Generation of RF by Atmospheric Filaments
Travis Garrett, Jennifer Elle, Michael White, Remington Reid, Alexander Englesbe, Ryan Phillips, Peter Mardahl, Andreas Schmitt-Sody, Erin Thornton, James Wymer, Anna Janicek, Oliver Sale
GGeneration of RF by Atmospheric Filaments
Travis Garrett, Jennifer Elle, Michael White, Remington Reid, Alexander Englesbe, Ryan Phillips, Peter Mardahl, Andreas Schmitt-Sody, Erin Thornton, James Wymer, Anna Janicek, and Oliver Sale Air Force Research Laboratory, Directed Energy Directorate, Albuquerque, NM 87123, USA Naval Research Laboratory, Plasma Physics Division, Washington, DC 20375, USA Leidos Innovations Center, Albuquerque, NM 87106, USA (Dated: February 26, 2021)Recent experiments have shown that femtosecond filamentation plasmas generate ultra-broadbandRF radiation. We show that a novel physical process is responsible for the RF: a plasma wake fielddevelops behind the laser pulse, and this wake excites (and copropagates with) a surface wave on theplasma column. The surface wave proceeds to detach from the end of the plasma and propagatesforward as the RF pulse. We have developed a four stage model of these plasma wake surface wavesand find that it accurately predicts the RF from a wide range of experiments.
The development of chirped pulse amplification [1] hasenabled an array of exotic physics [2–4], including atmo-spheric filamentation [5]. In turn filamentation producesa range of interesting and potentially useful effects, fromsupercontinuum generation [6, 7], pulses of THz radiation[8, 9] and ultra-broadband RF pulses [10], which haverecently been explored in great detail [11, 12]. Duringtypical λ = 800nm filamentation the laser source pro-vides more than 3GW of power so that Kerr self focusingcauses the core of the pulse to collapse [13, 14]. Whenthe laser pulse E field grows to roughly ∼ n e ∼ m − which counterbalances theKerr effect.This electron density corresponds to a plasma fre-quency of ω pl (cid:39) × rad/s and is closely linked tothe generation of the THz radiation. In the single colorTHz theory of [8] the freshly ionized electrons receivea ponderomotive push over the remainder of the laserpulse, thus exciting a coherent longitudinal current I z inthe plasma which is quickly damped by the high collisionfrequency. This short current pulse translates at c be-hind the laser pulse, thereby producing a conical shell ofradially polarized THz radiation.The RF radiation resembles the THz radiation in someways, and differs in others. It has a similar conical shellspatial profile and radial polarization, which indicatesthat a GHz scale longitudinal current pulse that trans-lates near c is also generated within the plasma. Howeverit also has a well defined broadband peak in the 5-15 GHzrange (and is not the tail of the THz radiation) and theRF amplitude grows strongly with decreasing pressure(see e.g. Fig. 5) [11]. A distinct physical mechanismthus must be responsible for the RF.Exploratory Particle In Cell (PIC) simulations [15, 16]provided the key insights that explain the RF generation.They revealed that a hot outer shell of electrons expandsoff the plasma into the surrounding atmosphere overroughly 50ps (as anticipated by [17]), and that the radialcurrent density J r of the expansion depends strongly on the electron-neutral collision frequency. Subsequent sim-ulations demonstrated that this Plasma Wake field (PW)excites a Surface Wave (SW) on the outer boundary ofthe plasma (namely, a long wavelength Surface PlasmonPolariton (SPP)[18, 19], with a speed approaching c ).Finally the SW was found to effectively detach from theend of the plasma, with most of the energy being thenconverted into a broadband RF pulse.The PWSW numerical model is split into 4 stages, aslength scales spanning 7 orders of magnitude need to beresolved. First the initial electron distribution function N e is determined. The experiments used a linearly po-larized, single color Ti:Sapphire laser source with a wave-length λ of 800nm, and a typical energy of 40 mJ. Thepulses were focused into a quartz cell to explore pres-sure dependence with a f /
60 mirror, which sets the beamwaist ω to 30 µ m. For this system the Keldysh parame-ter γ = ω laser (cid:112) m e I pot / ( q e E ) is close to 1, so we use thegeneral ionization rate W from [20], which is supportedby experimental data [21] near atmospheric pressure. At100 Torr about 2% of the O gas is ionized, and with a 50fs pulse length this corresponds to an ionization rate W ofroughly 10 s − , which occurs for electric field strengthsof E (cid:39) × V/m. This sets the rough length of theplasma column L pl to be ∼
27 cm, and the radius r pl to ∼ and much of the N isionized at a field strength of E (cid:39) × V/m, and thedimensions are reduced to L pl (cid:39)
13 cm and r pl (cid:39) . a r X i v : . [ phy s i c s . p l a s m - ph ] F e b clamping [13], and then a 100 µ m radial ramp down tozero density at the outer radius r pl .The velocity distribution is determined by our PICcode: given E ( (cid:126)x, t ) the air is ionized at the W rate [20]and the new electrons are accelerated through the re-mainder of the pulse [25]. In general the velocity distri-bution is influenced by both the details of strong fieldionization near γ = 1 (e.g. [26]), and by the deformationof the laser pulse during filamentation. For this work wefurther simplify and assume that the electrons are ion-ized with zero initial velocity and are then acceleratedby the remainder of a Gaussian pulse (with ˆ x polariza-tion and propagating in + z ). As a whole the initial N e is highly non-Maxwellian, and at 100 Torr has a peakkinetic energy K tail (cid:39) K avg (cid:39) . K tail (cid:39) K avg (cid:39) N e ,with periodic boundary conditions used in the longitu-dinal direction ˆ z (which is valid to leading order as theelectron velocities are a small fraction of c ) and simu-late the radial evolution with our PIC code. The Debyelength is quite small λ Debye (cid:39) ν eN depends on the cross sec-tions for O and N , which for our energies is roughly 10˚A . In turn the electron-ion momentum-transfer collisionfrequency is given by ν ei = 7 . × − n e ln(Λ C ) /K / eV ,where Λ C = 6 πn e λ Debye [28]. We find things are wellconverged with mesh resolutions of ∆ x = ∆ y = 2 µ m,and with macro-particle weights on the order of ∼
10 inthe low density outer radius region of the plasma. Theradial current density J r and electron density n e are thenmeasured as functions of radius and time.The electron number density for resulting simulationsof the PW at 100, 10, and 1 Torr over 100ps can beseen in Fig. 1. In these simulations the outer edge of theplasma at t = 0 has been given a step function profile,with an internal electron density of 10 m − at a plasmaradius of r pl = 0 . ∼ ∼ r > r pl ) is relatively insensitive to thedensity of the outer edge of the plasma (see Fig. 2), asopposed to the excited surface and volume plasmons.Analytic approximations provide a useful complementto the simulations. Consider the late time electron den-sity at a small distance r ∆ off the surface of the plasmaat radius r pl . Using Gauss’s law and approximatingln(( r pl + r ∆ ) /r pl ) as r ∆ /r pl for r ∆ (cid:28) r pl we find the thatthe number of electrons per unit length n L that escape Time, ps R a d i u s , mm
100 Torr
Time, ps10 Torr
Time, ps1 Torr E l e c t r o n D e n s i t y , m FIG. 1. Log plots of the electron density n e ( r, t ) showing theplasma wake variation at pressures of 100, 10, and 1 Torr.The plasma outer radius has been set to r pl = 0 . R a d i u s , mm Radial current density, n e = 10 m Time, ps R a d i u s , mm Radial current density, n e = 10 m R a d i a l c u rr e n t d e n s i t y , A / m FIG. 2. Log plots of the radial current density J r ( r, t ) at1 Torr, as it evolves in a transverse slice of plasma. Theinitial density at the edge of the plasma has been given astep function profile, with a value of n e = 10 m − in thetop plot and n e = 10 m − on the bottom. The profiles ofthe plasma wakes outside the column ( r > r pl = 0 . ∼
20 and ∼
60 GHz respectively, along with higher frequencyvolume plasmons in the interior. Over the length of a plasmawith density variations any higher frequency waves excited byinternal plasmons will tend to cancel out, while the ∼ to r ∆ scales as: n L ( r ∆ ) (cid:39) π(cid:15) K eV q e r pl r ∆ , (1)where K eV (cid:39) K tail is the kinetic energy of the hotelectrons that lead the escape. The radial electrostaticfield E r,stat that corresponds to n L has a simple form: E r,stat ( r ∆ ) = K eV /r ∆ . At 1 Torr K tail (cid:39) r ∆ (cid:39) µ m, giving E r,stat (cid:39) . × V/m. This allowsfor a rough estimate of the PW evolution timescale: anelectron with energy K tail = 16eV in an electric field ofthis magnitude follows a parabolic trajectory over 100ps.This compares well with the spread of electrons seen inFig. 1. The magnitude of the radial current density canlikewise be approximated as: J r (cid:39) v eff ( t ) q e n L πr pl r ∆ ; v eff ( t ) = min (cid:40) v tail v diff ( t ) (2)where the effective velocity v eff ( t ) is the minimum of thetail velocity and the Fickian diffusion speed v diff ( t ) =( K tail / ( m e νt )) / . At 1 Torr this gives J r (cid:39) × A/m . Finally we note that caution is needed for small r ∆ values: Eq.s (1) and (2) only hold up to densities thatare comparable to the original plasma edge density.We next switch from a transverse PIC simulation toa continuum FDTD-Drude model in a 2D axisymmet-ric coordinate system [29] that spans the length of theplasma. The Drude plasma model depends on boththe collision ν pl and plasma frequencies ω pl and is ex-pressed in the time domain with a polarization current: ∂ t (cid:126)J pl + ν pl (cid:126)J pl = (cid:15) ω pl (cid:126)E which is integrated into the codevia an auxiliary differential equation [30]. A mixed res-olution of ∆ r = 2 µ m and ∆ z = 50 µ m suffices to resolvethe SWs. The PW current profile J r is driven across thesurface of the plasma column in the longitudinal direc-tion at the speed of light. In this work only the externalradial currents J r>r pl are used to drive the SW and sub-sequent RF. The higher frequency volume plasmons canalso excite waves in neighboring lower density plasma,but we suspect that including fine scale plasma struc-ture will act as surface roughness [31] and cause these tocontinuously detach.As expected the PW current excites a broadband sur-face wave with similar frequency content. An exampleSW near the midpoint of the plasma column for a 10 Torrsimulation can be seen in Fig. 3. The frequency ω of thewave is considerably lower than the limit SPP frequencyand it thus travels at velocity c ((2 ω − ω pl ) / ( ω − ω pl )) / which approaches the speed of light. The resulting phase-matched copropagation leads to a steady growth in thesurface wave intensity (see also [32, 33]).The SW well approximated by the Sommerfeld-Goubau [34, 35] solution for SPPs on a cylinder of fi-nite conductivity. The external radial component of theelectric field E r has the form: E r ( r, z, t ) = − πr pl E r outer e i ( ωt − hz ) H (1)1 ( r (cid:112) k − h ) , (3)for waves of frequency ω propagating in the + z direction,with an amplitude of E at the surface. The complex h wavenumber describes the SW wavelength and attenua-tion length scale, k is the free space wavenumber k = ω/c , H (1)1 is a Hankel function of the first kind with order 1,and r outer = 1 / |√ k − h | . For radii r pl < r < r outer the Hankel function with complex argument is approxi-mately H (1)1 (cid:39) − r outer / ( πr ), and for larger radii it fallsoff exponentially (as is typical for SPPs). Length, cm R a d i u s , c m
10 Torr Surface Wave E l e c t r i c F i e l d S t r e n g t h , V / m FIG. 3. Log plot of the electric field strength for a 10 Torrsurface wave near the midpoint of the plasma column. Thesecond primary lobe of the final RF pulse develops as the SWdetaches from the end of the plasma.
Sommerfeld and Goubau developed iterative numericaltechniques for calculating h , and recent work [36, 37] hassimplified the solution through the use of the Lambert W function. Given a typical plasma conductivity σ acomplex ξ can be defined: ξ = − ke − γ euler (cid:114) (cid:15) ω σ r pl (1 + i ) , (4)(where γ euler is the Euler-Mascheroni constant), and h isthen given by: h = (cid:115) k + 4 ξe − γ euler r pl W ( ξ ) , (5)where the first branch of W is used. At 1 Torr the plasmacolumn roughly has a radius r pl (cid:39) . σ (cid:39) × S/m, and thus for a 10 GHzSW we find h (cid:39)
210 + 3 . i . The dispersive SW thustravels at roughly 0 . c , attenuates by a factor of 1 /e every 30cm (if not being actively excited), and has a 1 /r transverse envelope out to about r outer (cid:39) . I along an antenna of length L willradiate electromagnetic waves at a wavelength λ withpower: P Dipole = ( πη / I L /λ . In our case we have aradial current distribution of approximate length r ∆ thatis normal to a conducting cylinder of radius r pl . Whenthere is a separation of length scales r ∆ (cid:28) r pl (cid:28) λ andan overall radial current of I r then the resulting powerthat is primarily radiated into SWs is: P SW (cid:39) πη I r r r / pl λ / , (6)(where we have simplified with λ = 2 π/ R e( h )). We notethat the dimensionless term r / ( r / pl λ / ) (compare to L /λ for the Hertzian dipole), has different asymptoticscalings in other length scale regimes. Equation (6) ap-plies both to a stationary radial antenna and the ± z wavetrains it emits, and to our case, where the antennatranslates at c with its comoving SW pulse.The total effective current I r ( z ) at a longitudinal posi-tion z is found by integrating the current density J r from0 to z , and absent dissipation the amplitude E of theSW in (3) would grow linearly (for constant J r ). Howeverthere is dissipation due to the finite conductivity of theplasma, and the attenuation coefficient α = I m( h ) is sim-ilar in magnitude to 1 /L pl . The SW thus grows linearlyinitially and then saturates after several 1 /α , and the ef-fective total radial current after propagating a distance z is: I r ( z ) = 2 πr pl J r − e − αz α . (7)We express the amplitude of the final SW in terms of itslongitudinal current, as I z = 2 πr pl E /η . Integrating (3)to find the SW peak power and setting this equal to (6)we find: I z ( z ) = 4 π J r r ∆ (120 ln( r outer /r pl )) / − e − αz α (cid:16) r pl λ (cid:17) / . (8)At 1 Torr this expression leads to a final magnitude for I z at the end of the filament of roughly 0 . .
2A at 10 Torr, and about 0 .
05A at 100 Torr,which are all fairly close to the measured currents in thecorresponding simulations.When the surface wave reaches the end of plasma themajority of the energy detaches and propagates forwardas the RF pulse, with the reflected wave decaying as ittravels back down the plasma. In this way the systemloosely resembles a surface wave end-fire antenna [38, 39].The second lobe of the primarily one cycle RF pulse formsas the SW separates from the plasma, with some varia-tion in the profile dependent on the abruptness of thedetachment. The recorded I z ( z, t ) currents from the SWsimulation are used in a ∼ laboratory scale axisym-metric FDTD model, with resolution ∆ r = ∆ z = 0 . V pp viaa frequency dependent calibration factor C E/V ( f GHz ) (cid:39)
50 + 10 f GHz (which captures the horn antenna response[11]). The analytic approximations can also be used toconstruct these voltage profiles. We adapt the equationfor energy spectral density radiated per unit solid angle derived for THz radiation [8]: d Udωd
Ω = | I z ( ω ) | sin θ π(cid:15) c (1 − cos θ ) sin (cid:18) L pl ω c (1 − cos θ ) (cid:19) , (9)to our GHz scale surface current. The average ampli-tude of I z (8) over the length of the plasma is appliedto a ∼ V pp . Length, cm R a d i u s , c m Far field RF, 10 Torr E l e c t r i c F i e l d S t r e n g t h , V / m FIG. 4. Log plot of electric field strength for the far field RFpulse in a laboratory scale simulation at 10 Torr. Residualelectric fields along the plasma column can also be seen inthe bottom left. V pp (V)0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 (deg.) RF voltage, laboratory data V pp (V)0.0 0.2 0.4 0.6 0.8 1.0 (deg.) RF voltage, simulated and Eqn. (9)
FIG. 5. Polar plots of the detector peak to peak Voltage V pp as a function of angle and pressure as measured in the lab(left), and as calculated by both the simulation framework(right, triangles), and by the associated analytic approxima-tions (right, × ticks) that culminate in (9). In general there is excellent agreement between theoverall shape and pressure dependence of the simulatedRF as compared to the laboratory measurements. Theabsolute magnitude of the simulated RF is a bit lowerthan the lab data for the runs shown in Fig. 5, butis within the errors for both the simulations and inthe experimental data. Subtle changes in Ti:Sapphirelaser alignment generates ±
50% amplitude variations ata given pressure (with the spatial and frequency contentbeing much more stable). It is suspected that the slightlaser pulse variations are amplified during filamentation,leading to fine scale changes in the plasma radius, density,and electron velocities. Plausible variations in the sameparameters lead to comparable changes in the simulatedRF amplitude as well.We have demonstrated that a plasma wake field thatexcites a surface wave is responsible for the generation offilament RF. The rich internal dynamics of the plasmamerits further research: we expect that stochastic higherfrequency RF may inform on the fine scale structure ofthe plasma and the nonlinear optics that generate it. Theionization model needs to be expanded to include morephysical effects (e.g. [26, 40]), and we plan on measuringfilament RF from Argon and Krypton to better charac-terize strong field ionization in the intermediate γ = 1regime. The plasma can also be pushed into the stronglycoupled regime with harder focusing, and we intend tomodel resulting RF with the use of [41]. Finally we notethat it may be possible to recreate the core mechanism –an antenna that copropagates with the wave that it ex-cites – in other settings and thereby drive strong coherentradiation.The authors thank Edward Ruden, Serge Kalmykovand Charles Armstrong for useful discussions, and the AirForce Office of Scientific Research (AFOSR) for supportvia contracts [1] D. Strickland and G. Mourou, Compression of amplifiedchirped optical pulses, Optics communications , 219(1985).[2] J. Faure, Y. Glinec, A. Pukhov, S. Kiselev, S. Gordienko,E. Lefebvre, J.-P. Rousseau, F. Burgy, and V. Malka, Alaser–plasma accelerator producing monoenergetic elec-tron beams, Nature , 541 (2004).[3] X. Liu, D. Du, and G. Mourou, Laser ablation and mi-cromachining with ultrashort laser pulses, IEEE journalof quantum electronics , 1706 (1997).[4] A. Di Piazza, C. M¨uller, K. Hatsagortsyan, and C. H.Keitel, Extremely high-intensity laser interactions withfundamental quantum systems, Reviews of ModernPhysics , 1177 (2012).[5] A. Braun, G. Korn, X. Liu, D. Du, J. Squier, andG. Mourou, Self-channeling of high-peak-power femtosec-ond laser pulses in air, Optics letters , 73 (1995).[6] A. Brodeur and S. Chin, Ultrafast white-light continuum generation and self-focusing in transparent condensedmedia, JOSA B , 637 (1999).[7] V. Kandidov, O. Kosareva, I. Golubtsov, W. Liu,A. Becker, N. Akozbek, C. Bowden, and S. Chin, Self-transformation of a powerful femtosecond laser pulse intoa white-light laser pulse in bulk optical media (or su-percontinuum generation), Applied Physics B , 149(2003).[8] C. Amico, A. Houard, S. Akturk, Y. Liu, J. Le Bloas,M. Franco, B. Prade, A. Couairon, V. Tikhonchuk, andA. Mysyrowicz, Forward thz radiation emission by fem-tosecond filamentation in gases: theory and experiment,New Journal of Physics , 013015 (2008).[9] V. Andreeva, O. Kosareva, N. Panov, D. Shipilo,P. Solyankin, M. Esaulkov, P. G. de Alaiza Mart´ınez,A. Shkurinov, V. Makarov, L. Berg´e, et al. , Ultrabroadterahertz spectrum generation from an air-based filamentplasma, Physical review letters , 063902 (2016).[10] B. Forestier, A. Houard, M. Durand, Y.-B. Andr´e,B. Prade, J.-Y. Dauvignac, F. Perret, C. Pichot, M. Pel-let, and A. Mysyrowicz, Radiofrequency conical emissionfrom femtosecond filaments in air, Applied Physics Let-ters , 141111 (2010).[11] A. Englesbe, J. Elle, R. Reid, A. Lucero, H. Pohle,M. Domonkos, S. Kalmykov, K. Krushelnick, andA. Schmitt-Sody, Gas pressure dependence of microwavepulses generated by laser-produced filament plasmas, Op-tics Letters , 4953 (2018).[12] A. Janicek, E. Thornton, T. Garrett, A. Englesbe, J. Elle,and A. Schmitt-Sody, Length dependence on broadbandmicrowave emission from laser-generated plasmas, IEEETransactions on Plasma Science , 1979 (2020).[13] A. Couairon and A. Mysyrowicz, Femtosecond filamen-tation in transparent media, Physics reports , 47(2007).[14] S. L. Chin, Femtosecond laser filamentation , Vol. 55(Springer, 2010).[15] C. K. Birdsall and A. B. Langdon,
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