition rates achieving higher channel capacity (higherbit rates).

Our argument is that the losses of plasma filament arraywaveguides operating in TE01 can be limited by optimizing theplasma parameters and the plasma filament array geometry.Figure 4 shows the dependence of the attenuation lengthon the effective electron density of the plasma layer. Asexpected, an increase in the electron density brings higherplasma conductivity and reflectivity, which leads to largerattenuation lengths. Figure 4 also shows that an increase inthe waveguide radius leads to higher attenuation lengths for agiven effective electron density. By increasing the waveguide’sradius the microwave diffraction angle decreases, and thenormal component of the wavenumber vanishes, which leadsto the fact that a smaller portion of the wave propagates insidethe plasma and lowers plasma absorption.

Figure 5 shows the wavelength dependences of theattenuation length for guided microwave radiation workingin the low-loss TE01 mode. As shown in this figure,the attenuation length decreases with an increase in theradiation wavelength. As the radiation wavelength becomesshorter, diffraction tends to decrease the fraction of radiation

power guided in the waveguide plasma wall, leading to asubstantial increase in the attenuation length. On the otherhand, decreasing the wavelength (increasing the operatingfrequency) results in higher plasma surface resistance, whichleads to higher plasma absorption. As seen from figure 5,the latter effect is less dominating than the first one. Thus,to optimize this parameter, we must increase the operatingwavelength and at the same time increase the ratio betweenthe plasma waveguide’s radius and wavelength.

Table 2 summarizes the results of the calculations for somepossible geometry of filament array waveguides providingtransmission of microwave radiation over about a hundredmetres and the required pulse energy to excite the filamentarray waveguides. The energy needed to excite such plasmawaveguides can be roughly estimated as follows [28]:

E(J) = EiAplLplN(e-eff) (15)

where Ei, Apl, Lpl and N(e-eff) are, respectively, the ionizationenergy, the area of the plasma layer, the length of the waveguideand the effective density of the electrons in the plasma. Fromfigure 1, the plasma layer area can be calculated as

Apl = π(Rwg + hwg)2 − πR2

wg ≈ 2πRwghwg (16)

where 2πRwghwg � h2wg. For a model of N2–O2 mixture

with an average ionization potential Ei = 14.6 eV [29], andsubstitution of (16) into (15) leads to

E(J m−1) ≈ 1.5 × 10−17RwghwgN(e-eff) (17)

The results of propagation losses calculated in table 2show much better enhancement than those obtained in [22]for TE11 mode and those of an equivalent dielectric plasmawaveguide discussed in [12–16]. As an example for a plasmawaveguide working in the conducting regime for TE01 modeat 3 mm wavelength and 5 cm radius, the attenuation length isabout 100 m, while the equivalent one working in the dielectricregime for the lowest loss mode EH11 has an attenuation lengthof about 20 m [14–16].

To demonstrate the advantage of wave-guiding by anarray of filaments, a comparison is performed betweenthe power of a microwave signal transmitted through theplasma waveguide and a microwave signal freely propagatingthrough the atmosphere as functions of the propagation length.Figure 6 shows the microwave radiation loss for two possibleconfigurations (the last two cases in table 2), with that ofthe freely propagating one as a function of the propagatingdistance. As shown in figure 6, in the case of the plasmawaveguide the power decreases inversely with the distancefrom the point-like source, i.e. a linear attenuation with thepropagating distance, while the free propagating power fallsoff in inverse proportion to the squared distance to the sourceas the free space path loss (FSPL) equation states [30, 31]:

FSPL(dB) = 20 log10(S) + 20 log10(fmw) − 147.55 (18)

where S is the distance from the transmitter (in metres).The calculated results show that, in spite of the attenuation

of microwave radiation induced by plasma absorption, a

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J. Phys. D: Appl. Phys. 45 (2012) 265401 M Alshershby et al

Table 2. Filament array waveguide (WG) (Ne-eff = 1015 cm−3) for transmission of TE01 mode.

RequiredSkin Required pulse energy Attenuation

Frequency depth plasma wall Rwg for exciting Loss length(GHz) (um) thickness (mm) (cm) WG (J m−1) (dB m−1) (m)

5 950 4.75 4 2.90 7.08 0.275 3.65 6.79 0.90

100 269 1.35 4 0.81 0.16 51.645 1 0.08 100

Figure 6. Loss as a function of propagating distances for free spacepropagation, and the plasma filament waveguide for three differentconfigurations.

circular waveguide of plasma filaments can provide aconsiderable enhancement of transmission relative to freelypropagating microwave radiation over a limited distance(815 and 1715 m); these distances depend on the plasmaparameters, waveguide geometry and the initial microwaveradiation energy. By increasing the waveguide radius to highervalues such as 10 cm, a propagation loss of 0.01 dB m−1 willbe obtained and the distance will reach about 15 km insteadof 1715 m, which is much higher than that obtained for theequivalent dielectric plasma waveguide in [12–16].

In the above discussion, a step density profile of the plasmawaveguide was assumed. A real plasma waveguide wall hasa gradual density profile with spatial spread which is causedmainly by the filament profile and the plasma diffusion process.The gradient in the plasma profile can affect and change thecharacteristics of the various modes and the propagation of thewave. We will evaluate the effect of the plasma wall densityspread on the microwave transmission for the lowest loss TE01

mode. We use procedures similar to those adopted in [16],which examine the effect of the plasma wall density spread onthe propagation of TM01 mode, and we extend it to the case ofTE01 mode.

Consider a plasma waveguide with radius Rwg, boundedby a conducting plasma layer whose thickness hwg significantlyexceeds the skin depth δ, and this plasma layer has a dielectricpermittivity εp. At the air–plasma interface, Rwg + d > r >

Rwg, the density profile tends to zero at r → Rwg, and theplasma permittivity approaches the air permittivity, ε(r) → 1,where d represents the size of the density inhomogeneity at theair–plasma interface, which is assumed to be sufficiently small,as shown in figure 7. One can search for the electromagnetic

Figure 7. Plasma waveguide cross-section of the inhomogeneouswall.

fields as a perturbated series with the parameter δε determinedby δε = εpl − ε(r).

For TE01 mode, the Maxwell equations can be written incylindrical coordinates as [24]

1

r

∂

∂r(rEφ) = jωµHz

∂Hz

dr= j

εk20 − h2

ωµEφ

Hr = − h

ωµEφ.

(19)

The solution of the Maxwell equations can be considered as aperturbated series:

Eφ = Eφ0 + Eφ1 + · · ·Hz = Hz0 + Hz1 + · · · (20)

Hr = Hr0 + Hr1 + · · ·where Eφ0, Hz0 and Hr0 are the unperturbated field amplitudesin the plasma wall in the case of step density profile, which canbe defined as [32]

Eφ0 = jωµ

k2AH

(1)0 (k2r), Hr0 = j

h

k2AH

(1)1 (k2r)

and Hz0 = AH(1)0 (k2r) (21)

where k2 = (k2pl − h2)1/2, and A is a constant. For first order

in δε, we get the following equations for the perturbations:

1

r

∂

dr(rEφ1) = jωµHz1

∂Hz1

dr= j

k22

ωµEφ1 − jδε

k20

ωµEφ0.

(22)

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J. Phys. D: Appl. Phys. 45 (2012) 265401 M Alshershby et al

Figure 8. Attenuation length of microwaves versus wavelength fordifferent inhomogeneity sizes d at the plasma–air interface for thelow-loss mode TE01 (Ne-eff = 1015 cm−3, Rwg = 10 cm).

As a result one can find the following first-order corrections tothe electromagnetic field component as:

Hz ≈ A

[H 1

0 (k2r) − k2

µ

(∫δε dr

)H

(1)1 (k2r)

]

Eφ ≈ jωµ

k2A

[H

(1)1 (k2r) +

k2

µ

(∫δε dr

)H

(1)0 (k2r)

] (23)

where the field amplitudes in air r < Rwg are given as [32]

Eφ = jωµ

k1BJ1(k1r), Hr = −j

h

k1BJ1(k1r)

and Hz = BJ0(k1r) (24)

where k1 = (k20 − h2)1/2, and B is a constant. Applying

the boundary condition at the air–plasma interface (r =Rwg), the modified dispersion equation becomes, instead ofequation (12),

1

u

J1(u)

J0(u)= 1

w

[µH(1)1 (w) + k2[(εpl − 1)d]H(1)

0 (w)]

[µH(1)0 (w) − k2[(εpl − 1)d]H(1)

1 (w)](25)

where the right-hand side includes the effect of the plasmadensity spread in the waveguide wall, and the inhomogeneitysize d is defined as

∫δε dr ∼= (εpl − 1)d . Equation (25)

tends to equation (12) when the inhomogeneity size tendsto zero (d = 0). We numerically solved equation (25) forthree different values of the parameter d and calculated theattenuation length as a function of the operating wavelengthin the limit of high electron density (conducting waveguideregime), as shown in figure 8.

As one can see from figure 8, the effect of the plasmadensity spread at the air–plasma interface of the plasmawaveguide wall on the microwave signal propagation distanceis sufficiently small and this effect will decrease when theinhomogeneity size gets smaller, which indicates that theassumption of step density profile is justified. When theinhomogeneity size vanishes (d = 0), the attenuation lengthwill be consistent with the result calculated in the case of stepdensity wall profile calculated with the same parameters as infigure 5 (red line). According to figure 8, the effect of theplasma wall density spread at the air–plasma interface is verysmall and does not affect the dominance of TE01 mode.

5. Conclusion

A virtual microwave waveguide based on the hollowconducting concept, which consists of a set of laser plasmafilaments produced in air by the propagation of femtosecondlaser pulses, is proposed. The conductivity, skin depth ofthe electromagnetic waves in these plasma filaments and theenergy need to excite such a waveguide are calculated fordifferent possible configurations. The microwave transmissionis caused by reflection on the overdense plasma walls.The waveguide can support a microwave radiation in themillimetre to centimetre wavelength range. The dependenceof attenuation length of the TE01 mode on the electrondensity, wavelength of the microwave and waveguide radius iscalculated. Based on the results of calculations, the parametersfor most efficient guiding of microwave radiation along thisvirtual system are as follows: working in TE01 mode operatesat the lowest ratio of wavelength to waveguide radius and withthe highest possible electron density. The effect of the spatialdensity spread of the plasma wall is investigated and found tohave a small influence on the propagation characteristics of theTE01 mode. Using the hollow conducting plasma waveguide,tens of dB of transmission enhancement over the case offree space is obtained. Thus, this work opens the possibilityof using the hollow conducting plasma waveguide to guidethe microwave energy over an extended distance achievinga higher channel capacity. The hollow conducting plasmawaveguide may find applications in directed energy deliveryand point-to-point communication, and may also be usefulfor transporting pulsed-modulated microwaves because of itswide-band feature and short duration.

Acknowledgments

This project was supported by the National Natural ScienceFoundation of China under Grant Nos 60978014, 11074027and 61178022, Funds from the Science & TechnologyDepartment of Jilin Province, Grant No 20111812, basic fundNo 9140c150302110c1501 and the project sponsored by SRFfor ROCS, SEM.

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