Long-distance directed transfer of microwaves in tubular sliding-mode plasma waveguides produced by KrF laser in atmospheric air V. D. Zvorykin, A. O. Levchenko, A. V. Shutov, E. V. Solomina, N. N. Ustinovskii et al. Citation: Phys. Plasmas 19, 033509 (2012); doi: 10.1063/1.3692090 View online: http://dx.doi.org/10.1063/1.3692090 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v19/i3 Published by the American Institute of Physics. Related Articles Terahertz emission from a two-color plasma filament in a slot waveguide Appl. Phys. Lett. 100, 091113 (2012) Global gyrokinetic particle simulation of toroidal Alfvén eigenmodes excited by antenna and fast ions Phys. Plasmas 19, 022507 (2012) Surface plasma source with saddle antenna radio frequency plasma generator Rev. Sci. Instrum. 83, 02A712 (2012) A close-coupling multi-antenna type radio frequency driven ion source Rev. Sci. Instrum. 83, 02B106 (2012) The dispersion relation and excitation of transverse magnetic mode electromagnetic waves in rippled-wall waveguide with a plasma rod and an annular dielectric Phys. Plasmas 19, 013109 (2012) Additional information on Phys. Plasmas Journal Homepage: http://pop.aip.org/ Journal Information: http://pop.aip.org/about/about_the_journal Top downloads: http://pop.aip.org/features/most_downloaded Information for Authors: http://pop.aip.org/authors PHYSICS OF PLASMAS 19, 033509 (2012) Long-distance directed transfer of microwaves in tubular sliding-mode plasma waveguides produced by KrF laser in atmospheric air V. D. Zvorykin, A. O. Levchenko, A. V. Shutov, E. V. Solomina, N. N. Ustinovskii, and I. V. Smetanina) P. N. Lebedev Physical Institute, Leninskii prospect 53, Moscow 119991, Russia (Received 7 October 2011; accepted 15 February 2012; published online 15 March 2012) A new regime of the sliding-mode propagation of microwave radiation in plasma waveguides in atmospheric air is studied both experimentally and theoretically. The mechanisms of air photoionization and relaxation under propagation of 25-ns pulses of KrF laser are investigated. It is shown that a tubular plasma waveguide of large radius (much larger than wavelength of the microwave signal) can be produced in the photoionization of air molecules by 248-nm radiation of KrF-laser. We experimentally demonstrate the laser-enhanced transfer of 38-GHz microwave signal to a distance of at least 60 m. The mechanism of the transfer is determined by total internal reflection of the signal on the optically less dense wall of the plasma waveguide. Analytical and numerical simulations performed for various waveguide radii and microwave radiation wavelengths show that the propagation length increases with decrease in the wavelength reaching a few kilometers for submillimeter waves. Medium-size KrF laser facility with about 400-J energy in a train of picosecond pulses is suggested for the directed transfer of microwave radiation to 1-km C 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3692090] distance. V I. INTRODUCTION The properties of plasma waveguides have been studied in a large number of works during the recent decades because those are closely related to issues of acceleration of charged particles in plasma, amplification, generation and transfer of microwave radiation, microwave heating and diagnostics of plasma, diffraction on plasma bodies, and a range of other problems. Transfer of electromagnetic (microwave and radiofrequency) radiation pulses in atmospheric air using the laser plasma as a guiding structure has been proposed,1 and the waveguide properties of the laser spark have been experimentally demonstrated.2–4 The development of plasma structures up to several tens and hundreds of meters long became possible with the discovery of the effect of filamentation of high-power ultrashort laser pulses.5–7 In the process of filamentation, when laser pulse propagates in atmospheric-pressure gases, a trace is formed in the shape of a thin plasma filament 100 lm in diameter and up to several hundred meters long, with electron density of 1015 1017 cm3. From the viewpoint of certain applications, it appears interesting to use such plasma formations with controlled geometry and characteristics (density profile, etc.) for electromagnetic-radiation transfer. The propagation of 3D modes of microwave radiation in hollow plasma waveguides, whose wall is formed by such filaments, was theoretically investigated.8 The optimum choice of the plasma waveguide radius was found to be of the order of the signal wavelength, Rk. The transfer regime is ensured by the high conductivity of the plasma in the channel; thus, the physical mechanism is quite similar to the conventional case of waveguides with a) Electronic mail: smetanin@sci.lebedev.ru. 1070-664X/2012/19(3)/033509/15/$30.00 metal wall. The conductivity of plasma, however, is several orders of magnitude lower than that of metal and, as a result, the signal propagates only to a small distance. The experiment9 was performed in a hollow waveguide 4.5 cm in diameter, whose wall is formed by multiple filaments, to demonstrate the transfer of a 10 GHz microwave signal to a distance of 16 cm. Numerical calculations10 show that some increase in the transfer length can be achieved in structures formed by orderly arranged plasma filaments of the type of photonic crystals. In this work, we investigate experimentally and theoretically an alternative mechanism of the sliding-mode propagation of microwave radiation inside a hollow plasma channel of large radius R k, which makes it possible to increase significantly the signal transfer length. Such a possibility with the use of a tubular UV laser beam was first indicated by Askar’yan,1 and some estimates for KrF laser were made.11 Physically, this mechanism is based on the effect of total reflection at the interface with an optically less dense medium. For waveguides of sufficiently large radius, the lowest-order modes become “sliding:” the transverse wavenumber is significantly smaller (with a multiplier of k=R) than the longitudinal wavenumber, and the “effective” angle of incidence on the reflection surface exceeds the critical angle determined by the ratio of the refractive indices of air and plasma. With this approach, high conductivity of the plasma is not necessary and wall plasma density can be 1011 1014 cm3, corresponding to a low degree of air ionization. We should emphasize the difference between plasma waveguides used in our work and large-radius dielectric waveguides.12,13 Those waveguides are, in particular, capillary tubes with dielectric wall of radius well exceeding the wavelength; high-power laser radiation propagating in the 19, 033509-1 C 2012 American Institute of Physics V 033509-2 Zvorykin et al. waveguides ionizes the gas filling the capillary tube and forms a plasma wave accelerated electrons.14–16 Analysis of the natural modes in such a structure has been done.13 In such dielectric waveguides, reflection occurs at the boundary of the wall, whose permittivity is larger than that inside the capillary, and the losses emerge because of leakage of the radiation through the capillary wall. In contrast, in plasma waveguides, the total internal reflection from the optically less dense wall takes place, and the modes are damped due to plasma conductance. We report here the results of further development of our approach17,18 to the sliding-mode microwave transport in plasma waveguides. The paper is organized as follows. In Sec. II, the results of experimental investigation of the main photoionization and recombination processes in KrF laserproduced air plasma are presented, we provide also the accurate measurements of the ionization yield dependence on the UV laser intensity. The revival effect and storage of electron density utilizing effective photodetachment in the scheme with subsequent laser pulses is demonstrated. In Sec. III, a detailed description of the experiment on efficient channeling and transfer of the sliding mode of a microwave signal at 8.5-mm wavelength in large-radius plasma waveguides being formed in the atmospheric air by powerful GARPUN KrF laser with 100-ns, 50-J pulse is presented. In Sec. IV, we theoretically investigate the propagation of the sliding modes in plasma waveguides and present the results of both numerical and analytical solutions to the dispersion equations describing the propagation of the lowest sliding axial-symmetric modes E01 and H01 of a plasma cylindrical waveguide (Subsection IV B), as well as of the hybrid mode EH 11 (Subsection IV C). The effect of a finite wall thickness of plasma waveguides on the attenuation increment is analyzed in the Subsection IV D. And in the Subsection IV E, the effect of the plasma density spread in the waveguide wall is investigated. In Sec. V, we discuss the prospects for the microwave transfer at distances up to 1 km and make estimates of laser parameters and availability with e-beampumped KrF lasers. A train of short UV laser pulses and photo-detachment of electrons, being captured by O2 molecules, by consequent pulses in the train are shown as the means how to revive lost primary electrons and to keep the required electron density in a plasma waveguide for long enough time. II. PHOTOIONIZATION AND PLASMA DECAY PROCESSES IN ATMOSPHERIC AIR UNDER UV LASER IRRADIATION Phys. Plasmas 19, 033509 (2012) sp ¼ 25 ns were focused with various lenses (F ¼ 0.5, 0.7, and 2 m) through ring electrodes (to eliminate photoemission from electrodes) into a drift gap, with length l varied from 0.5 to 12 mm matching the lenses caustic (Fig. 1). DC voltage U ¼ 0.2 2.5 kV was applied to the gap with a caution that no spark breakdown does occur in the air. Distribution of laser radiation across laser beam was measured by OPHIR profilometer. For the 70-cm focus lens, an average diameter of the beam in a focal plane (at FWHM of energy fluence distribution) was d 50 lm, and this core contained about 20% of the total energy. The rest of energy was contained in the 250-lm diameter spot. For the other lenses, the laser profiles were similar, the core diameter being dependent on focal length. In all, laser intensity was varied in the range I ¼ 4 106 2 1011 W/cm2 by attenuating laser energy, changing the focal length, and by defocusing laser beam (moving the focus beyond the inter-electrode gap). The maximum laser intensity was just a bit below the threshold for an avalanche air breakdown (laser spark) which was determined to be Ith ¼ ð2 3Þ 1011 W/cm2. Photocurrent response was measured using either matched impedance input of the oscilloscope, Rosc , of 50 X, or high-impedance input of 1 MX (Fig. 2). In the former case, the plasma conductivity r was proportional to the oscilloscope signal u, rðtÞ ¼ ðl=USRosc ÞuðtÞ, l is the drift gap length, S is the plasma cross section. In the latter case, the photocurrent was integrated over the circuit time Rosc Cosc 150 ls, where Cosc ¼ 150 pF is a total capacitance of the oscilloscope input and the cable, the Ð t oscilloscope signal being expressed by uðtÞ ¼ ðUS=lCosc Þ 0 rðt0 Þdt0 . Such measurement mode allowed us to operate with lower laser intensities. Analyzing the signals in Fig. 2, one can find that there are three characteristic time intervals with different carrier mobility in reduced electric field E ¼ U/l and with different decay time: (1) the fast photocurrent component (a) starts simultaneously with the laser pulse (b) and lasts slightly longer (Fig. 2(a))—it is attributed to electrons; (2) the slower current component attains maximum value by 100ls (Fig. þ 2(b))—it is attributed to positive and negative ions Oþ 2 , O4 , þ N2 , O2 , and so on, being formed from the main air species, oxygen and nitrogen in ionization and electron attachment processes and going down due to ion-ion recombination; (3) long-lived current lasting few ms (Fig. 2(c))—it might be caused by non-compensated ions in the presence of microscopic charged aerosol particles (or very large cluster A. Measurements of the specific conductivity and electron density in KrF laser-produced plasma Specific conductivity and electron density were derived from the measurements of photocurrent induced by KrF laser. The experiments were performed with a dischargepumped KrF laser (master oscillator of GARPUN multistage KrF laser system19) to investigate atmospheric air ionization and to measure specific plasma conductivity and electron density ne in dependence on laser intensity.20 Laser pulses with variable energy Q ¼ 5 200 mJ and FWHM duration FIG. 1. (Color online) Layout of plasma conductivity measurements. 033509-3 Long-distance directed transfer of microwaves Phys. Plasmas 19, 033509 (2012) law was checked first for Du versus applied voltage U (for a fixed drift gap l) and Du versus l (for a fixed voltage U) dependencies. Those shown in Fig. 3 evidence that (1) there is no contribution of photoelectron emission from the electrodes (dependence Du / 1=l) and (2) specific electron conductivity does not depend on the applied voltage (dependence Du / U). The electron density was then calculated as ne ¼ re =ele with assumed electron mobility at atmospheric pressure le 600 cm2 (V s)1 in accordance with Refs. 21 and 22. Dependence of the electron density ne on laser intensity I is shown in Fig. 4. Note that the obtained data are tenfold higher than the previous results;17,18,20 the reason is that they take into account more accurately the intensity distribution over the laser spot and a contribution of low-intensity wings into measured electron conductivity. For relatively high intensities I ¼ 3 108 7 1010 W/cm2, experimental data are well approximated by a power law ne / I2 . Similar dependencies were observed in experiments23,24 for short UV laser pulses sp ¼ 35 ps with I ¼ 1010 1011 W/cm2 and for even shorter pulses25 sp ¼ 400 fs and I 1011 W/cm2. Such dependence manifests the resonance enhanced multiphoton ionization (REMPI) proceeding via two-photon excitation of high-energy Rydberg molecular levels which is followed by one-photon ionization. The REMPI processes can lead to significant enhancement of the ionization yield compared with direct multi-photon ionization (MPI) of air molecules from the ground state. A rich vibration spectrum of molecular nitrogen and oxygen opens a number of REMPI channels, the most probable of which are those which FIG. 2. (Color online) Oscillograms of the laser pulse (2) and photocurrent signals (1) for different input impedances and laser intensities Rosc ¼ 50 X, 10 ns/div (a); Rosc ¼ 1 MX, 50 ls/div (b); and Rosc ¼ 1 MX, 500 ls/div (c). ions) formed in the photoemission processes. The latter component was blown off if air flow organized perpendicular to the discharge gap. As electrons have at least 3 orders of magnitude higher mobility than the ions, we easily select their contribution re into plasma conductivity. In the case of highimpedance oscilloscope input, photoelectron current integrated over electron lifetime produced the initial steep jump Du, while the following gradual signal growth and decline corresponded to time-resolved ion current. As the laser beam profile has broad wings (they are not distinctly visible with the profilometer because of its limited dynamic range), special attention was paid to prevent photoelectron emission from ring electrodes. Fulfillment of Ohm’s FIG. 3. (Color online) Photoelectron signal jump Du vs. applied DC voltage U for a fixed drift length l ¼ 4 mm (a) and vs. l for a fixed U ¼ 250 V (b). 033509-4 Zvorykin et al. Phys. Plasmas 19, 033509 (2012) P ðK j Þ j Here, Se ¼ N0 j aj rj IK is a rate of electrons production in different MPI processes; N0 is total molecular concentrationj in air, aj are concentration fractions of various species, ðK Þ are corresponding MPI cross sections, and K j is the rj MPI degree (i.e., the number of photons needed to ionize the specific air molecule). For KrF laser, a direct 3-photon MPI of O2 molecules (with ionization potential Ii ¼ 12:06 eV) is usually suggested as a primary ionization process, while direct 4-photon ionization of N2 molecule (Ii ¼ 15:6 eV) is assumed significantly less probable (see e.g., Refs. 31 and 32). However, our experiments and also23–25 evidence that (2 þ 1) REMPI dominates for KrF laser pulses of subpicosecond to nanosecond length at least in the intensity range below 1011 W/cm2. Disappearance of electrons in the air plasma of medium density is mostly related to a three-body electron attachment to O2 molecule O2 þ e þ O2 ðH2 OÞ ! k1 ðk2 Þ O 2 þO2 ðH2 OÞ: FIG. 4. (Color online) Electron density ne vs. laser intensity I. Different symbols correspond to experiments with various focusing conditions. include a two-photon excitation of metastable states. The (2 þ 1) and (3 þ 1) REMPI in N2 with ionization potential Ii ¼ 15:6 eV through the 1 Rþ g molecular state were observed in the field of ArF (h ¼ 6:3 eV)26–28 and KrF laser radiation (h ¼ 5:0 eV),28 respectively. Multi-photon ionization of molecular oxygen by KrF laser radiation28 occured via twophoton excitation of resonant nsrg ; ndrg , and ndpg Rydberg levels. For lower laser intensity I ¼ 3 106 3 108 W/cm2, a linear dependence ne / I was observed which could be explained by a (1 þ 1) REMPI of impurity molecules with low ionization potential Ii 10 eV or by photoemission from aerosol microscopic particles.29 A strong influence of impurities on the conductivity was observed in our experiments: adding of some organic hydrocarbon vapors, e.g., kerosene, to laboratory air increased the photoionization signal by up to 3 orders of magnitude, and the signal was linear in the whole intensity range 3 106 7 1010 W/cm2. Similar effect was also observed in air with cigarette smoke. (R1, R2) In atmospheric-pressure air, the characteristic time for this attachment process se ¼ ðk1 NO2 2 þ k2 NO2 NH2 O Þ1 reported in literature varies from few nanoseconds to several tens or even hundred nanoseconds (see e.g., Refs. 22, 33, and 34). Alternative process of the electron loss, two-body dissociative attachment, O2 þ e ! k3 O þO (R3) is less efficient33 at our experimental conditions with reduced electric field E=N0 20 Td (E ¼ 5 kV/cm and N0 ¼ 2:5 1019 cm3). Both the three-body and two-body attachment processes are dependent on electric field and air pressure. For higher E=N0 and correspondingly increased mean electron energy, the two-body process begins to dominate over the three-body one.22,33 As the binding energies of electron in negative molecular ion O 2 eb 0:5 eV and in atomic ion O eb 1:4 eV are significantly less than energy quantum of KrF laser radiation h ¼ 5 eV, a reverse process of electrons photodetachment35 takes place O 2 þ h ! O2 þ e: (R4) Actually, it is accompanied36 by photodissociation of O 2 B. Evolution of electron component in the photoionized air plasma O 2 þ h ! O þO; (R5) Although a comprehensive description of the air plasma evolution under action of ionizing radiation includes a huge amount of reactions and ions (see e.g., Ref. 30), we will follow a bulk of works (see e.g., Ref. 22) in our simple analysis, and the basic rate equations are as follows: where atomic ion O after absorption of photon gives again free electron dne ¼ Se ne ðk1 NO2 2 þ k2 NO2 NH2 O þ k3 NO2 Þ dt Da I k7 ne NOþ2 2 ne þ NO2 rph ; O 2 h K dN O2 I ¼ ne ðk1 NO2 2 þ k2 NO2 NH2 O Þ NO2 rph : O 2 dt h Cross sections for the processes (R4), (R5), and (R6) depend on the laser wavelength. For KrF laser (k ¼ 248 nm), they ¼ 9:5 1018 were measured37,38 to be enormously high rph O 2 2 cm (total cross-section for reactions (R4) and (R5)) and 18 cm2. The photo-detachment reduces or rph O ¼ 11:3 10 completely compensate loss of electrons as the result of (1) O þ h ! O þ e: (R6) 033509-5 Long-distance directed transfer of microwaves Phys. Plasmas 19, 033509 (2012) attachment during the laser pulse if its intensity I h 2 106 W/cm2. Besides, photodetachment by =se rph O 2 another consequent laser pulse enables one to revive electrons captured by O2 molecules.39–41 This effect might be very useful to keep the electron density in a plasma channel for a long time using a train of pulses with a relatively low intensity. An ambipolar electron-ion diffusion is taken into account in Eq. (1) for plasma cylindrical geometry with an appropriate diffusion coefficient22 Da 200 cm2 s1 where K ¼ Rpl =2:4 and Rpl is the plasma radius equal to the laser beam radius. Although diffusion leads to electron escape from the initial plasma channel of radii Rpl 50 lm, the photocurrent signal in the drift gap being integrated over the plasma cross section was not sensitive to the diffusion. Dissociative recombination of electrons with positive molecular oxygen ions Oþ 2 (their concentration is assumed to be NOþ2 ne ),22 k7 Oþ 2 þ e ! O þO (R7) is the only relaxation process where electrons are irretrievably lost and the laser energy spent until then for ionization of O2 molecules is redistributed between oxygen atoms in the ground and excited states. The rate constant for this process is in the range k7 1:3 108 3:1 107 cm3 s1 in dependence on electron temperature. Dissociative recombination prevails at higher electron density ne ne ¼ ðk1 NO2 2 þ k2 NO2 NH2 O Þ=k7 ¼ 1014 1015 cm3 which corresponds in our case to laser intensities I 5 109 W/cm2 (Fig. 4). This recombination process can be seen in Fig. 2(a) obtained at high laser intensities: photocurrent signal follows the laser pulse-form being slightly longer at the trailing edge. C. Photodetachment of electrons from O 2 ions To demonstrate the revival effect and storage of electrons via the processes (R4)-(R6), we used a double pulse scheme (Fig. 5). Laser radiation was split into two beams, one of them of lower intensity passed an optical delay line and spatially superimposed with a non-delayed beam. The combined beam was focused by F ¼ 70 cm lens in the drift gap to measure a photocurrent response. Overlapping of both beams in the focal plane was controlled by the profilometer. Because of a slight astigmatism of the focusing lenses and vibration of optical mirrors of the delay line, the beam overlapping factor was measured to be f ¼ 0:8 on average. FIG. 5. (Color online) Layout of the double pulse experiments. Typical oscilloscope signals of the combined laser pulse and photoelectron current at matched oscilloscope input are presented in Fig. 6 for a time delay between two pulses sd ¼ 80 ns. The intensity of the first pulse was set I1 ¼ 3:3 109 W/cm2 (averaged value over a full focal spot distribution of 250 lm diameter) so that the condition ne ne was satisfied and a three-body attachment (R1) prevailed over dissociative recombination (R7). It is seen that photocurrent decreases just slightly to the end of the laser pulse and lasts for about 100 ns. This manifests slowdown of electron attachment to O2 molecule under the action of laser radiation. By the arrival of the second delayed pulse, the most of electrons produced by the first pulse are already attached. Amplitude of the second photocurrent pulse appears to be larger than of the first one in contrast to the ratio of amplitudes of laser pulses. Such behavior indicates that the second laser pulse not only produces photoelectrons in the primary REMPI process but, additively, photodetaches them from negative molecular ions O 2 , left after the first laser pulse. The revival effect of the second pulse was observed also for a 40-ns time delay between the laser pulses. To characterize what fraction of electrons is revived by the second laser pulse of different intensities, the electron revival photodetachment ratio Re is introduced Re ¼ Du1þ2 ðDu1 þ Du2 Þ : sd sp Du1 f 1 exp se (2) Here, Du1þ2 is the photoelectron signal jump produced by the combined laser pulse, Du1 and Du2 are the photoelectron signals produced separately by two single pulses (when the other pulse is blocked) with the same amplitudes as in the combined pulse, all being measured independently on a high-impedance oscilloscope input; f ¼ 0:8 is the spatial overlapping in the combined laser beam. The numerator in expression (2) is proportional to the total amount of revived electrons flowed through a drift gap. The denominator is proportional to the amount of O 2 ions formed in the overlapped volume after termination of the first laser pulse in electron attachment till the beginning of the second pulse. It was FIG. 6. (Color online) Oscillograms of the combined laser pulse (1) and photoelectron current on matched oscilloscope input (2). 033509-6 Zvorykin et al. Phys. Plasmas 19, 033509 (2012) FIG. 7. (Color online) Electron photodetachment ratio Re vs. averaged intensity of the second laser pulse I2 . Intensity of the first pulse I1 ¼ 3:3 109 W/cm2 and se ¼ 50 ns. assumed that under the action of laser radiation attachment is completely balanced by the photodetachment process. The value se in Eq. (2) can be considered as fitting parameter to attain independence of parameter Re on sd varying in the range up to at least a few hundreds of microseconds (that is characteristic time of ion-ion recombination). For our two experimental marks of sd 40 ns and 80 ns, the best fit for se is 50 ns (see Fig. 7). Our preliminary experiments with a short subpicosecond laser pulse44 confirmed that characteristic time for electron attachment is indeed 50 ns. Figure 7 represents the photodetachment ratio Re for the fixed intensity of the first pulse I1 ¼ 3:3 109 W/cm2 as a function of average intensity of the second pulse which was varied by a stepped diffraction attenuator in the range I2 ¼ 2 107 2 109 W/cm2. For average intensity in the laser spot higher then 109 W=cm2 Re tends to unity—that is, almost all attached electrons are revived. III. EXPERIMENTAL REALIZATION OF SLIDING MODES IN PLASMA WAVEGUIDES Below, we present the results of the first experiments on efficient channeling and transfer of the sliding mode of a microwave signal in a plasma waveguide of a comparatively large radius R 5 cm formed in atmospheric air by 248-nm radiation of GARPUN KrF excimer laser.45 In the unstableresonator injection control regime, the laser generated pulses of 70 ns FWHM, energy of 50 J, and radiation divergence 104 rad. To obtain a parallel, convergent, or divergent “tubular” beam, we used various optical setups (Fig. 8). In setup (a), the central part 100–120 mm in diameter was shut in the initial laser beam 180 160 mm in transverse size using a round closure. The radiation intensity (in the unblocked beam) did not exceed I ¼ 2 106 W/cm2, and photoelectron density in air according to the plasma conductivity measurements (Fig. 4) ne 2 109 cm3. In setups (b)–(d), the initial laser beam was compressed by a two-lens telescope (Fig. 9, left). In setup (b), with the use of two axicons (conical lenses), the compressed laser beam was transformed without energy loss into a tubular beam 120 mm in outer diameter and 10 mm “wall” thickness (Fig. 9, right). Due to a small difference in the apex angles of the axicons, the tubular beam in these experiments was FIG. 8. (Color online) Various setups of experiments on the MW radiation propagation in plasma waveguides. convergent: its diameter decreased by half at a distance of about 15 m along the propagation axis. In setup (c), we used one axicon, and the tubular beam diverged with an angle of 2.4 . In setup (d), use was made of a combination of an axicon and a lens, which decreased the divergence angle of the tubular beam down to 1 . In setups (b)–(d), radiation intensity in the wall ring was I ¼ 107 W/cm2 and electron density in air increased compared to the case (a) up to ne 8 109 cm3. To further increase the electron density, a ready-to-ionize hydrocarbon vapor was added to atmospheric air. Among a few tested hydrocarbons, kerosene proved to be the best photoionized and provided a three orders of magnitude increase in the electron density. The vapor concentration was maintained only along the 15-m distance from the microwave (MW) emitter by a tissue strip impregnated with kerosene and laid on the floor immediately below the propagation track. Thus, FIG. 9. (Color online) Laser-beam prints on a photo paper after a two-lens telescope (left) and a two-axicon telescope (right). 033509-7 Long-distance directed transfer of microwaves adding the hydrocarbon vapor enlarged the electron density in the laser-irradiated tubular layer of air up to ne 1012 cm3. As a source of microwave radiation, we used a pulsed magnetron with a peak output power of 20 kW at a frequency of 35.3 GHz (k 8:5 mm). The microwave emitter, equipped with a conical horn transmitter antenna 25 mm in diameter, had a total angle of radiation divergence of about 30 . The microwave radiation receiver with similar horn antenna was positioned at various distances L from the emitter. In pure air, no noticeable change of the microwave signal was observed in the presence of a tubular laser beam with either of the optical schemes employed. The reason is insufficient photoelectron density responsible for the formation of a virtual plasma waveguide. Upon addition of hydrocarbon vapor along the propagation route, there occurs an interaction between the microwave radiation and photoionized plasma of the waveguide revealing itself in change of the microwave signal. The laser pulse and the signal from the microwave receiver are shown in Fig. 10. The oscilloscope traces on the left ((a) and (c)) correspond to the case of the laser beam switched off (blocked by an opaque screen). Depending on the scheme of the experiments and the laser radiation intensity (consequently, the electron density in the waveguide wall), we observed that during the laser pulse, the microwave signal was either absorbed (in setup (a), cf. Figs. 10(a) and 10(b)) or increased (in setups (c) and (d), cf. Figs. 10(c) and 10(d)). The largest increase in the microwave-signal amplitude, up to 6 times, was observed in setup (d) at a distance L ¼ 60 m (the largest distance available in our experiments) from the microwave emitter to the receiver. In setup (b), in which the tubular laser beam converged, the microwave signal almost did not change with switching on the laser beam. Phys. Plasmas 19, 033509 (2012) The mechanism of microwave radiation channeling in a weakly ionized plasma waveguide is related to the reflection of radiation from the electron-density gradient at the plasmaair interface. This effect is similar to total internal reflection of optical radiation in optical fibers, but differs in that microwave radiation penetrates into the waveguide plasma. The sign of the effect (signal enhancement or reduction) is governed by the balance between the microwave radiation reflection from and penetration through the waveguide plasma. The absorption predominates at low laser intensities and electron densities (the case of setup (a)). At higher laser intensities in setups (c) and (d), the microwave radiation is amplified due to its channeling. A qualitative condition for the microwave-radiation channeling in geometric approximation (strictly speaking, true when the waveguide diameter D k) is the condition that the diffraction angle of divergence of microwave radiation b k=D is smaller than the critical sliding angle H (which is complementary to the angle of total internal reflection) determined by the relation cos H ¼ n, where n is the refractive index of ionized gas with respect to air (Fig. 11(a)) at the microwave wavelength. For small sliding angles, H can be written in the form (see Eq. (3) below) pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ H2 X2p =ðx2 þ 2T Þ, where Xp ¼ 4pne e2 =me is the plasma frequency and T is the characteristic transfer frequency of electron collisions. Figure 12 presents the values of the diffraction angle and the critical sliding angle for pure air and air with volatile hydrocarbon vapor versus the microwave radiation wavelength. It is seen that for k ¼ 8 mm (shown by the vertical line), b H in pure air, and radiation channeling is impossible. With the addition of hydrocarbon, the diffraction angle b is just slightly larger than the angle H. Owing to this, the microwave radiation channeling in a convergent waveguide FIG. 10. (Color online) Signals from the MW receiver (upper beam) and laser pulse (lower beam): upper ((a) and (b)) for the setup in Fig. 8(a) and lower ((c) and (d)) for the setup in Fig. 8(d). The distance to the receiver L ¼ 12 m. 033509-8 Zvorykin et al. Phys. Plasmas 19, 033509 (2012) sider modes of a round waveguide of constant radius R k; the density of the plasma is assumed to be uniform over its cross section and along the propagation length, which is a good approximation under the conditions of our experiment. The diffuse spreading of the waveguide walls is assumed to be small (in comparison with the microwave-signal wavelength) during the microwave pulse action. The permittivity of a weakly ionized air plasma in the microwave electromagnetic-wave field is approximated by the relation22 ep ¼ eair FIG. 11. (Color online) Schematic view of the MW radiation propagation in a cylindrical (left) and conical (right) plasma waveguide. (setup (b) in Fig. 8), whether it takes place, is compensated by the radiation absorption in the plasma-waveguide wall. In a slightly divergent conical plasma waveguide, the condition of channeling is easier to satisfy; in this case, it takes the form b a < H (see Fig. 11(b)). As a consequence, in experiments with setups (c) and (d) (Fig. 8), we observed enhancement of the microwave signal. The interaction length of microwave radiation and plasma in a weakly diverging waveguide was evaluated experimentally by closing the laser beam with a dielectric screen at various distances from the microwave source; the interaction length was found to be about 10 m. IV. THEORY OF THE SLIDING-MODE PROPAGATION IN PLASMA WAVEGUIDES A. Dielectric properties of weakly ionized atmospheric air plasma To consider theoretically the implemented sliding mode of the microwave radiation propagation in plasma waveguides, we make use of a simplest model. We assume a plasma waveguide to be an air cylinder of radius R, encircled by a plasma layer whose thickness significantly exceeds the field-penetration depth. We con- (3) The permittivity of atmospheric air for the centimeter– submillimeter wavelength range is eair 1 104 , so one can assume eair ¼ 1. In the experiments under consideration, plasma is formed as a result of direct or stepwise multiphoton ionization of air molecules in the radiation field of a KrF laser (kL ¼ 248 nm and hxL 5 eV). The average energy defect in the photoionization of air molecules is 1 eV, and since the electronic component of the plasma is rapidly thermalized (of the order of the electron–electron collision time), it can be assumed that in our case, the characteristic electron temperature Te is within the range of 0.03–1 eV. An estimate22,46,47 of the effective momentum transfer frequency of electron collisions with air molecules at atmospheric pressure is T 1012 s1. With increase in the degree of ionization, the electron– ion collisions may play a significant role. The effective frequencies of the electron–electron and electron–ion collisions are assessed by the following relations:22,48 ee ½s1 ¼ 3:7ne ½cm3 3=2 Te ½K lnK; ee ei pﬃﬃﬃ ; 2 (4) where the Coulomb logarithm lnK ¼ 7:47 þ 3=2logT½K 1=2logne ½cm3 . It is easy to see that, under the considered conditions, the electron–ion collisions are insignificant from the viewpoint of the dielectric properties of the plasma within the density range ne 1015 1016 cm3. Thus, within the centimeter–submillimeter wavelength range, extended plasma waveguides of the sliding modes in atmospheric air are characterized by the permittivity, for which the following relation between the real and imaginary parts is fulfilled: Reð1 ep Þ ¼ FIG. 12. (Color online) Diffraction angle and angles of total internal reflection from the plasma–air interface vs the MW radiation wavelength. The MW radiation diffraction angle (1), atmospheric air (2), and hydrocarbons added (3). X2p : xðx þ i T Þ n n T Imð1 ep Þ ¼ : 2 2 2 2 1 þ x = T 1 þ x = T Xp (5) Here, n ¼ X2p = 2T 3:2 103 ðne ½cm3 =1012 Þ at the plasma density ne 1015 cm3. The condition (5), as we discussed above, distinguishes plasma waveguides from dielectric waveguides.12,13 In the considered waveguides with R k, the conditions of internal reflection are, obviously, fulfilled more easily at a smaller sliding angle of the mode (i.e., the closer the 033509-9 Long-distance directed transfer of microwaves Phys. Plasmas 19, 033509 (2012) angle of “incidence” on the waveguide wall to p=2); due to this fact, we choose as operating modes the lowest axialsymmetric transverse magnetic (TM) (E0n ) and transverse electric (TE) (H0n ) modes, as well as the EH 11 mode, which is known to be the main operating mode in dielectric waveguides.12 B. Propagation of axial-symmetric sliding modes First, we consider the axial-symmetric modes E0n and H0n . The transverse distribution of the electromagnetic-field longitudinal components (Ez for TM and Hz for TE modes) has the form J0 ðj1 rÞexp½iðhz xtÞ inside the waveguide ð1Þ and H0 ðj2 rÞexp½iðhz xtÞ in the wall plasma, where r and z are the transverse and longitudinal coordinates (the cylindrical coordinate system is used). Correspondingly, the electromagnetic field of the TM mode E0n contains three components (Ez ; Er ; H/ ), which inside the cylinder (air, r < R) have the form12,49 Ez ¼ E0 J0 ðj1 rÞ; H/ ¼ i Er ¼ i h E0 J1 ðj1 rÞ; j1 k0 E0 J1 ðj1 rÞ; j1 (6) ð1Þ plasma, C=E0 ¼ C0 =H0 ¼ J0 ðj1 RÞ=H0 ðj2 RÞ, and the dispersion equation is49 ð1Þ 1 J1 ðj1 RÞ v H1 ðj2 RÞ ¼ ; j1 R J0 ðj1 RÞ j2 R H ð1Þ ðj2 RÞ where v ¼ ep stands for the TM axial-symmetric modes, and v ¼ 1 for the TE modes. Since R=k 1, dispersion Eq. (11) have, generally speaking, a number of roots corresponding to various transverse axial-symmetric modes. The greatest propagation length corresponds, apparently, to the minimum value of the transverse wavenumber j1 , i.e., to the lowest transverse mode. In the Figure 13, the characteristic propagation length ðImhÞ1 of the sliding axial-symmetric modes (E01 and H01 ) of a plasma waveguide versus the signal wavelength is presented. Also, the results for the axially non-symmetric hybrid mode EH 11 are shown (see Sec. IV C). The calculations were carried out at the plasma density ne ¼ 1012 cm3 for various values of the waveguide radius R ¼ 5, 10, and 30 cm; the characteristic momentum transfer frequency was taken to be T ¼ 1012 s1. Under these conditions, jep 1j 1 and the results for the TE and TM modes are practically the same; some discrepancy appears only at wavelengths k 1 cm. For further analysis, it is convenient to introduce the dimensionless and in ambient plasma, r > R, parameter ðX = Þ2 p T 2 l2 ¼ 1þðx= 2 ðk0 RÞ . TÞ 2 ð1Þ Ez ¼ CH 0 ðj2 rÞ; H/ ¼ i Er ¼ i h ð1Þ CH1 ðj2 rÞ; j2 ep k 0 ð1Þ CH 1 ðj2 rÞ: j1 (7) For the mode H0n , the field is represented by the components Hz ; Hr ; E/ and has the form Hz ¼ H0 J0 ðj1 rÞ; E/ ¼ i Hr ¼ i h H0 J1 ðj1 rÞ; j1 k0 H0 J1 ðj1 rÞ j1 (8) in air and ð1Þ Hz ¼ C0 H0 ðj2 rÞ; E/ ¼ i Hr ¼ i h 0 ð1Þ C H1 ðj2 rÞ; j2 k0 0 ð1Þ C H1 ðj2 rÞ j1 (11) 0 Setting 2 j1 R ¼ xl and j2 R ¼ yl, we have x y ¼ 1 i T =x, and the solution of dispersion Eq. (11) is, in fact, determined by two parameters, l and T =x. Numerical analysis makes it possible to determine the threshold value of the parameter l at which effective propagation of the sliding mode is possible, lth 1: (12) In the range of parameters l 0:5 1, absorption sharply increases, and at l 0:5, the propagation almost terminates: the characteristic propagation length ðImhÞ1 gets limited to several wavelengths. In the domain l > lth ¼ 1, the propagation regime is stabilized. As l2 / ne R2 , relation (12), in fact, determines the lower limits of plasma densities (9) in the plasma. Here, the functions Jn ðxÞ and Hnð1Þ ðxÞ are Bessel and Hankel functions of the first kind, and E0 ; C and H0 ; C0 are the amplitudes of the fields in air and plasma for the TM and TE modes, respectively. The transverse wavenumbers are determined by the dispersion relations in air and plasma, j21 ¼ k02 h2 ; j22 ¼ ep k02 h2 ; (10) where k0 ¼ x=c is the wavenumber in vacuum. The boundary conditions (continuity of the tangential components at r ¼ R) determine the amplitudes of the field in FIG. 13. Characteristic attenuation length ðImhÞ1 of the lower axialsymmetric E01 ; H01 and axial-asymmetric hybrid EH11 sliding modes of MW radiation vs. its wavelength at plasma density ne ¼ 1012 cm3 and waveguide radius R ¼ 5 cm (a), 10 cm (b), and 30 cm (c). 033509-10 Zvorykin et al. Phys. Plasmas 19, 033509 (2012) and waveguide radii at which the sliding propagation regime is feasible. In Fig. 14, the indicated limit for the sliding mode is presented for microwave-signal wavelengths k ¼ 8 mm and k ¼ 3 mm. In the limit of large values, l 1; T =x 1; (13) we succeed in obtaining simple analytical relations for the solutions of dispersion Eq. (11), by analogy with those found for large-radius dielectric waveguides.13 It is easy to see that the root of the dispersion equation occurring near the point x a=l, where a 3:83 is the first root of the Bessel function J1 ðaÞ ¼ 0, corresponds to the sliding regime for the lowest axial-symmetric modes E01 and H01 . For this, it is necessary that pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jep j k0 R n T =x; (14) which indicates smallness of the coefficient v=j2 R 1 on the right-hand side of Eq. (11). Indeed, then y2 ¼ i T =x þ x2 1 whence, for the characteristic transverse wavenumber in the plasma of the waveguide wall, we obtain rﬃﬃﬃﬃﬃﬃ ly l T 6ð1 þ iÞ j2 ¼ ; (15) R R 2x where the plus sign should be chosen to provide the decay of the field Eq. (7) in the plasma with the radius. In accordance with this relation, in the considered limit Eq. (13), the argument of the Hankel functions on the right-hand side of dispersion relations (11) is large, jj2 Rj 1, and making use of the corresponding asymptotic,50 we have ð1Þ ð1Þ H1 ðlyÞ=H0 ðlyÞ i. The Bessel functions in the neighborhood of the given root x ¼ a=l þ dx; dx a=l are approximated as follows:50 ldx J0 ðaÞ; J1 ðlxÞ ldx 1 2a (16) 1 J0 ðlxÞ 1 ðldxÞ2 J0 ðaÞ; 2 and since y ð1 þ iÞ ﬃ pﬃﬃﬃﬃ T 2x 1 i 2xT h a2 l2 i 1 i la xT dx , in the lowest order from dispersion Eq. (11), we obtain " # av x 3=2 1 2ð1 iÞ dx l 2 T 2 rﬃﬃﬃﬃﬃﬃﬃﬃ va x a x : 1þi 21 ð1 þ iÞ 2 l l 2 T 2 T (17) Thus, as we demonstrated numerically above (see Fig. 13), for plasma with a low degree of ionization, such that jep 1j 1 (i.e., n T =x 1), the solutions for the E01 and H01 modes practically coincide, and for the attenuation coefficient, we have Imh k02 R3 a2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : 2n T =x (18) The characteristic propagation length increases with increase in the signal frequency and plasma waveguide ra1=2 dius, ðImhÞ1 / R3 x3=2 n1 . Such behavior is confirmed by numerical investigation of the roots of dispersion Eq. (11): the simulation results are presented in Fig. 15 for the dependence of ðImhÞ1 on the plasma density within the range of 1010 1014 cm3. The calculations were carried out for the wavelength k ¼ 8 mm, corresponding to the experiment, at the plasma waveguide radii R ¼ 5, 10, and 30 cm. With increase in the plasma density in the domain pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ k0 R n T =x 1 (in this case, condition (14) is still valid), the solutions for various modes split as shown in Fig. 15. For the mode H01 , the root of the dispersion equation remains near x a=l, and relation (18) is preserved. For the shifting mode E01 , the root is gradually pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ with increase in electron density, ðlx aÞ=a / n T =x=k0 R, and for the increment of attenuation of this mode, we derive rﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃ! a2 n T 1 n T 1þ ; (19) Imh 2 3 2x k0 R 2x k0 R i.e., its characteristic propagation length begins to decrease with density, / n1=2 . e Transition to the regime of a metal waveguide occurs with further density increase and shifts to the domain pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ n T =x k0 R, when relation Eq. (14) ceases to be fulfilled. In this regime of high conductance of the wall, the coefficient on the right-hand side of Eq. (11) becomes large, jn=j2 Rj 1, and as a consequence, the root of the dispersion equation for the E01 mode shifts to the value lx b 2:405 (J0 ðbÞ ¼ 0), which is characteristic of the E01 mode of a metal waveguide. In accordance with the theory of metal waveguides, the increment of attenuation of such a TM mode pﬃﬃﬃﬃ increases with frequency, Imh / x, and its minimum value is achieved near the cutoff frequency.12 In this range of parameters, the optimum propagation conditions are realized8,9 at R k. C. Propagation of axial-asymmetric sliding mode EH 11 FIG. 14. Characteristic threshold values (relation (12)) of the waveguide radius vs. the density of wall plasma for MW radiation wavelengths k ¼ 8 mm (a) and k ¼ 3 mm (b). The lowest axial-asymmetric mode EH11 is the main operation mode of dielectric waveguides.12 This is a hybrid mode, i.e., it contains all six components of the electromagnetic field; herewith, all longitudinal components of the 033509-11 Long-distance directed transfer of microwaves Phys. Plasmas 19, 033509 (2012) Imh b2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; k02 R3 2n T =x (24) similar to Eq. (18). Thus, the characteristic propagation length of the hybrid mode EH11 exceeds the propagation 2 length of the axial-symmetricp mode ða=bÞ 2:54 times. ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ At the densities k0 R n T =x 1, we arrive, similarly to Eq. (19), at the estimate rﬃﬃﬃﬃﬃﬃﬃﬃ b2 n T : (25) Imh 2 3 2x k0 R FIG. 15. Characteristic attenuation length ðImhÞ1 of the lower axialsymmetric E01 ; H01 and axial-asymmetric hybrid EH11 sliding modes of MW radiation with the wavelength k ¼ 8 mm vs. the density of waveguide wall plasma at waveguide radii R ¼ 5 cm (a), R ¼ 10 cm (b), and R ¼ 30 cm (c). This result is confirmed by the numerical solution of dispersion Eq. (20) under the condition of realizing the sliding regime, see Figs. 13 and 15. D. Effect of the thickness of plasma waveguide wall electric and magnetic fields have the form Ez ; Hz J1 ðj1 rÞ ð1Þ inside the waveguide and Ez ; Hz H1 ðj2 rÞ in plasma of the walls. The dispersion equation for the mode EH11 can then be written as49 " ! # ð1Þ 1 J0 ðj1 RÞ ep H0 ðj2 RÞ 1 ep 2 2 j1 J1 ðj1 RÞ j2 J1 ðj2 RÞ j1 R j2 R " ! # ð1Þ 1 J0 ðj1 RÞ 1 H0 ðj2 RÞ 1 1 2 2 j1 J1 ðj1 RÞ j2 J1 ðj2 RÞ j1 R j2 R 1 ep 1 1 : (20) ¼ j21 R j22 R j21 R j22 R By analogy with the case of axial-symmetric modes, this equation can be approximately solved in the limit (13) of large values of the parameter l 1 and transfer frequency T =x 1. It is easy to see that with condition (14) imposed, for plasma with a low degree of ionization jep 1j 1, the root of dispersion Eq. (20) is near lx b 2:405 (J0 ðbÞ ¼ 0). Indeed, in this case, 1 ep 1 1 2 2 2 ; 2 j1 R j2 R j1 R j2 R (21) The previous consideration was carried out in the limit of infinitely thick walls of plasma waveguides, such that the wave practically does not escape to the external space—the characteristic depth of field-into-plasma penetration is small in comparison with the wall thickness d, j2 d ¼ lyd=R 1. Nevertheless, the wall thickness apparently significantly affects the characteristic length of the sliding mode propagation—the losses increase in thin walls due to “outflow” (reemission) of the sliding modes “sideways.” At the same time, an increase in the thickness of the plasma walls of extended waveguides implies, in fact, a significant increase in the energy spent by the ionizing laser pulse. Thus, from the viewpoint of optimizing the efficiency of the transfer, of interest is the study of the dependence of the attenuation length of the sliding mode of microwave radiation on the thickness of plasma waveguide walls. In this work, we restrict ourselves to the case of the axialsymmetric mode E01 . We consider the plasma waveguide to be a cylinder; the radius of the internal boundary of the waveguide is R1 , and that of the external boundary R2 . The field structure of the E01 mode inside the waveguide, r < R1 , is determined by expression (6), and in the region occupied by plasma, R1 < r < R2 , we have and Eq. (20) gets significantly simplified, ð1Þ 1 J0 ðj1 RÞ ep 1 J0 ðj1 RÞ i i 0: j2 j1 J1 ðj1 RÞ j2 j1 J1 ðj1 RÞ Ez ¼ AJ0 ðj2 rÞ þ BH0 ðj2 rÞ; i ih h ð1Þ AJ1 ðj2 rÞ þ BH1 ðj2 rÞ ; Er ¼ j2 i iep k02 h ð1Þ H/ ¼ AJ1 ðj2 rÞ þ BH1 ðj2 rÞ ; xj2 (22) Here, we made use of the asymptotics of the Hankel funcð1Þ ð1Þ tions H0 ðlyÞ=H1 ðlyÞ i at large values of the argument ly 1. Expanding the dispersion equation in the neighborhood of this root x ¼ b=l þ dx; dx b=l, we obtain J0 ðlxÞ J1 ðbÞldx, and for the approximate value of the root (22), we have rﬃﬃﬃﬃﬃﬃﬃﬃ b x : (23) dx ð1 þ iÞ 2 l 2 T As a result, for the increment of attenuation of the EH 11 mode, we obtain the relation (26) and ð1Þ Ez ¼ CH 0 ðj1 rÞ; Er ¼ C ik2 ð1Þ H/ ¼ C 0 H1 ðj1 rÞ; xj1 ih ð1Þ H ðj1 rÞ; j1 1 (27) in the ambient space, r > R2 . Herewith, for the transverse wavenumbers, relation (20) is preserved. 033509-12 Zvorykin et al. Phys. Plasmas 19, 033509 (2012) Making use of the boundary conditions, i.e., the continuity of the tangential components of the field at r ¼ R1 and r ¼ R2 , we obtain for the amplitudes of the field components in plasma ð1Þ A¼E0 ð1Þ ð1Þ J0 ðj2 R1 ÞH1 ðj2 R1 ÞJ1 ðj2 R1 ÞH0 ðj2 R1 Þ ð1Þ ¼C ð1Þ J0 ðj1 R1 ÞH1 ðj2 R1 Þ½j2 =ðep j1 Þ J1 ðj1 R1 ÞH0 ðj2 R1 Þ ð1Þ ð1Þ ð1Þ H0 ðj1 R2 ÞH1 ðj2 R2Þ½j2 =ðep j1Þ H1 ðj1 R2ÞH0 ðj2 R2Þ B¼E0 ð1Þ ð1Þ ð1Þ ð1Þ J0 ðj2 R2 ÞH1 ðj2 R2 ÞJ1 ðj2 R2 ÞH0 ðj2 R2 Þ J0 ðj1 R1 ÞJ1 ðj2 R1 Þ½j2 =ðep j1 Þ J1 ðj1 R1 ÞJ0 ðj2 R1 Þ J0 ðj2 R1 ÞH1 ðj2 R1 ÞJ1 ðj2 R1 ÞH0 ðj2 R1 Þ ð1Þ ¼C ; ð1Þ H0 ðj1 R2 ÞJ1 ðj2 R2 Þ½j2 =ðep j1 Þ H1 ðj1 R2 ÞJ0 ðj2 R2 Þ ð1Þ ð1Þ J0 ðj2 R2 ÞH1 ðj2 R2 ÞJ1 ðj2 R2 ÞH0 ðj2 R2 Þ : (28) Finally, the dispersion equation is obtained in the form FIG. 16. Characteristic attenuation length ðImhÞ1 of the axial-symmetric mode E01 of MW radiation with wavelength k ¼ 8 mm vs. the relative plasma-waveguide-wall thickness DR=R. The result is obtained by numerically solving the dispersion Eq. (29) at plasma density ne ¼ 1013 cm3 and waveguide radius R ¼ 10 cm. thickness of 10%, i.e., 1 cm for the given parameters. The local maxima on the curves are, obviously, of an interference character. ð1Þ J1 ðj1 R1 Þ ep j1 H1 ðj2 R1 Þ j2 H ð1Þ ðj2 R1 Þ J0 ðj1 R1 Þ 0 ð1Þ H0 ðj2 R2 ÞJ0 ðj2 R1 Þ J1 ðj1 R1 Þ ep j1 J1 ðj2 R1 Þ ¼ ð1Þ j2 J0 ðj2 R1 Þ H0 ðj2 R1 ÞJ0 ðj2 R2 Þ J0 ðj1 R1 Þ ( ) ð1Þ ð1Þ H1 ðj1 R2 Þ ep j1 H1 ðj2 R2 Þ ð1Þ j2 H ð1Þ ðj2 R2 Þ H0 ðj1 R2 Þ 0 ( )1 ð1Þ H1 ðj1 R2 Þ ep j1 J1 ðj2 R2 Þ : ð1Þ j2 J0 ðj2 R2 Þ H ðj1 R2 Þ E. Effect of the plasma wall density spread (29) 0 The left-hand side of the dispersion equation coincides with dispersion Eq. (11) for the sliding axial-symmetric TM mode in the limit of a thick waveguide wall. The effect of the finite wall thickness is taken into account on the right-hand side. This effect can be understood qualitatively by making use of the following considerations. ð1Þ At sufficiently large Req and Imq, the ratio H1 ðqÞ= ð1Þ H0 ðqÞ ! i, and the right-hand side of the dispersion equation is determined mainly by the first multiplier. According to the asymptotic behavior of the Hankel functions,50 ð1Þ H0 ðj2 R2 Þ ð1Þ H0 ðj2 R1 Þ exp½Imj2 ðR2 R1 Þ : (30) Thus, with increase in the waveguide wall thickness, the effect of the second boundary decreases exponentially. The result of numerical analysis of the dispersion equation is shown in Fig. 16, in which the dependence of the attenuation length of the sliding mode on the wall thickness of the plasma waveguide for values of parameters close to the experimental values is presented. The calculations were performed at wavelength k ¼ 8 mm for the case of plasma density ne ¼ 1013 cm3 and waveguide internal radius R1 ¼ 10 cm. The curves correspond to the characteristic transfer frequency of collisions T ¼ 1012 s1; herewith, the parameter l 13:86. The calculations show that the effect of the external boundary of the plasma waveguide is insignificant up to a relative wall In the above consideration, a step density profile of the plasma waveguide has been assumed. A real plasma waveguide wall has a gradual density profile (spread) which is caused by several reasons, including the UV laser intensity profile and diffraction, plasma diffusion process, etc. In this section, we will evaluate the effect of the wall density spread on the microwave transport characteristics. Let us consider the same as in Subsection IV B infinitely thick hollow plasma waveguide of a radius R which is characterized by the plasma dielectric permittivity ep . At the airplasma interface R þ d > r > R, the density profile tends to zero at r ! R, and the dielectric permittivity approaches the dielectric permittivity of air, eðrÞ ! 1. We will assume the characteristic size d of the density inhomogeneity at the airplasma interface is sufficiently small, j2 d 1, and one can seek the electromagnetic fields amplitudes as perturbation series with the parameter of smallness determined by de ¼ eðrÞ ep . For the particular case of the axial-symmetric sliding mode E01 , the Maxwell equations can be written as 1@ ðrH / Þ ¼ ik0 eEz ; r @r @Ez ek2 h2 ¼i 0 H/ ; @r ek0 h Er ¼ H/ : ek0 (31) We will seek solution to the Maxwell equations as the perturbation series H/ ¼ H/0 þ H/1 þ …; Ez ¼ Ez0 þ Ez1 þ …; (32) Er ¼ Er0 þ Er1 þ …: Here, the unperturbed amplitudes H/0 ; Ez0 ; and Er0 coincide with the field amplitudes inside the plasma wall given by Eq. (7) found in the case of the stepwise density profile, and to 033509-13 Long-distance directed transfer of microwaves Phys. Plasmas 19, 033509 (2012) the first order in de, we get the following equations for the perturbations: 1@ ðrH /1 Þ ¼ ik0 ep Ez1 þ ik0 deEz0 ; r @r @Ez1 j2 h2 ¼ i 2 H/1 þ ide 2 H/0 : @r ep k 0 ep k 0 (33) As a result, we find the following first order corrections to the electromagnetic field amplitudes (7): ð h2 ð1Þ ð1Þ Ez C H0 ðj2 rÞ þ dedr H1 ðj2 rÞ ; ep j2 ð ep k 0 j2 ð1Þ ð1Þ dedr H0 ðj2 rÞ ; (34) H/ i C H1 ðj2 rÞ j2 ep and the E01 mode amplitudes in air r < R are given by Eq. (6). The boundary conditions at the air-plasma interface r ¼ R determine the amplitude C of the field in plasma, and the modified dispersion equation now becomes, instead of Eq. (11), ð1Þ 1 J1 ðj1 RÞ ep H1 ðj2 RÞ j1 J0 ðj1 RÞ j2 H ð1Þ ðj2 RÞ " 0 # ð1Þ 1 ep h2 J1 ðj1 RÞ H1 ðj2 RÞ ; ¼ d ep þ ep j1 j2 J0 ðj1 RÞ H ð1Þ ðj2 RÞ (35) where the right hand side determines the effect of the plasma density spread in the waveguide wall, and the characteristic Ð depth d has been defined as dedr ðep 1Þd. Under the conditions (13), effect of the plasma density spread can be easily estimated analytically. To the first order in small parameter j2 d, we find by analogy to used in the Subsection IV B procedure that the root of dispersion Eq. (35) is ldx ep T i d þ 1 in i : j2 R x R a In this section, we estimate the required parameters of UV laser radiation to transfer the MW up to about 1 km and discuss the availability of KrF laser for such a goal. According to the results of Sec. II, under the conditions of moderate laser intensity and suppressed electron attachment, the electron density in the atmospheric-pressure air in the field of UV laser radiation with intensity I and pulse duration sp can be estimated as ne K2 I2 sp þ K3 I3 sp : which can be solved in an implicit form 1 I 1 I0 þ K2 z; ¼ þ ai ln þ ai ln 1 þ ai I0 I 1 þ ai I I0 (36) With an increase in the plasma density, at the condition k0 R n T =x 1, we find the following correction to the attenuation coefficient (19): rﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃ! a2 n T n T 1 þ 2k0 d ; (38) Imh 2 3 2x 2x k0 R According to Eqs. (37) and (38), as well as one can readily see from Eq. (36), the effect of the plasma density spread at the air-plasma interface of the waveguide wall on the microwave signal propagation distance is sufficiently small at reasonable characteristic depths of this spread, j2 d 1. (40) (41) where ai ¼ K3 =K2 and I0 ¼ Iðz ¼ 0Þ is the initial value of laser pulse intensity. If REMPI process prevails, one can find the following intensity distribution along the propagation distance: IðzÞ ¼ In the low density limit Eq. (14), n T =x 1 and the corrected with respect to Eq. (18) attenuation coefficient now is " # a2 k0 d 2n T 3=2 : (37) Imh 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ x 2 k0 R3 2n T =x (39) The first term on the right hand side of this equation describes the (2 þ 1) REMPI of O2 molecules with the rate coefficient K2 and the second term corresponds to the direct MPI with the coefficient K3 . One can neglect the ionization of nitrogen molecules because of higher ionization potential of N2 which requires 4-photon ionization (instead of 3photon for O2 ). A depletion of laser pulse energy during its propagation caused by consumption for the ionization can be expressed by the equation dI ne ðIÞ ¼ 3h ; dz sp 0 h V. DISCUSSION I0 ; 1 þ 3hK2 I0 z (42) and if the direct MPI takes place, the distribution is IðzÞ ¼ I0 : ð1 þ 6hK3 I02 zÞ1=2 (43) It has been shown in Sec. II that in the intensity range I ¼ 3 108 7 1010 W/cm2, the REMPI dominates for KrF laser radiation. The corresponding rate coefficient determined from these experiments (sp ¼ 25 ns) is K2 100 cm/ (W2s). It is important that similar values of K2 were also measured24,25 for the intensities I 1011 W/cm2 in the short UV laser pulses of sp ¼ 35 ps and sp ¼ 0:4 ps, respectively. As we have found in Sec. IV, the propagation distance of MW radiation with the wavelength k ¼ 8 mm as high as 1 km can be achieved for EH11 mode in the plasma waveguide of a large radius R ¼ 30 cm and with the electron density in the waveguide wall ne ¼ 1013 cm3. To attain such concentration with a long laser pulse of splong ¼ 100 ns duration (which is typical for e-beam pumping and a free-running operation of powerful KrF lasers19), one needs laser intensity 033509-14 Zvorykin et al. I0long ¼ ðne =K2 splong Þ1=2 ¼ 109 W/cm2. It corresponds to energy fluence in the pulse Flong ¼ I0long splong ¼ 100 J/cm2 and total laser energy Qlong ¼ Flong S ¼ 19 kJ, where S ¼ 2pRDR ¼ 190 cm2 is a cross section area of the tubular laser beam of radius R ¼ 30 cm and ring width DR ¼ 1 cm. It was assumed that attachment of electrons to O2 molecules was completely suppressed. It is very difficult to obtain such high energy with KrF laser of a reasonable size. The attractive alternative is the use of a train of short laser pulses of the duration spshort ¼ 1 ps, which can effectively extract the pumping energy from KrF amplifier if a time interval between pulses exceeds the population inversion recovery time of the gain medium, that is, Dt 2 ns.19 The required electron density ne ¼ 1013 cm3 can be build up by such a train in which each individual pulse (1) effectively produces electrons via REMPI as its intensity I0short is significantly higher than I0long and (2) effectively revives the electrons which has been attached before by the oxygen molecules. During the same pumping time of KrF amplifier (100 ns), Np ¼ 50 pulses can be amplified having a peak intensity of I0short ¼ ðne =K2 Np spshort Þ1=2 ¼ 4:5 1010 W/cm2 and energy fluence in each pulse Fshort ¼ I0short spshort ¼ 45 mJ/cm2. The energy in the pulse Qshort ¼ Fshort S ¼ 8:5 J and the total energy in the train Np Qshort ¼ 425 J seem to be quite realistic. In the optimum regime of a short-pulse amplification (at the highest extraction efficiency), KrF amplifier produces output energy fluence Fopt ¼ Fs lnðg0 =ans Þ 5 6 mJ/cm2, where Fs 2 mJ/cm2 is a saturation fluence for KrF gain medium, g0 =ans 10 20; the ratio of the small-signal gain coefficient to the nonsaturated absorption coefficient slightly varies with the pumping conditions.19 To produce the required energy in the train of short pulses, the aperture of the supposed KrF laser should be A ¼ ðQshort =Fopt Þ1=2 40 cm. Several such single-shot KrF laser facilities have been already realized (see e.g., Ref. 19 and references therein). The Electra laser42 with a 30-cm aperture and up to 700-J output energy in a long 140-ns pulse is a continuously operating facility with a repetition rate of a few Hz being a medium-scale driver prototype for the inertial fusion energy.43 The characteristic depletion distance for short KrF laser pulses (at which its intensity becomes the half of initial value), estimated according to Eq. (42), is L ¼ ð3hK2 I0 Þ1 930 m, which is well comparable with the MW attenuation length. VI. CONCLUSION The results of theoretical and experimental study of the sliding mode of microwave radiation transfer in tubular UV laser-produced plasma waveguides in atmospheric air are presented. The mechanisms of air photoionization and relaxation under propagation of 25-ns pulses of KrF laser are investigated. The electron photodetachment from negative oxygen ions is shown to be capable of keeping the electron density for longer time. The sliding-mode microwave transfer is based on the effect of total reflection at the interface with an optically less dense medium and does not require Phys. Plasmas 19, 033509 (2012) high plasma density. The transfer of a microwave pulse with a wavelength k ¼ 8:5 mm to a distance of at least 60 m is experimentally demonstrated. The results of calculations are in good agreement with the experiment and convincingly demonstrate the advantage of the sliding-mode propagation in comparison with high-density plasma waveguides—the laser power inputs for a waveguide prove to be lower, and the range of the directed microwave radiation transfer increases with decrease in wavelength and reaches several kilometers for submillimeter waves. It is shown that the directed transfer of microwave radiation in laser-produced waveguides can be implemented using medium-scale ebeam-pumped KrF laser facilities. 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