Meter-Scale, Conditioned Hydrodynamic Optical-Field-Ionized Plasma Channels
A. Picksley, A. Alejo, R. J. Shalloo, C. Arran, A. von Boetticher, L. Corner, J. A. Holloway, J. Jonnerby, O. Jakobsson, C. Thornton, R. Walczak, S. M. Hooker
MMeter-Scale, Conditioned Hydrodynamic Optical-Field-Ionized Plasma Channels
A. Picksley, A. Alejo, R. J. Shalloo, C. Arran, A. von Boetticher, L. Corner, J. A.Holloway, J. Jonnerby, O. Jakobsson, C. Thornton, R. Walczak, and S. M. Hooker ∗ John Adams Institute for Accelerator Science and Department of Physics,University of Oxford, Denys Wilkinson Building,Keble Road, Oxford OX1 3RH, United Kingdom Cockcroft Institute for Accelerator Science and Technology,School of Engineering, The Quadrangle, University of Liverpool,Brownlow Hill, Liverpool L69 3GH, United Kingdom Central Laser Facility, STFC Rutherford Appleton Laboratory, Didcot OX11 0QX, United Kingdom (Dated: September 1, 2020)We demonstrate through experiments and numerical simulations that low-density, low-loss, meter-scale plasma channels can be generated by employing a conditioning laser pulse to ionize the neutralgas collar surrounding a hydrodynamic optical-field-ionized (HOFI) plasma channel. We use particle-in-cell simulations to show that the leading edge of the conditioning pulse ionizes the neutral gascollar to generate a deep, low-loss plasma channel which guides the bulk of the conditioning pulseitself as well as any subsequently injected pulses. In proof-of-principle experiments we generateconditioned HOFI (CHOFI) waveguides with axial electron densities of n e0 ≈ × cm − ,and a matched spot size of 26 µ m. The power attenuation length of these CHOFI channels is L att = (21 ±
3) m, more than two orders of magnitude longer than achieved by HOFI channels.Hydrodynamic and particle-in-cell simulations demonstrate that meter-scale CHOFI waveguideswith attenuation lengths exceeding 1 m could be generated with a total laser pulse energy of only1 . I. INTRODUCTION
Many fields exploiting laser-plasma interactions re-quire the laser pulse to propagate through plasma formany Rayleigh ranges, which means that the laser pulsemust be guided via relativistic self-focusing effects [1–3]or by an external waveguide. Examples of such appli-cations include Raman amplification [4], high-harmonicgeneration in ions, x-ray lasers [5], and laser-drivenplasma accelerators [6–8]. The last of these is particu-larly challenging since the multi-GeV accelerator stagesneeded to drive compact light sources, or future parti-cle colliders [9, 10], require guiding of laser pulses withpeak intensities of order 10 W cm − over distances inthe range 0 . cm − .Many methods for guiding high-intensity laser pulseshave been investigated. These include grazing-incidenceguiding in capillaries [11], and many varieties of plasmachannels generated by hydrodynamic expansion [12–15],capillary discharges [16–18], Z-pinches [19, 20], open-geometry discharges [21], and laser-heated capillary dis-charges [22, 23]. To date, the most successful approachesfor driving laser-plasma accelerators are capillary dis-charges and its laser-heated variant, which have beenused to generate electron beams with energies up to4 . . ∗ [email protected] damage to the capillary structure, which would limit theworking lifetime of the waveguide in future high repeti-tion rate plasma accelerators.Hydrodynamic plasma channels [12, 13] have the ma-jor advantage of being free-standing, and hence immuneto laser damage. In this approach a column of plasmais created, and heated collisionally, by a picosecond-duration laser pulse. Rapid radial expansion of theplasma column drives a strong shock wave into the sur-rounding gas, forming a plasma channel — i.e. a re-gion of radially-increasing electron density — betweenthe axis and the shock front. In order to achieve suf-ficiently rapid heating the initial density must be high,which limits the on-axis density of the subsequent chan-nel to n e0 (cid:38) × cm − .We recently proposed [26] that optical field ionizationcould generate much lower density channels since thetemperature to which it heats the ionized electrons is in-dependent of plasma density. In previous work [27, 28] wehave generated these hydrodynamic optical-field-ionized(HOFI) plasma channels with lengths as long as 100 mm,with axial densities as low as n e0 (cid:46) × cm − , andat repetition rates of up to 5 Hz.A drawback of HOFI channels is that they are shal-lower than collisionally-heated hydrodynamic channels,since the energy per unit length deposited in the initialplasma column is nearly an order of magnitude lower[29], and since the channel depth is roughly proportionalto the axial density [27]. As a consequence, the powerattenuation lengths achieved to date in HOFI channels[28] are of order L att (cid:46)
100 mm, which is too short for a r X i v : . [ phy s i c s . acc - ph ] A ug many applications.In the present paper we build on a phenomenon ob-served in our 2018 experiments on HOFI channel for-mation with axicon lenses [27]: the transverse extent ofthe plasma immediately after the passage of the guidedpulse is greater than before that pulse arrives. This sug-gests that the neutral species surrounding the plasmacolumn are ionized by the electric fields of the guidedpulse leaking through the channel walls [30]. In this pa-per we present that data in detail. We show, throughexperiments and simulations, that guiding a “condition-ing pulse” in a HOFI channel can create a deep, thick-walled plasma channel by ionization of the collar of neu-tral gas which surrounds the initial HOFI channel. Wepresent measurements showing the evolution of the trans-verse electron density profile of these conditioned HOFI(CHOFI) channels, and demonstrate the formation of16 mm-long CHOFI channels at a repetition rate of 5 Hz(limited only by the repetition rate of the laser system),with power attenuation lengths of ∼
20 m. We use hy-drodynamic and particle-in-cell (PIC) simulations to un-derstand the formation of the CHOFI channels, and todemonstrate that they could be generated at the metrescale with modest laser pulse energies.We note that this work is closely related to ear-lier work by Spence and Hooker [17], who investigated‘quasi-matched’ guiding in which ionization of a partially-ionized parabolic plasma channel by the leading edgeof an intense laser pulse creates a fully-ionized plasmachannel which can guide the bulk of the pulse with lowlosses. More recently, Morozov et al. [31] explored sim-ilar effects to those reported here in their investigationof ionization-assisted guiding in plasma channels formedin 3 . × cm − . An alternative approach in whichthe neutral gas collar is ionized by a coaxial high-orderBessel beam has also recently been described [32]. II. EXPERIMENTAL SETUP
The experiments were undertaken with the 5 Hzrepetition-rate Astra-Gemini TA2 laser at the CentralLaser Facilty, UK. The set-up employed has been de-scribed previously [27], and is described in detail in theSupplemental Material, so here we provide only an out-line.The compressed, collimated beam from the Astra-Gemini TA2 laser was split into a channel-forming beamand a guided or “conditioning” beam by an annulardielectric mirror. The channel-forming beam reflectedfrom this mirror was directed to a retroreflecting delaystage, focused by a fused silica axicon lens of base angle ϑ = 5 . × W cm − .The conditioning pulse transmitted through the annu- Probe BeamTo interferometer Channel-forming Beam Conditioning Beam from OAPTo forward diagnostics AxiconGas Cell Holed Mirror(a)
Channel-Forming (b)-200 0 200∆ z ( µ m)-200-1000100200 y ( µ m ) Both Pulses (c)-200 0 200∆ z ( µ m) Conditioning (d)-200 0 200∆ z ( µ m) 0.00.20.40.6 φ (r a d ) Figure 1. (a) Schematic of the experimental interaction re-gion. (b-d) Phase shifts measured by the transverse probebeam at z ≈ . τ = 1 . lar mirror was sent to a retro-reflecting delay stage andfocused through the hole in the turning mirror by an off-axis paraboloid (OAP) mirror of focal length f = 750 mmto the entrance pinhole of the gas cell. The measuredbeam waist and Rayleigh range of the conditioning beamwere w = (22 ± µ m and z R = (1 . ± .
4) mm respec-tively. The conditioning pulse could be operated at twointensities: (i) high-intensity, with a peak axial intensityat focus of I highpeak = 4 . × W cm − ; and (ii) low inten-sity, with a peak intensity of I lowpeak ≈ . × W cm − ,achieved by replacing one of the mirrors in the condi-tioning pulse beamline with an uncoated, optically flatwedge.The channel-forming and conditioning pulses enteredand left the gas cell via entrance and exit pinholes ofdiameter 1 . µ m respectively, spaced by16 mm. The cell pressure could be varied in the range5 mbar to 120 mbar.Optically flat fused silica windows mounted on thesides of the cell allowed optical access by a separatelycompressed, 1 mJ, 800 nm probe pulse used for transverseinterferometry of the plasma channels. The delay τ be-tween the arrival of the channel-forming beam and theprobe beam could be varied in the range 0 ns to 6 ns by adelay stage in the probe beam line. A Keplerian telescopemagnified the transmitted probe beam by a factor of 6,and a Mach-Zehnder interferometer located within thetelescope generated a fringe pattern, which was recordedby a 12-bit CMOS detector located at the image planeof the telescope. The measured spatial resolution in theobject plane of this interferometer was (8 . ± . µ m. r ( µ m)00.511.52 n e ( c m − ) ∆ τ = 0 . n e ( c m − ) r ( µ m)00.511.5 0.8 ns(f) n e ( c m − ) r ( µ m) 1.0 ns(g) 0 50 100 150 r ( µ m) 1.5 ns(h) 0 50 100 150 r ( µ m) 2.0 ns(i)1 Figure 2. Formation and temporal evolution of the transverse electron density profiles n e ( r ) in the CHOFI waveguide. (a)Measured electron density profile immediately before (dashed, from [27]) and after (solid) the arrival of the conditioning pulseat τ = 1 . τ = 0 ns). The horizontal, grey line shows the density corresponding to full ionization of the ambient gas. (b)-(i)Electron density profiles measured at different additional delays ∆ τ (indicated) after the arrival of the channel-forming pulse,for a conditioning pulse arriving at τ = 1 . z ≈ . .
75 mm. The shaded region is one standard deviation wide.
A mode imaging system was used to record the exitmode of the conditioning pulse, and the energy transmis-sion of this pulse was measured by directing a fraction ofthe transmitted beam to a pyroelectric energy meter.
III. EXPERIMENTAL RESULTS
Figure 1(b-d) captures the moment when the condi-tioning pulse arrives, at τ = 1 . z = 0, so that the region of posi-tive z contains plasma generated by the channel-formingpulse alone, whereas the region ∆ z < τ = 1 . r shock ≈ µ m, cre-ated by the channel-forming pulse. However, at largerradii the electron density is increased substantially to form a deep, thick-walled CHOFI plasma channel: thedepth of the channel was increased by a factor of 10to ∆ n e = (1 . ± . × cm − ; whereas, within theexperimental uncertainty, the axial density remained at n e0 = (2 . ± . × cm − . The radial extent of theplasma was increased by the conditioning pulse from r shock ≈ µ m to r max ≈ µ m.As shown in Fig. 2, the electron density in the region r > r shock is comparable to that expected for full ion-ization of the initial ambient gas, consistent with fieldionization of the neutral gas by the transverse wings ofthe conditioning pulse. It is noticeable that in the regionclose to r ≈ µ m the electron density is greater thanthat which would be generated by ionization of the neu-tral gas at its ambient density; this is consistent with abuild up of neutral gas at the shock front as it pushes thegas outwards.The evolution of the plasma at various delays ∆ τ af-ter the arrival of the conditioning pulse at τ = 1 . τ = 0 . τ , electrons in the high density ring spread radiallyoutwards and inwards, which causes the position of thepeak density to increase to r = 100 µ m by ∆ τ ≈ τ ≈ τ . The evo-lution of the total number of electrons per unit lengthof waveguide N e = 2 π (cid:82) r max n e ( r ) r d r , is shown in in Fig.3(a). Upon arrival of the conditioning pulse, N e increasedfrom N e = 7 . × cm − to N e = 5 . × cm − in- N e ( c m − ) (a)02468 n e ( c m − ) (b)0 0.5 1 1.5 2∆ t (ns)10203040 w m ( µ m ) (c)0 0.5 1 1.5 2∆ t (ns)10 − L a tt ( m ) (d)1 Figure 3. Temporal evolution of the properties of the CHOFIplasma channels as a function of delay ∆ τ after the arrivalof the conditioning pulse: (a) Measured total number of elec-trons per unit length of waveguide, N e ; (b) the measured axialdensity n e0 ; (c) the calculated matched spot size w m of thelowest order mode; and (d) the calculated attenuation lengthof the lowest order channel mode. The white circles show thesame parameters of the HOFI channel immediately before theconditioning pulse arrives, indicated by the red region. In allcases, the error bars reflect the uncertainties in n e ( r ) depictedin Fig. 2. stantaneously. Following this, the number of electronsper unit length increased for the first 0 . . τ ≈ . et al. [34],and the power attenuation lengths L att of these modeswere calculated by solving the paraxial Helmholtz equa-tion [28]. Figure 3(c) shows the evolution of the matchedspot size w m . It can be seen that w m remains close tothat of the HOFI channel, which was measured to be w m , HOFI = (24 ± µ m, up to delays of ∆ τ ≈ . w m = (35 ± µ m over thenext nanosecond. The effect of the conditioning pulse µ m (a) z = 0 mm (b) z = 16 mm Low Intensity (c) z = 16 mm High Intensity F l u e n ce ( n o r m . ) I peak (10 W cm − )0.20.40.60.81.0 E n e r g y T r a n s m i ss i o n (d) 1 Figure 4. Properties of the transmitted conditioning pulse.The transverse fluence profile of the conditioning pulse at theinput plane of the gas cell (a), and at the exit of the cell for thelow- (b) and high-intensity (c) configurations. (d) the pulseenergy transmission T of the conditioning pulse as a functionof input input I peak . on the propagation losses is striking: immediately beforethe arrival of the conditioning pulse, L att = (15 ±
8) mm,the relatively large uncertainty arising from the smallfringe shifts ( φ (cid:46)
100 mrad) generated by the HOFIchannel [27]. Immediately after the arrival of the con-ditioning pulse, L att = (0 . ± .
01) m. In the following200 ps, the increase in N e increases the thickness of thechannel wall, further increasing the attenuation length.The largest L att was achieved at ∆ τ = 1 . L att = (21 ±
3) m, almost four orders of magnitude largerthan achieved by the channel-forming pulse alone.Figure 4 shows the transverse fluence profile of the low-and high-intensity conditioning pulses at the entranceand exit of the 16 mm-long gas cell for P = 40 mbarand τ = 1 . . z R . Forthe low-intensity case, the measured energy transmis-sion and output spot radius was T = (21 ±
4) % and w out = (23 ± µ m respectively. For the high-intensityconfiguration, T = (60 ±
3) % and w out = (25 ± µ m.It is clear that the mode profiles of the transmitted pulseswere essentially the same for the two cases. However, theenergy transmission was substantially higher for the high-intensity pulse. Figure 4(d) shows the measured varia-tion of T as the peak input intensity of the pulse wasincreased, keeping the other parameters constant. It isclear that T increases substantially as the intensity of thepulse is increased. This suggests that the leading edge ofthe conditioning pulse improved the channel propertiesvia ionization, as in Fig. 2, allowing the body of the pulseto propagate with low losses. The coupling efficiency ofthe conditioning pulse, deduced from the overlap inte-gral of the input spot with the calculated HOFI chan-nel modes is estimated to be T (0) = 65 %, and henceat the highest input intensities the propagation losses ofthe conditioning pulse were low. It should be noted thatsince the channel-forming beam and conditioning beamenergies could not be varied independently, the intensityof the channel-forming pulse also varied with the inten-sity of the conditioning pulse. However, interferomet-ric measurements indicated that properties of the HOFIchannel created by the channel-forming pulse were notsignificantly affected for the intensity range consideredhere. IV. HYDRODYNAMIC SIMULATIONS
Hydrodynamic simulations were undertaken to under-stand in detail the distribution of plasma and neutralgas prior to the arrival of the conditioning pulse. Thesewere performed in two dimensions, using the Euleriancode FLASH [35]. A 3-temperature model was employed,which allowed for independent evolution of the electron,ion, and radiation species. The simulations included en-ergy diffusion, thermal conductivity, heat exchange be-tween electrons and ions and atoms, and radiation trans-port. Tabulated values for the equations of state wereemployed. The initial conditions of the simulations werea plasma column with a super-gaussian transverse elec-tron temperature surrounded by neutral atomic hydrogenat a pressure of 50 mbar and a temperature of 298 K.Figure 5 summarizes the results of these simulations.Figure 5(a) shows, for a delay τ = 1 . Z , the density of neutralhydrogen n H , and the electron temperature. The forma-tion of a high-density shock front is clearly observed, asexpected for a Sedov-Taylor-like expansion. Also evidentis significant cooling of the plasma as it expands. Forthis delay the plasma cools on-axis from 10 eV to 0 .
44 eV,which results in the fractional ionization decreasing from¯ Z (cid:39)
100 % to ¯ Z (cid:39)
44 %. The cooling is even more pro-nounced at the shock front, where T e,shock (cid:39) . − . Z (cid:39)
15 %.Figure 5(b) shows the temporal evolution of the elec-tron and neutral gas transverse density profiles. Al-though the initial temperature is high enough to allowfor full ionization of the plasma column, a neutral gascollar appears in the early stages of the expansion. Thesimulation shows the number of free electrons to remainapproximately constant throughout the expansion (seeSupp. Material), indicating that the accumulation ofneutral gas in the region of the shock is caused by thehigh pressure in the inner regions of the the channel.
V. PROPAGATION SIMULATIONS
Particle-in-cell (PIC) simulations were performed inquasi-3D cylindrical geometry using FBPIC [36] to pro-vide insight into the formation of CHOFI channels, and − − −
10 0 10 20 30 x ( µ m) − − − y ( µ m ) t = 1.5 ns (a) . e V . e V . e V . e V . . . Z n H (10 cm − )0 10 20 30 40 50 r ( µ m)0 . . . . . . . n ( c m − ) × (b) n e n H τ ( n s ) Figure 5. Hydrodynamic simulation of a HOFI channel. (a)Properties of the plasma channel at τ = 1 . x < µ m is shown the ionization fraction of the hydrogenatoms ( ¯ Z ), and in the region x > µ m the neutral hydro-gen density ( n H ). Superimposed on these plots are contoursof the electron temperatures T e ; the contours are spaced by0 . T e = 0 . n e , straight) and neutral hydrogen( n H , dashed) transverse density profiles at various delays τ given by the color scale. to explore the prospects for generating metre-scale chan-nels. We note that use of a PIC code means that pon-deromotive effects are included. The PIC simulationsused the transverse electron density and neutral densityprofiles calculated by the FLASH simulations for a delay τ = 1 . (a) 1.96 2.00 z (mm)-150-100-50050100150 y ( µ m ) (b)101.76 101.80 101.84 z (mm) (c)321.36 321.40 321.44 z (mm) 0.00.51.01.52.02.5 n e ( c m − ) I ( W c m − ) Figure 6. Transverse profiles of the electron density and the laser intensity when the peak of the conditioning pulse, indicatedby the dashed line, has reached (a) z = 2 mm, (b) z = 101 mm, and (c) z = 321 mm. The input intensity of the conditioningpulse was I peak = 6 . × W cm − . closely matching the experimental values.Figure 6 shows the transverse electron density andlaser intensity profiles at three points in the channelfor a conditioning pulse with a peak input intensity I peak = 6 × W cm − . Close to the channel entrance,the far leading edge of the pulse ionizes the neutral gassurrounding the HOFI channel, creating a deep CHOFIchannel in which the main body of the pulse is guided.The position at which ionisation first occurs correspondsto the position at which I ( r, z ) first exceeds I th where I th , is the threshold laser intensity for ionization [26].Since the transverse intensity profile of the condition-ing pulse is not perfectly matched to that of the lowest or-der mode of either the HOFI or CHOFI waveguides, thespot-size of the conditioning pulse oscillates by ±
10 %during the first few centimetres of propagation. How-ever, this variation primarily affects the wall thickness ofthe CHOFI channel, not its depth or matched spot size.Further, the higher-order modes excited by the condi-tioning pulse are attenuated with propagation, and theconditioning pulse adopts a stable configuration. Thissituation is shown in Fig. 6(b-c), which shows that theradial extent of the additional ionization is reduced withpropagation distance z . However, the high walls of theCHOFI channel remain, and are sufficiently thick, to en-sure low-loss propagation of the bulk of the condition-ing pulse. The conditioning pulse continues to propagatewith low loss, and to generate a low-loss channel, un-til it can no longer ionize the neutral gas close to theshock front. Some temporal compression of the condi-tioning pulse is also evident, caused by leaky guiding ofthe leading edge in the HOFI channel and loss of energyto ionization, but this also has little effect on the gener-ated CHOFI channel. We note that, close to the channelentrance, a plasma wakefield with a relative amplitude of δn e /n e = 10 % is driven by the conditioning pulse. Thiswakefield has no observable effect on the properties of theCHOFI channel, and in any case is expected to decay ina time [37] which is much shorter than the timescale ob-served in Fig. 2 for significant evolution of the transverseelectron density profile. To summarize, the CHOFI chan-nel is relatively stable to variations in the properties of z (mm)0.20.40.60.81.0 T ( z ) I peak = 6 × W cm − I peak = 6 × W cm − Figure 7. Calculated pulse energy transmission T ( z ) of the conditioning laser pulse as a function of propagation distance z , for peak input intensities of 6 × W cm − (dashed, red)and 6 × W cm − (solid, black). the conditioning pulse as it propagates, as well as to adegree of mis-matching to the HOFI channel.Figure 7 shows the pulse energy transmission T ( z ) ofthe conditioning pulse as a function of propagation dis-tance z , for various input intensities. For intensities lowerthan ∼ × W cm − , the conditioning pulse is not in-tense enough to ionize the neutral collar of gas, and henceit propagates in the HOFI channel and experiences highlosses. For I peak = 6 × W cm − , only the peak of thepulse is intense enough to ionize the neutral collar andgenerate a deep, thick channel. Thus, the leading edgeetched away at a significantly rapidly, and the length ofthe CHOFI channel is limited to ≈
100 mm. In contrast,the highest intensity conditioning pulse shown in Fig. 7propagates with low losses over the length of the sim-ulation. The calculated energy loss of the conditioningpulse is ∼ . − , and hence the energy loss is dominated byetching of the leading edge in the leaky HOFI waveguide,and to driving a wakefield.The guiding properties of the CHOFI channels gener-ated by the channel-forming and conditioning pulses werecalculated by the numerical propagation code. First, thelowest-order mode was found by numerical propagationof an initially un-matched beam until the higher-ordermodes had been damped. Second, the power attenua-tion length was found by numerical propagation of thismode through the entire length of the CHOFI channel,including longitudinal variations in transverse profile ofthe CHOFI channel. For the conditions of Fig. 6, theattenuation length was found to be L att = (2 . ± .
1) m,which is nearly three orders of magnitude higher thanthat of the unconditioned HOFI channel.
VI. DISCUSSION
The experimental and simulation results above demon-strate that the leading edge of a conditioning laser pulsecan ionize the neutral gas surrounding HOFI plasmachannels to form a conditioned HOFI channel which canguide the bulk of the conditioning pulse, and any subse-quently injected laser pulse, with very low losses. Thismechanism is likely to have played a role in our recentexperimental demonstration of guiding of high-intensitylaser pulses in 100 mm long channels [28]. The simu-lations shown in Fig. 6 demonstrate the generation ofCHOFI channels up to 325 mm long, limited by erosion ofthe leading edge of the channel-forming pulse and pumpdepletion to the wakefield. We note that since the lossesof the CHOFI channels are so low, it would be straight-forward to extend the channel beyond this limit by em-ploying a longer channel-forming pulse, or by using twoor more channel-forming pulses.Control over the axial density and matched spot size ofCHOFI channels is in principle possible by adjusting thedelay τ between the channel-forming and conditioningpulses, and the initial gas density. For example, increas-ing the delay τ from 1 . n e = 2 . × cm − to n e ≈ × cm − whilst maintaining an attenuation length L att ≈ shape of the CHOFIchannel is possible by adjusting the delay ∆ τ betweenthe conditioning pulse and the pulse to be guided. Asshown in Fig. 2, for small values of ∆ τ the CHOFI chan-nel has a core of approximately uniform electron den-sity, surrounded by high and thick walls; this profile is agood approximation to the near-hollow plasma channel,which can provide independent control of the focusingand accelerating fields for electron and positron accelera-tion [38]. For larger values of ∆ τ the core of the channeldevelops an approximately parabolic profile.An important feature of the work presented here isthat the conditioning pulse is self-guided by ionizationof the neutral gas at the edge of the initial HOFI chan-nel. This self-guiding has three important consequences.First, as discussed below, the energy of the conditioning pulse is used efficiently, which reduces the total laser en-ergy required to create the CHOFI channel. Second, thegenerated CHOFI channel is rather robust to variationsof the parameters or pointing of the conditioning pulse.Thirdly, a wide range of channel parameters and shapesare accessible with a simple experimental setup. We notethat the conditioning pulse does not require transverseshaping, and should have the same spot size and centralwavelength as the main pulse to be guided; as such itcould be generated very simply by introducing a smallpre-pulse to the main pulse.It is useful to estimate the total laser energy whichwould be required to generate a 1 m long CHOFI chan-nel. In our experiments approximately 5 mJ per centime-ter of HOFI channel was required for the channel-formingpulse. Simulations of the propagation of the condition-ing pulse show that efficient self-guiding is achieved if itionizes the neutral gas out to approximately 2 w m , CHOFI ,which requires I peak,cond (cid:38) . × W cm − for the23 µ m spot size used here. The calculations showed thatwhen this is satisfied, the propagation losses are approx-imately 7 mJ cm − , from which we deduce that an inputpulse energy of 700 mJ is required for the conditioningpulse. Hence a total laser energy of 1 . × W cm − . A power atten-uation length of L att ≈ .
15 m has been reported for cap-illary discharge waveguides [18], and that for the lowest-order mode of hollow capillary waveguides [11, 40] is L att ≈ . µ m diameter capillary. Gonsalves etal. [41] have reported an energy transmission of 85 % forlow-intensity pulses guided in 90 mm long laser-assistedcapillary discharges, corresponding to L att ≈ . L att (cid:29) L pd , where L pd is the pump depletion length of the wakefield driver.A drive laser pulse of energy 1 J, and with a spot sizematched to the CHOFI channel, has a normalized peakvector potential a = 1 .
13. Guiding this over the de-phasing length (0 . . a = 1 . . n e0 ≈ × cm − and a matched spot size of ∼ µ m. The work described in the present paper sug-gests that these challenging parameters could be achievedby CHOFI channels. VII. CONCLUSIONS
We have demonstrated through experiments and sim-ulations that low-density, low-loss, metre-scale plasmachannels can be generated by ionization of the neutral gascollar surrounding a hydrodynamic optical-field-ionizedplasma channel. Channels with axial electron densi-ties of n e0 ≈ × cm − , a matched spot size of w m ≈ µ m, and a power attenuation length of up to L att = (26 ±
2) m were generated experimentally. Thechannel depth was increased by a factor of ten comparedto the unconditioned HOFI plasma channel.Hydrodynamic simulations of the dynamics of theplasma formed by the channel-forming pulse show thata low-density HOFI plasma channel is created by thechannel-forming pulse, and that this is surrounded bya collar of neutral gas. Particle-in-cell simulations showthat the leading edge of a conditioning pulse injected intothis structure ionizes the neutral gas in its transversewings, to form a deep conditioned HOFI (CHOFI) chan-nel which can guide the bulk of the conditioning pulse —and any trailiing laser pulses — with very low propaga-tion losses.The properties of CHOFI channels, including theirshape, can be controlled by adjusting the initial gas den- sity, and the delays between the channel-forming, con-ditioning, and guided pulse. Further, the channels arefree-standing, which makes them immune to laser dam-age, and they can be generated with a total laser pulseenergy of order 1 J per metre of channel. These prop-erties would seem to make them ideally suited to manyapplications in high-intensity light-matter interactions,including multi-GeV plasma accelerator stages operatingat high pulse repetition rates.
ACKNOWLEDGEMENTS
The authors thank G. Hine for experimental assis-tance, and L. Feder, B. Miao and H. M. Milchberg forongoing discussions. This work was supported by theUK Science and Technology Facilities Council (STFCUK) [grant numbers ST/P002048/1, ST/P002056/1,ST/N504233/1, ST/R505006/1]; the Engineering andPhysical Sciences Research Council [studentship No.EP/N509711/1]; and the Central Laser Facility of theUnited Kingdom. This material is based upon work sup-ported by the Air Force Office of Scientific Research un-der award number FA9550-18-1-7005. This work wassupported by the European Union’s Horizon 2020 re-search and innovation programme under grant agreementNo. 653782. [1] P. Sprangle, C.-M. Tang, and E. Esarey, Relativistic Self-Focusing of Short-Pulse Radiation Beams in Plasmas,IEEE Transactions on Plasma Science , 145 (1987).[2] G.-Z. Sun, E. Ott, Y. C. Lee, and P. Guzdar, Self-focusingof short intense pulses in plasmas, Phys. Fluids , 526(1987).[3] P. Sprangle, E. Esarey, and A. Ting, Nonlinear-Theoryof Intense Laser-Plasma Interactions, Phys Rev Lett ,2011 (1990).[4] R. M. G. M. Trines, F. Fiuza, R. Bingham, R. A. Fonseca,L. O. Silva, R. A. Cairns, and P. A. Norreys, Simulationsof efficient Raman amplification into the multipetawattregime, Nat Phys , 87 (2010).[5] A. Butler, A. J. Gonsalves, C. M. McKenna, D. J.Spence, S. M. Hooker, S. Sebban, T. Mocek, I. Bettaibi,and B. Cros, Demonstration of a Collisionally ExcitedOptical-Field-Ionization XUV Laser Driven in a PlasmaWaveguide, Phys Rev Lett , 205001 (2003).[6] T. Tajima and J. M. Dawson, Laser Electron-Accelerator,Phys Rev Lett , 267 (1979).[7] E. Esarey, C. B. Schroeder, and W. P. Leemans, Physicsof laser-driven plasma-based electron accelerators, Rev.Mod. Phys. , 1229 (2009).[8] S. M. Hooker, Developments in laser-driven plasma ac-celerators, Nature Photon , 775 (2013).[9] P. A. Walker, P. D. Alesini, A. S. Alexandrova,M. P. Anania, N. E. Andreev, I. Andriyash, A. As-chikhin, R. W. Assmann, T. Audet, A. Bacci, I. F.Barna, A. Beaton, A. Beck, A. Beluze, A. Bern-hard, S. Bielawski, F. G. Bisesto, J. Boedewadt, F. Brandi, O. Bringer, R. Brinkmann, E. Br¨under-mann, M. B¨uscher, M. Bussmann, G. C. Bussolino,A. Chance, J. C. Chanteloup, C. M, E. Chiadroni,A. Cianchi, J. Clarke, J. Cole, M. E. Couprie, M. Croia,B. Cros, J. Dale, G. Dattoli, N. Delerue, O. Delfer-riere, P. Delinikolas, J. Dias, U. Dorda, K. Ertel, A. Fer-ran Pousa, M. Ferrario, F. Filippi, J. Fils, R. Fiorito,R. A. Fonseca, M. Galimberti, A. Gallo, D. Garzella,P. Gastinel, D. Giove, A. Giribono, L. A. Gizzi, F. J.Gr¨uner, A. F. Habib, L. C. Haefner, T. Heinemann,B. Hidding, B. J. Holzer, S. M. Hooker, T. Hosokai,A. Irman, D. A. Jaroszynski, S. Jaster-Merz, C. Joshi,M. C. Kaluza, M. Kando, O. S. Karger, S. Karsch,E. Khazanov, D. Khikhlukha, A. Knetsch, D. Kocon,P. Koester, O. Kononenko, G. Korn, I. Kostyukov, L. La-bate, C. Lechner, W. P. Leemans, A. Lehrach, F. Y.Li, X. Li, V. Libov, A. Lifschitz, V. Litvinenko, W. W.Lu, A. R. Maier, V. Malka, G. G. Manahan, S. P. D.Mangles, B. Marchetti, A. Marocchino, A. Martinezde la Ossa, J. L. Martins, F. Massimo, F. Mathieu,G. Maynard, T. J. Mehrling, A. Y. Molodozhentsev,A. Mosnier, A. Mostacci, A. S. Mueller, Z. Najmudin,P. A. P. Nghiem, F. Nguyen, P. Niknejadi, J. Osterhoff,D. Papadopoulos, B. Patrizi, R. Pattathil, V. Petrillo,M. A. Pocsai, K. Poder, R. Pompili, L. Pribyl, D. Pu-gacheva, S. Romeo, A. R. Rossi, E. Roussel, A. A. Sahai,P. Scherkl, U. Schramm, C. B. Schroeder, J. Schwindling,J. Scifo, L. Serafini, Z. M. Sheng, L. O. Silva, T. Silva,C. Simon, U. Sinha, A. Specka, M. J. V. Streeter, E. N.Svystun, D. Symes, C. Szwaj, G. Tauscher, A. G. R. Thomas, N. Thompson, G. Toci, P. Tomassini, C. Vac-carezza, M. Vannini, J. M. Vieira, F. Villa, C.-G.Wahlstr¨om, R. Walczak, M. K. Weikum, C. P. Welsch,C. Wiemann, J. Wolfenden, G. Xia, M. Yabashi, L. Yu,J. Zhu, and A. Zigler, Horizon 2020 EuPRAXIA designstudy, J. Phys.: Conf. Ser. , 012029 (2017).[10]
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The experiment was carried out on the 5 Hz repetition-rate TA2 Astra-Gemini Laser at the Central LaserFacilty, UK, providing a peak power of 9 . I collax ≈ . × W cm − . This was reflected from a reterof-electing delay stage to allow for timing control, then fo-cused by a fused silica axicon lens of base angle ϑ =5 . × W cm − inside the linefocus, and a transverse profile that matched well to the- Figure 8. Detailed experimental layout showing vacuumchamber. ory [43], I ( r ) ∝ J ( βr ) where β = k [arcsin( η sin ϑ ) − ϑ ], η is the refractive index, and k = 2 π/λ . The line focusextended throughout the length of the gas cell.The conditioning pulse was transmitted through HM1and passed through a retro-reflecting delay stage to allowthe time between the channel-forming and conditioningpulses to be varied. It was then focused by an off-axisparaboloid (OAP) mirror of focal length f = 750 mmused at f /25 to the entrance pinhole of the gas cell. Themeasured beam waist and Rayleigh range were w =(22 ± µ m and z R = (1 . ± .
4) mm respectively. By re-placing the final turning mirror for an uncoated, opticallyflat wedge, the conditioning pulse could be operated attwo intensities. The high-intensity pulse had peak inten-sity I highpeak = 4 . × W cm − at the entrance pinholecorresponding to a ≈ .
43, whilst the lower intensitymode had a peak intensity I lowpeak ≈ . × W cm − .The gas cell was constructed from aluminium and filledwith hydrogen gas through a 5 mm inlet located 10 mmfrom the entrance pinhole. Optically-flat, fused silicawindows were placed on each face to allow for transverseprobing of the plasma channel. The entrance pinholehad a diameter of 1 . µ m. The distance between the two pinholes wasmeasured to be 16 mm. The pressure in the cell could bevaried in the range 5 mbar to 120 mbar and was measuredusing a pressure transducer located in a gas reservoir be-fore the gas cell.1A separately compressed, 1 mJ probe pulse with thesame central wavelength as the main pulse was used toprobe the interaction transversely. The probe was passedtransversely through the plasma at a delay τ after thearrival of the channel-forming beam; this delay could bevaried in the range 0 ns to 6 ns by adjusting a retrore-flecting delay stage. Scattered light from the interac-tion region was collected and collimated by a 50 . f = 250 mm,and subsequently imaged onto a 12-bit CMOS detec-tor by a second, planoconvex lens of diameter 50 . f = 1500 mm. Before reaching the detector, thisbeam was passed through a Mach-Zehnder interferome-ter to form equally spaced, straight fringes on the detec-tor. The measured spatial resolution of the detector was(8 . ± . µ m.After the gas cell, light from the conditioning pulsewas collected and collimated by a 76 . f =500 mm focal length achromatic lens. A retro-reflectingstage placed before this lens allowed the object plane tobe varied from the entrance to the exit pinhole of thegas cell. The collimated light was subsequently reimagedonto a 12-bit CMOS detector and an pyroelectric energymeter by f = 750 mm, achromatic lens. Light from thechannel-forming beam was blocked from reaching thesediagnostics by an aperture. HYDRODYNAMIC SIMULATIONSSimulation parameters
In order to study the gas and plasma densities prior tothe interaction with the conditioning pulse, the channelexpansion was modelled with hydrodynamic simulations.The 2D simulations were performed using the Euleriancode FLASH 4.6.2[35], capable of performing adaptive-mesh refinement inside the 100 µ m × µ m simulationbox. A 3-temperature model was employed in the calcu-lations, which allowed for independent evolution of theelectron, ion, and radiation species. Heat exchange wasenabled to allow for energy exchange between electronsand ions. The energy diffusion and thermal conductivitymodules were also activated, to improve the modelling ofthe species propagation.Initially, the simulation box was filled with neutral hy-drogen gas, at a pressure of 50 mbar and a temperatureof 298 K. The plasma column was initialised by config-uring a region of higher temperature. As described inRef. [26], the initial electron temperature profile for theplasma column created by the axicon can be describedas a super-Gaussian, T e = T e, · exp (cid:2) − ( r/r ) (cid:3) , where T e, =10 eV is the initial maximum temperature, and r = 4 µ m is the initial radius. Tabulated values wereused for the equation of state (EoS) and ionization frac-tion of atomic hydrogen, with tables obtained from thecommercially available PROPACEOS package [44]. Itshould be noted that atomic hydrogen was used since the hydrodynamic properties of molecular hydrogen werenot available in the code. Radiation transport was mod-elled using multi-group diffusion, with a total of 6 energygroups distributed between 10 meV and 12 eV. Tabu-lated values were used for the Rosseland and Planck opac-ities, also obtained from the PROPACEOS package. Additional simulation results
The temporal evolution of the ionized and neutral com-ponents in the simulation was obtained by integrating az-imuthally the electron and neutral densities, respectively.As shown in Fig. 9(a), the fraction of each component re-mains constant throughout the expansion. t (ns)0 . . . N ( a . u . ) × (a) ElectronNeutral r ( µ m)10 − − T ( e V ) (b) ElectronIon0 10 20 30 40 50 r ( µ m)10 P i o n ( m b a r ) (c) τ ( n s ) τ ( n s ) Figure 9. Additional results from the hydrodynamic simula-tions. (a) Temporal evolution of the number of free electrons(black) and neutral atoms (blue) throughout the plasma ex-pansion. (b) Temporal evolution of the electron ( T e , solid)and ion ( T i , dashed) temperature profiles at various delays τ given by the color scale. (c) Temporal evolution of ion pres-sure profiles at various delays τ given by the color scale. Figure 9(b) shows the temporal evolution of the elec-tron and ion temperature profiles. The temperature ofboth species exhibits an initial rapid evolution, until2electron-ion thermalisation is reached at τ (cid:39) PROPAGATION SIMULATIONS
The simulations of the propagation of the conditioningpulse were initialized with an electron and neutral den-sity profile equal to that simulated for τ = 1 . . τ = 40 fs, and aGaussian spatial profile with w = 23 µ m, closely match-ing those in the experimental setup.The simulation window was 400 × r, z ) cylindrical grid, and co-propagated with the con- ditioning laser pulse. The grid resolution was 0 . µ mand 1 µ m in the longitudinal and transverse directions re-spectively. Each cell was initialized with 4 particles percell. At the transverse boundaries, convolutional per-fectly matched layers (CPML) were used to absorb out-ward traveling radiation from leaky modes. A linear den-sity ramp of length 100 µ m was included at the entranceto the channel, to prevent unwanted reflections from adensity discontinuity. The laser pulse was focused at thetop of the ramp, although since the length of the rampwas much shorter than the Rayleigh length of the laserpulse, the longitudinal position of the focus would nothave had a significant effect.The laser transmission was calculated by isolating theelectric field of the laser from the plasma and integratingat each timestep T ( z ) = 1 E init π(cid:15) (cid:90) R (cid:90) Z + Z − E ( r, z ) r d r d z, (1)where (0 , R ) and ( Z − , Z + ) are the edges of the simula-tion box in r and z respectively, and E init is the initiallaser energy given by E init = (cid:16) π (cid:17) / τ w I peak ..