Gas-filled capillary discharge waveguides - OSA Publishing

138

J. Opt. Soc. Am. B / Vol. 20, No. 1 / January 2003

Spence et al.

Gas-filled capillary discharge waveguides D. J. Spence, A. Butler, and S. M. Hooker Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK Received March 30, 2002; revised manuscript received September 4, 2002 We describe in detail the operation of the gas-filled capillary discharge waveguide for high-intensity laser pulses and discuss measurements and magnetohydrodynamic simulations that show that the plasma channel produced is parabolic and essentially fully ionized. We present the results of experiments in which laser pulses with a peak input intensity of 1.2 ⫻ 1017 W cm⫺2 were guided through hydrogen-filled capillary discharges with lengths of 30 and 50 mm. The pulse energy coupling and transmission losses were determined to be ⬍4% and (7 ⫾ 1) m⫺1 , respectively. We discuss the application of waveguides of this type to driving short-wavelength lasers and laser wakefield accelerators. © 2003 Optical Society of America OCIS codes: 230.7370, 140.7240.

1. INTRODUCTION Many applications of high-intensity laser pulses require some form of optical guiding to increase the length of interaction with the laser radiation. Examples of such applications include high-harmonic generation,1 laser wakefield accelerators,2,3 and x-ray lasers.4,5 In the absence of guiding, the interaction length is fundamentally limited by diffraction to distances of the order of the Rayleigh range, Z R ⫽ ␲ W 0 2 /␭, where W 0 and ␭ are the spot size and the wavelength of the beam, respectively. In addition, many applications of high-intensity laser pulses involve the propagation of laser pulses through partially ionized plasmas, in which case the interaction length is further limited by ionization-induced refractive defocusing. To date a wide range of methods for guiding highintensity laser pulses has been investigated, and these may be divided into two broad classes: hollow dielectric waveguides and plasma waveguides. Hollow dielectric waveguides guide intense laser pulses by grazing-incidence reflection at the walls of a hollow capillary tube. Jackel et al. investigated6 the propagation of 1-J, 1-ps optical pulses through evacuated 30-mmlong capillaries with internal diameters of 266 and 100 ␮m. For an input pulse intensity of 2 ⫻ 1017 W cm⫺2 those authors measured pulse-energy transmissions of 26% and 16%, respectively, for the 266- and 100-␮mdiameter capillaries. In the research reported in Ref. 6 the guiding was multimode; the numbers of modes transmitted by the waveguide were approximately 8 and 2 for the wider and the narrower capillaries, respectively. Multimode guiding is associated with a highly structured transverse intensity profile, temporal dispersion, and low group velocity.7 More recently, Dorchies et al.8 investigated single-mode propagation of intense laser pulses through hollow capillaries. For 30-mJ, 120-fs laser pulses focused to an input intensity of 5 ⫻ 1016 W cm⫺2 they measured a pulseenergy transmission of approximately 15% through 105mm-long, 50-␮m-diameter capillaries. However, the introduction of gas into the capillary was found to decrease 0740-3224/2003/010138-14$15.00

the transmission significantly. For example, for a 40mm-long capillary the introduction of 20 mbars (1 mbar ⫽ 100 Pa) of He reduced the transmission by a factor of 6, owing to decreased coupling into and increased damping within the capillary. Some of those authors and others also studied monomode guiding at higher laser intensities and reported9 significant decreases in pulse-energy transmission for pulses with input intensities greater than 1017 W cm⫺2 . At these higher intensities the capillary was destroyed in a single laser shot. The reduction in pulse transmission and capillary lifetime was interpreted as being due to the formation of a plasma on the front face of the capillary entrance by a nanosecond-long pedestal to the laser pulse. Avoiding such difficulties requires that the input laser beam have excellent spatial and temporal contrasts, and it may also be necessary to employ capillaries with tapered entrances. Plasma waveguides operate by forming a plasma with a refractive index that decreases with radial distance from the propagation axis, resulting in beam focusing that can counteract diffraction and ionization-induced refraction. The refractive index ␩ experienced by a laser pulse propagating through a plasma may be written as ␩ ⫽ (1⫺N e e 2 / ␥ m e ␧ 0 ␻ 2 ) 1/2, where N e is the electron density, ␻ is the angular frequency of the laser, and ␥ describes the relativistic increase in electron mass that arises from the electron’s quiver motion in the electric field of the laser. A suitable refractive-index profile can be formed by the radial variation in ␥ created by a laser pulse with a transverse intensity profile that is peaked on axis. This relativistic self-focusing is greater than diffraction for laser powers that exceed a critical power10 Pc 2 2 ⫽ 17( ␻ / ␻ p ) GW, where ␻ p ⫽ (N e e /m e ␧ 0 ) 1/2 is the plasma frequency. For sufficiently intense pulses, expulsion of electrons from the axial region by the ponderomotive force can reduce the axial electron density, thereby enhancing the refractive-index profile. The contribution of the ponderomotive effect reduces11 the threshold for relativistic focusing to P c ⫽ 16.2( ␻ / ␻ p ) 2 GW. Relativistic and ponderomotive channeling have been observed experimentally10,12 and are attractive guiding mecha© 2003 Optical Society of America

Spence et al.

nisms because they require nothing more than the laser beam itself. However, both channeling mechanisms depend critically on the laser intensity and are not efficient at guiding short laser pulses owing to the finite response time of the plasma.13 One may also generate a plasma waveguide by preforming a channel with an axial minimum in the radial electron density profile. In an ideal preformed plasma waveguide the radial electron density profile is parabolic: N e (r) ⫽ N e (0) ⫹ ⌬N e (r/r ch) 2 , where N e (r) is the electron density at a radial distance r from the axis and ⌬N e is the increase in the electron density at r ⫽ r ch . In the absence of further ionization of the plasma by the guided laser pulse, and where ponderomotive and relativistic effects can be neglected, a Gaussian laser beam will propagate though the guide with a constant spot size W M , provided that W M ⫽ 关 r ch2 /( ␲ r e ⌬N e ) 兴 1/4, where r e is the classic electron radius.14 A number of techniques for producing a plasma waveguide have been investigated. A suitable channel may be formed during the hydrodynamic expansion of a laserproduced cylindrical spark created by focusing of a picosecond laser pulse to a line focus with an axicon lens.15–17 For example, Nikitin et al. employed this technique to form highly ionized channels in Ar. They injected 50-mJ, 100-fs pulses from a Ti:sapphire laser system into the plasma channel approximately 20 ns after the creation of the spark and demonstrated guiding of pulses with a peak intensity of 5 ⫻ 1016 W cm⫺2 over distances of 15 mm and with a pulse-energy transmission of 52%.16 In an extension of this research, Volfbeyn et al. described17 an ignitor–heater technique that is able to form plasma channels in low-Z gases. In their approach a line focus produced by a short, intense laser pulse creates an initial spark that is heated by a long, energetic laser pulse. Volfbeyn et al. have used this approach to channel laser pulses with a peak input intensity of ⬃5 ⫻ 1017 W cm⫺2 in one-dimensional guides of a few millimeters’ length. Gaul et al. have reported18 the use of a pulsed electrical discharge to provide seed electrons for a laser-produced spark in helium. Those authors reported a pulse-energy transmission of 50% through 15-mm-long channels in fully ionized helium for laser pulses with an input intensity of the order of 1017 W cm⫺2 . Recently a new method of using laser pulses to generate a plasma channel was reported by Fan et al.19 They used an axicon lens and a phase plate to generate a hollow Bessel beam in 700 Torr of argon gas. The resultant plasma was 8 mm long, with a tubular electron density profile. The steep walls in the electron density were calculated to be able to guide intense radiation with a spot size of approximately 3 ␮m. To date, however, there have been no reports of guiding with this technique. Plasma channels may also be formed following ablation of a capillary wall by a slow electrical discharge.20–22 In this approach an arc is passed through an initially evacuated capillary, which ablates and ionizes material from the wall. Radial heat conduction to the capillary wall results in the formation of a stable, long-lived guiding channel. With this approach, laser pulses with peak input intensities greater than 1017 W cm⫺2 have been channeled21 through 20-mm-long polypropylene capillaries, with a

Vol. 20, No. 1 / January 2003 / J. Opt. Soc. Am. B

139

pulse energy transmission of 75%. However, for discharge-ablated polypropylene capillaries the plasma comprises low ionization states of carbon and fully stripped hydrogen. As a result, whereas high pulseenergy transmission may be observed, ionization-induced defocusing can cause severe modulation of the laser spot size with propagation distance, leading to large variations in the axial intensity of the guided pulse.23 Transient channels have been observed during the rapid collapse of a plasma driven by fast, Z-pinch capillary discharges.24,25 For example, Hosokai et al. have observed24 guiding of laser pulses with an input intensity of ⬃1017 W cm⫺2 through 20-mm-long channels with a pulse-energy transmission of 64%. A disadvantage of this approach is that the plasma channel exists for only a few nanoseconds, which necessitates low timing jitter between the laser system and the Z-pinch discharge. In this paper we describe the operation and some potential applications of a new type of waveguide for highintensity laser pulses: the gas-filled capillary discharge waveguide.26 As discussed below, the plasma channel is essentially fully ionized, minimizing spatial or temporal distortion of the guided laser pulse, and the channel is long lived, permitting straightforward synchronization with the laser system. Butler et al. recently27 used this device to guide laser pulses with a peak input intensity of 1.2 ⫻ 1017 W cm⫺2 through 30- and 50-mm-long waveguides with pulse-energy transmissions of approximately 90% and 80%, respectively. For conditions that produced the greatest axial intensity of the transmitted beam the coupling and propagation losses of the waveguide were determined to be ⬍4% and (7 ⫾ 1) m⫺1 , respectively, which, to our knowledge, are the lowest losses for guiding of laser pulses with a peak intensity of 1017 W cm⫺2 or greater. The paper is organized as follows. In Section 2 we describe the construction and operation of the gas-filled capillary discharge waveguide. We discuss the results of interferometric measurements of the plasma channel and compare them with the results of magnetohydrodynamic simulations of the capillary discharge. In Section 3 we describe experiments to characterize guiding of highintensity laser pulses and discuss the results in detail. In Section 4 we discuss two possible applications of gasfilled capillary discharge waveguides: driving novel short-wavelength lasers and laser-driven plasma accelerators. We present simulations of the propagation of an intense driving laser pulse through gas-filled capillary discharge waveguides containing gas mixtures suitable for short-wavelength lasers driven by optical field ionization. In that section we also discuss the application of hydrogen-filled capillary discharge waveguides to laser wakefield accelerators. In Section 5 we conclude.

2. GAS-FILLED CAPILLARY DISCHARGE WAVEGUIDE Figure 1 shows schematically our most recent design of a gas-filled capillary discharge waveguide. This design differs somewhat from that employed in our earlier experiments in that the discharge is double-ended; the main advantages of this design are improved shielding of the

140


Fig. 1. Schematic diagram of the gas-filled capillary discharge waveguide.

high-voltage electrode and increased discharge length. The capillary comprises an alumina tube of inner and outer diameters 400 ␮m and 1 mm, respectively. Four slots, each 0.8 mm long and approximately 75 ␮m wide, are laser machined near each end of the capillary to introduce gas into the capillary. Hydrogen gas is flowed through the gas injection slots and out into the surrounding vacuum chamber, such that the steady-state pressure in the region of the capillary between the injection slots is uniform. A stainless-steel earth electrode is located coaxially at each end of the capillary, and a stainless-steel cathode is located at the center of the capillary. Four slots, 0.6 mm long and 75 ␮m wide, allow current to flow through the wall of the capillary to the cathode. The capillary and electrodes are located in a transparent plastic [poly(methyl methacrylate)] housing held within an earthed aluminum can. O-ring seals between the capillary, the electrodes, the housing, and the aluminum can ensure vacuum integrity while they allow the capillary to be replaced easily. The discharge circuit comprises a 7.5-nF capacitor charged to 17–30 kV that can be connected across the electrodes by use of a thyratron switch. The temporal profile of the current pulse may be measured with a Rogowski coil placed around the wire connecting the capacitor to the cathode. A five-axis stage, bolted to the aluminum can, permits accurate alignment of the axis of the capillary with the propagation axis of the incoming laser pulse. A. Interferometric Measurements of the Plasma Channel Our measurements of the radial electron density profile generated in a hydrogen-filled capillary discharge waveguide were reported previously.26 In the research reported in Ref. 26 the discharge was a conventional singleended configuration; the discharge current was approximately sinusoidal, with a half-period of 200 ns and a peak of 300 A. The capillaries were of 300-␮m inner diameter. Electron density profiles were measured as a function of time during the discharge-current pulse by longitudinal interferometry of short capillaries with

Spence et al.

8-ns pulses at 355 nm from a Nd:YAG laser. By making measurements for 3- and 5-mm-long capillaries it was possible to deduce the electron density profile in the central section between the gas injection holes. Figure 2 shows the results of such measurements for an initial hydrogen pressure in the capillary of 67 mbars at a delay time t ⫽ 60 ns after the onset of the discharge current. A parabolic fit to the measured profile, shown as a dashed curve in Fig. 2, yields N e (0) ⫽ 2.7 ⫻ 1018 cm⫺3 and ⌬N e ⫽ 1.2 ⫻ 1018 cm⫺3 (with r ch ⫽ 150 ␮ m), corresponding to a matched spot size of 37.5 ␮m. By integrating the measured electron density profile to find the number of electrons per unit length of capillary and comparing this result to the initial density of hydrogen atoms, we calculated the average ionization of ⫹0.01 the hydrogen plasma to be Z * ⫽ 0.99⫺0.12 . Figure 3 shows the temporal evolution of the plasma channel during the discharge pulse. The figure plots the measured radial profile of the phase shift introduced by the discharge plasma for a 300-␮m-diameter capillary and an initial hydrogen pressure of 67 mbars. Unlike in Fig. 2, these data have not been corrected for end effects.

Fig. 2. Interferometric measurement of the electron density profile at t ⬇ 60 ns in the uniform central section of a 5-mm long, 300-␮m-diameter capillary for an initial hydrogen pressure of 67 mbars.

Fig. 3. Interferometric measurement of the evolution during the discharge pulse of the plasma channel in a 5-mm-long, 300-␮mdiameter capillary for an initial hydrogen pressure of 67 mbars. The measured phase shift introduced by the plasma column is shown; it has not been corrected for end effects. Inset, measured temporal profile of the discharge-current pulse.

Spence et al.

Fig. 4. Comparison of measured and simulated electron density profiles for the conditions of Fig. 2. Open circles, measured electron density. Simulations are shown for delays of 55 ns (filled circles), 60 ns (solid curve), and 65 ns (dashed curve).


141

steady-state equilibrium in which the plasma pressure and the electric field were uniform. Radiation cooling of the plasma was found to be negligible and, instead, the temperature profile of the plasma was determined by a balance between ohmic heating and thermal conduction to the capillary wall. As a result the temperature of the plasma was greatest on axis, and, as the plasma pressure was uniform, the plasma density had an axial minimum, corresponding to a guiding electron density profile. It was shown that during the quasi-steady-state regime the electron density was parabolic in the axial region that corresponded, for the plasma conditions of the interferometric measurements, to a plasma channel with a matched laser spot size of W M 关 ␮ m兴 ⫽ 1.48 ⫻ 105 a 关 ␮ m兴 1/2共 Z * N i0 关 cm⫺3 兴兲 ⫺1/4, (1)

It is clear that the plasma channel was formed within approximately 50 ns of the initiation of the discharge and remained essentially unchanged for the duration of the current pulse. For delays greater than approximately 300 ns, that is, shortly after the end of the main current pulse, rapid changes in the measured phase shift were observed for regions close to the capillary wall as a result of electron–ion recombination. The measured phase shift on axis was found to be highest approximately 30 ns after the initiation of the discharge pulse. Thereafter the measured phase shift decreased monotonically over an interval of approximately 650 ns, indicating the time scale for longitudinal flow of the plasma out of the capillary. The fact that the guiding channel was observed as soon as 50 ns after the initiation of the discharge current demonstrates that longitudinal flow is not required for forming a guiding channel. B. Magnetohydrodynamic Simulations of the Capillary Discharge Together with Bobrova et al. we compared the results of the interferometric measurements with magnetohydrodynamic (MHD) simulations.28 These calculations showed that, in contrast to capillary discharges operated at higher current densities,24,25 no pinching of the plasma was observed. The plasma was found to become essentially fully ionized for t greater than approximately 55 ns, and ablation of the capillary wall was found to be negligible. Figure 4 compares calculated electron density profiles for t close to 60 ns with the measured electron density profile for t ⫽ 60 ns presented in Fig. 2. It can be seen that the simulated profiles for t ⫽ 55 ns and t ⫽ 65 ns bracket the measured profile reasonably well. We conclude that, allowing for the fact that the measured electron density profiles are averaged over the ⬃8-ns duration of the probe pulses, the MHD simulations are in good agreement with the interferometric measurements. The MHD simulations also show that for times shortly after 60 ns the electron density profile evolves from the rather flat profiles shown in Fig. 4 to one that is approximately parabolic. It was found that for t ⬎ 80 ns the plasma, acoustic, thermal conduction, and skin times are small compared with the time scale at which the discharge current changes. As a result the discharge reached a quasi-

where a is the radius of the capillary and N i0 is the initial density of hydrogen atoms.

3. GUIDING EXPERIMENTS We have demonstrated guiding of high-intensity laser pulses in both single-ended29 and double-ended27 hydrogen-filled capillary discharge waveguides. Here we briefly describe the results of the more recent of those experiments, which used the double-ended capillary discharge design shown schematically in Fig. 1. The apparatus employed in the guiding experiments is shown schematically in Fig. 5. Radiation from the Astra Ti:sapphire laser at the Rutherford Appleton Laboratory was focused onto the entrance plane of the capillary with an f ⫽ 1.6 m parabolic mirror used off axis at f/27. The intensity of the radiation leaving the capillary was reduced by reflection from a wedged optical flat and collimated by an achromatic f ⫽ 700 mm, f/14 lens (L1; Fig. 5). Reflections from two more optical flats reduced the intensity of the transmitted beam further before the beam was refocused with an f ⫽ 500 mm, f/8 lens (L2) and

Fig. 5. Schematic diagram of the apparatus used to demonstrate guiding of high-intensity laser pulses in a hydrogen-filled capillary discharge waveguide. ND, neutral density.

142


magnified onto a 12-bit CCD camera by a microscope objective. The object plane of the imaging system was set to be coincident with the front focal plane of lens L1 such that we could vary the object plane by translating L1 without changing the transverse magnification. The energy of the laser pulses input to and transmitted by the capillary were recorded by pyroelectric energy meters located, respectively, behind a turning mirror before the vacuum compressor of the Astra laser system and behind wedge 2. The spectra of the laser pulses leaving the capillary were measured by a spectrograph located behind wedge 3. The mean full width at half-maximum duration of the laser pulses from Astra, as measured with a single-shot autocorrelator, was 70 ⫾ 20 fs, assuming a Gaussian temporal profile. The average energy of the laser pulses input to the capillary during the guiding experiments was 162 ⫾ 18 mJ. A. Results Figure 6 shows the measured pulse-energy transmission T as a function of delay t for a 30-mm-long hydrogen-filled capillary and for initial hydrogen pressures of 110–330 mbars. For these experiments the discharge-current pulse, as shown in Fig. 6, had a damped, approximately sinusoidal profile, with a quarter-period of approximately 190 ns and a peak current of approximately 550 A per arm. The measured pulse-energy transmission T can be seen to have the following form. The transmission was low for laser pulses injected before the onset of the discharge current; the transmission decreased as the initial hydrogen pressure was raised. These low values of the transmission were caused by strong ionization-induced defocusing of the incident laser pulses in the initial, un-ionized hydrogen gas. Immediately after the onset of the discharge current, T decreased to low values and remained low for a time that increased with the initial hydrogen pressure. At longer delays the pulse-energy transmission increased rapidly to a plateau of high energy transmission with T ⬇ 90%. The duration of the rise in T was seen to increase as the initial hydrogen pressure was increased. The initial decrease in the transmission after the onset of the discharge current and the subsequent relatively slow rise of T are associated with the time required first to establish a uniform discharge in each arm of the discharge and then to fully ionize the hydrogen. Finally, we note that several pronounced minima of the transmission were observed in the plateau region at times that correspond closely to a zero discharge current. Comparison with earlier guiding experiments29 and an analysis of the spectra and transverse spatial profiles of transmitted laser pulses injected at delays close to a zero in the discharge current suggest that these dips are caused by the formation of nonuniformities in the discharge near the cathode rather than by effects that result from cooling of the discharge plasma. For 50-mm-long capillaries the results were qualitatively similar; the transmission reached approximately 80% in the plateau region. For a given initial hydrogen pressure the delay before, and the duration of, the rise in T was longer than for the 30-mm-long capillaries.

Spence et al.

Figure 7 shows the measured transverse intensity profiles in the entrance plane of the capillary and in the exit plane of the 30-mm-long capillary for various delays t and for an initial hydrogen pressure of 330 mbars. We normalized the fluence profiles recorded by the CCD camera to the mean input pulse energy of 162 mJ and converted them to intensity profiles by using the measured pulseenergy transmission, assuming that the temporal profiles of the input and the transmitted laser pulses were the same. The laser pulses input to the waveguide had a spot size of approximately 31 ␮m and a peak axial intensity of 1.2 ⫻ 1017 W cm⫺2 . For laser pulses injected before the onset of the discharge current, strong defocusing in the neutral hydrogen gas caused the transmitted radiation to fill the diameter of the capillary, resulting in a peak intensity more than 3 orders of magnitude below that of the input pulses. However, as the discharge developed, the transmitted radiation became constrained to the axial region, which resulted in a dramatic increase in the peak axial intensity. For example, the transmitted laser pulse recorded at t ⫽ 731 ns had a pulse-energy transmission of 83%, a spot size of approximately 41 ␮m, and a peak axial intensity of 0.4 ⫻ 1017 W cm⫺2 , some 36% of that of the input pulse. Similar results were obtained with 50-mmlong capillaries; the peak axial intensities reached 23% of that of the input pulse for t ⬇ 730 ns. We emphasize that these data are typical results, not the best profiles recorded. Indeed, for both capillary lengths several transmitted laser pulses were recorded with peak axial intensities as much as 50% higher than described here. Figure 8 shows the spectra of pulses transmitted by a 30-mm-long capillary for an initial hydrogen pressure of 330 mbars and t ⫽ 730 ns, corresponding to the conditions that produce the highest axial intensity of the transmitted pulses. The spectrum recorded when the capillary was removed is also shown. It can be seen that the spectral widths of the transmitted laser pulses were approximately equal to that of the input laser pulses. However, the spectra were shifted to shorter wavelengths by (12 ⫾ 2) nm; the shift was essentially independent of the initial hydrogen pressure. Similar results were obtained for 50-mm-long capillaries; the wavelength shift was (22

Fig. 6. Measured pulse-energy transmission T as a function of pulse injection delay t for a 30-mm-long hydrogen-filled capillary and for various initial hydrogen pressures. Also shown is the temporal profile of the discharge current.

Spence et al.


143

Fig. 7. Normalized transverse intensity profiles in the entrance plane of the capillary and in the exit plane of the 30-mm-long capillary for various delays t and for an initial hydrogen pressure of 330 mbars. In all cases the spatial scale is in micrometers and the vertical scale is in units of 1017 W cm⫺2 . The intensity profile recorded for t ⫽ ⫺82 ns has been multiplied by 100, as indicated.

Fig. 8. Measured spectra of the transmitted laser pulses for t ⫽ 731 ns for a 30-mm-long capillary and initial hydrogen pressures in the range 70–330 mbars. The spectrum recorded by the output spectrometer with the capillary removed is also shown. The spectra have been displaced vertically to prevent overlapping.

⫾ 2) nm. We note that the spectra of pulses transmitted through the waveguide show some modulation. This modulation varied on a shot-to-shot basis under otherwise identical conditions. B. Discussion of the Results of the Guiding Experiments The data presented in Figs. 6 and 7 show that, for suitable delays t, intense laser pulses transmitted by a hydrogen-filled capillary discharge waveguide can have a high pulse-energy transmission and a peak axial intensity equal to a significant fraction of that of the input pulses. These results agree well with measurements and

simulations presented in Section 2, which show that the discharge can produce an essentially fully ionized parabolic channel. It is expected, therefore, that for optimum delays ionization-induced refractive defocusing of the propagating laser pulses will have been negligible. For a parabolic plasma channel, a mismatch between input spot size W 0 of the laser pulses and matched spot size W M of the waveguide causes the spot size of the guided laser pulse to oscillate between W 0 and W M 2 /W 0 , with a wavelength14 along the propagation axis of Z s ⫽ ␲ 2 W M 2 /␭. It is unlikely that the spot size of the input laser pulses would have been perfectly matched to that of the plasma channel. However, even with mismatched guiding, the oscillations in the axial intensity of the guided beam are not necessarily particularly severe. To illustrate this point, for the 400-␮m-diameter capillaries employed, Eq. (1) predicts that the matched spot size of the plasma channel would have decreased from approximately 43 to 33 ␮m as the initial hydrogen pressure was increased from 110 to 330 mbars. For the worst case of W M ⫽ 43 ␮ m, and a known spot size of the input beam of W 0 ⫽ 31 ␮ m, the spot size of the guided beam would have oscillated from 31 to 60 ␮m. If other losses are ignored, this oscillation in spot size would have corresponded to an oscillation in the peak axial intensity of the propagating pulse of 100–27% of that of the input laser pulses. At higher initial hydrogen pressures the oscillation of the spot size of the plasma channel will have been much less severe. As shown in Fig. 8, for times that correspond to good guiding the pulses transmitted by the waveguide were not broadened spectrally but were shifted to shorter wavelengths. The measured wavelength shift was independent of the initial hydrogen pressure but was approximately proportional to the length of the capillary. We

144


conclude, therefore, that the observed blueshift was the result not of laser-induced ionization of residual neutral hydrogen but of ionization of low levels of species from the capillary wall. The wavelength shift experienced by a laser pulse propagating through a length l of ionizing plasma is given by30 ⌬␭ ⫽ ⫺(1/2␲ )(e 2 ␭ 0 3 / 3 4␲ ␧ 0 m e c )l ⳵ N e / ⳵ t, where ␭ 0 is the central (vacuum) laser wavelength and the other symbols have their usual meanings. From the wavelength shift per unit length deduced from the linear fit, we found that ⳵ N e / ⳵ t ⫽ 7 ⫻ 1029 cm⫺3 s⫺1 . Assuming that the laser-induced ionization occurred for a time equal to the 70-fs duration of the laser pulse, the increase in electron density is calculated to be ⌬N e ⫽ 5 ⫻ 1016 cm⫺3 . This increase corresponds to only 0.3% of the total electron density for an initial hydrogen pressure of 330 mbars, indicating that the level of impurities in the plasma channel was relatively small. This simple analysis has been confirmed by numerical simulations of the propagation of intense laser pulses through plasma channels containing low levels of aluminum and oxygen ions from the capillary wall.27 The simulations show that the observed blueshift arises from a slight steepening of the leading edge of the temporal profile of the laser pulse but that the peak axial intensity and the trailing edge of the pulse are little affected. Thus the procedure adopted for converting the measured CCD profiles into intensity profiles is expected to be valid. The losses of the waveguide can be estimated from the measured pulse-energy transmission. For both capillary lengths the peak axial intensity of the transmitted pulses was found to be greatest for t ⬇ 730 ns and an initial hydrogen pressure of 330 mbars. Under these conditions the pulse-energy transmission was (82 ⫾ 4)% and (70 ⫾ 4)% for the 30- and 50-mm-long capillaries, respectively. Assuming that the transmission of the capillary can be modeled by T ⫽ T 0 exp(⫺␣l), we found that T 0 ⫹0 ⫽ (100⫺4 )% and ␣ ⫽ (7 ⫾ 1) m⫺1 . To our knowledge, these results show that the hydrogen-filled capillary discharge waveguide has the lowest coupling and propagation losses reported for any waveguide that is able to guide laser pulses with peak input intensities of 1017 W cm⫺2 or greater.

4. APPLICATION TO SHORT-WAVELENGTH LASERS AND PLASMA ACCELERATORS To illustrate the potential of hydrogen-filled capillary discharge waveguides, we now consider two important applications to which it would seem to be well suited. A. Short-Wavelength Lasers Lasers operating at x-ray wavelengths require pumping at extremely high power densities for achieving the necessary population inversion. The first x-ray lasers to be demonstrated employed large visible or infrared laser systems to produce a plasma from a solid target and to heat it sufficiently to produce a high density of the lasant ion. Lasing then occurred by electron collisional excitation31 or electron–ion recombination.32 Ever since these first demonstrations considerable effort has been devoted to reducing the size, and increasing the repetition rate, of the laser systems that are used to

Spence et al.

pump x-ray lasers. Significant progress toward this goal was achieved by use of visible or infrared pump pulses from picosecond and femtosecond laser systems. In particular, for collisionally excited lasers the total pump laser energy required has been reduced dramatically by use of a long, nanosecond laser pulse to produce and heat a plasma sufficiently to generate a large fraction of the lasant ion and by subsequent pumping with intense picosecond laser pulses to produce high transient gain.33–35 Pumping of short-wavelength lasers in fast capillary discharges was reported36 by Rocca et al. In that research a current pulse with a peak of the order of 40 kA was driven through a capillary of 4-mm diameter containing approximately 1 Torr of argon gas to form a hot, compressed plasma column that comprised Ar8⫹ ions. Electron collisional excitation then resulted in lasing on the 3p – 3s transition of Ar8⫹ at 46.9 nm. This concept was recently extended by use of a combination of a fast capillary discharge to produce the initial lasant ions and subsequent heating by a picosecond laser pulse to produce high transient gain.37 The recent development of femtosecond laser systems that produce peak output powers of several terawatts has provided the possibility of pumping short-wavelength lasers with a compact, high-repetition-rate source. At intensities of the order of 1016 W cm⫺2 and above, the electric field of electromagnetic radiation is comparable with that which binds valence electrons within an atom. As such, atoms subjected to laser pulses with intensities of this order are ionized by distortion of the intra-atomic potential, a process known as optical field ionization (OFI). Further, the energy of the ionized electrons can be controlled by adjustment of the polarization of the driving laser: Circularly polarized radiation generates hot electrons; linearly polarized radiation produces relatively cold electrons.38 Hence OFI offers considerable control over both the ion stage and the electron temperature of the plasma that is produced, permitting, in principle, the generation of population inversions by either electron collisions or electron–ion recombination. For short-wavelength lasers driven by picosecond or femtosecond optical pulses the duration of the population inversion produced is usually short compared to l g /c, where l g is the gain length, in which case the pumping must be in a traveling-wave configuration. The most convenient way to achieve traveling-wave pumping is for the pump radiation to propagate along the axis of the short-wavelength laser. However, the gain length produced by longitudinal pumping of this type will be limited by diffraction and by ionization-induced refraction. In as much as the output of such lasers consists of single-pass amplified spontaneous emission, the production of a long gain length is crucial for obtaining sufficient amplification for saturated extraction of the energy stored in the population inversion and for the amplified spontaneous emission output to have a low beam divergence. It is apparent, therefore, that short-wavelength lasers pumped by picosecond or femtosecond laser pulses would benefit greatly from guiding of the pump laser radiation to produce longer gain lengths. Driving shortwavelength lasers within a waveguide should greatly increase the output energy of existing laser systems as well

Spence et al.


145

as permit many new transitions to reach the threshold single-pass gain for lasing to occur. To illustrate the application of gas-filled capillary discharge waveguides to driving short-wavelength lasers, we consider two OFI laser systems: the collisionally excited lasers operating in eight-times-ionized argon, krypton, and xenon that were proposed by Lemoff et al.,39 and a recombination laser operating40 in sodiumlike argon. We simulate the propagation of the driving laser pulses through uniform gas cells and doped plasma channels and compare the distribution of lasant ion stages produced in each case. The propagation code that was described previously23 solves the paraxial wave equation for a laser pulse propagating in a plasma, with an arbitrary transverse profile, that is undergoing optical field ionization. Other nonlinear effects are neglected.

1. Collisionally Excited Optical Field Ionization Lasers The Xe8⫹ laser was the first collisionally excited OFI laser to be demonstrated experimentally. The laser operates at 41.8 nm on the 4d 9 5d 1 S 0 – 4d 9 5p 1 P 1 transition in Xe8⫹ produced by OFI with circularly polarized driving radiation. In the first demonstration5 of this system, 70mJ, 40-fs, 800-nm pulses from a Ti:sapphire laser were focused into a differentially pumped gas cell that had been backfilled with xenon to a pressure of several millibars. Refraction limited the length over which Xe8⫹ was generated, and hence l g , to less than 8.5 mm. Saturation of this laser transition was recently achieved by Sebban et al.,41 who used 330-mJ, 35-fs laser pulses from a Ti:sapphire laser system. Figure 9(a) shows the calculated distribution of xenon ions produced by focusing of the pump radiation at the entrance plane of a cell of pure xenon gas with an initial pressure of 5 mbars. The pump laser radiation was taken to have a sech-squared temporal profile, with a full width at half-maximum of 50 fs, and to be focused to a waist of 25 ␮m at a peak axial intensity of 8 ⫻ 1016 W cm⫺2 . It can be seen that the lasant ion stage is generated over a length of only 4 mm before ionizationinduced refraction causes the intensity of the pump radiation to fall below the threshold intensity for producing Xe8⫹. Of course, slightly longer lengths of the lasant ion could be generated with the same pump laser pulse by focusing slightly into the gas cell, as implemented by Lemoff et al. and Sebban et al. in their experiments. Figure 9(b) shows the calculated distribution of xenon ions produced by the same driving laser pulse propagated through a hydrogen-filled capillary discharge waveguide doped with 5 mbars of xenon. The plasma channel was assumed to be parabolic, with a matched spot size of 25 ␮m and an axial electron density of 6 ⫻ 1018 cm⫺3 . The temperature of the discharge plasma was taken to be 6 eV, in agreement with the MHD simulations discussed above. It was assumed that the initial ionization state of the plasma was determined by the condition for Saha equilibrium at a plasma temperature of 6 eV: The hydrogen was fully ionized, and the xenon was ionized to Xe3⫹. The initial radial distribution of the xenon ions established by the discharge was assumed to have the same parabolic profile as the total electron density.

Fig. 9. Calculated xenon ion stage as a function of radius and propagation distance z after passage of an 800-nm, 50-fs laser pulse focused to a peak intensity of 8 ⫻ 1016 W cm⫺2 and waist of 25 ␮m at the entrance to (a) a gas cell containing 5 mbars of xenon gas and (b) a hydrogen-filled capillary discharge waveguide doped with 5 mbars of xenon. Note that in (a) the simulation extends only to z ⫽ 70 mm.

For the doped plasma channel the total length over which Xe8⫹ is calculated to be produced is approximately 84 mm, more than a factor of 20 longer than for the static gas cell. Figure 9(b) shows evidence of oscillation of the spot size of the driving laser pulse with propagation distance as a result of ionization-induced defocusing by the partially ionized xenon and of subsequent refocusing by the plasma channel. This perturbation of the propagating laser pulse results in several small regions in which the Xe is overionized to Xe9⫹. These regions are unlikely to have a deleterious effect on the operation of the Xe8⫹ laser but in any case could be avoided by optimization of the intensity of the pump laser pulses and the initial partial pressure of xenon. It is important to consider not only the increase in gain length achieved but also whether suitable conditions for creating gain on the Xe8⫹ laser transition would exist for the case of the xenon-doped plasma channel. Whereas detailed modeling of the gain generated under such conditions is beyond the scope of the present paper, we may make some general observations. The main differences from the case of the static gas fill are the partial preionization of the xenon by the discharge and the large density of electrons from ionization of the hydrogen by the discharge. Inverse bremsstrahlung heating of the discharge electrons by the femtosecond driving laser pulse is expected to be insignificant,42 so immediately after the passage of the driving laser pulse the distribution of electron energies will consist of two groups: the hot electrons

146


produced by OFI of Xe3⫹ ions and the relatively cold electrons from the discharge. Figure 10 shows the calculated distribution of electron energies produced by OFI of neutral Xe by the pump laser pulse considered above, together with a Maxwellian distribution with a mean energy of 6 eV. The OFI produces eight classes of electron energy, each of which corresponds to ionization of one of the ions in the series from Xe to Xe8⫹, with mean energies of approximately 17, 43, 94, 160, 250, 370, 810, and 1100 eV. The upper and lower energy levels of the Xe8⫹ laser lie 104 and 75 eV above the ground state of the ion, and hence only the higher six OFI electron classes have sufficient energy to contribute to the collisional pumping of either of the laser levels. When the Xe laser is driven within a doped hydrogenfilled capillary discharge waveguide, the plasma will contain a large density of discharge electrons with an energy of approximately 6 eV, and the lowest three OFI electron classes will be missing owing to the partial preionization of the Xe by the discharge. For a density of discharge electrons equal to 6 ⫻ 1018 cm⫺3 the times required for the five remaining OFI electron classes to equilibrate their energy with the discharge electrons were calculated43 to be 20, 38, 68, 220, and 340 ps in order of increasing electron energy. The duration of the population inversion on the short-wavelength laser transition is expected to be a few tens of picoseconds.39 Clearly the evolution of the electron energy distribution in this system is a complex matter. However, this simple calculation suggests that on the time scale of the duration of the population inversion the discharge electrons are not expected to equilibrate significantly with the OFI electrons, especially not the with the most energetic OFI electrons, which are responsible for pumping the upper laser level. We note also that preionization of the xenon by the discharge to Xe3⫹ will mean that only the five highest-energy OFI electron classes will be available for pumping the laser levels. However, this difference will decrease the rate of excitation of the upper and lower laser levels only slightly compared with operation in a pure-xenon gas cell. The linewidth of the Xe8⫹ laser transition will increase as results of increased Stark broadening by the discharge

Fig. 10. Calculated electron energy distribution following OFI of neutral xenon with a circularly polarized, 800-nm laser pulse of 50-fs duration (FWHM) and peak intensity 8 ⫻ 1016 W cm⫺2 . Also shown are a Maxwellian distribution of electron energies for a temperature of 6 eV and the energies of the Xe8⫹ laser levels.

Spence et al.

electrons and of an increase in the Doppler width of the transition caused by discharge heating of the xenon ions to 6 eV. We used the Cowan code44 to calculate the radiative lifetimes of the upper and lower laser levels to be 28.7 and 10.4 ps, respectively. We calculated the Stark broadening of the laser transition within the impact approximation by using classic Kramers–Gaunt factors45,46 as described previously by Hooker.47 At a pressure of 5 mbars, following OFI of neutral Xe, the natural (⌬ ␯ N ), Stark (⌬ ␯ S ), and Doppler (⌬ ␯ D ) widths of the laser transition were found to be 21, 55, and 8.8 GHz, respectively. Taking the linewidth ⌬␯ of the laser transition to be given by ⌬ ␯ ⬇ (⌬ ␯ N 2 ⫹ ⌬ ␯ S 2 ⫹ ⌬ ␯ D 2 ) 1/2, we found that ⌬ ␯ ⬇ 60 GHz. At a density of 6 ⫻ 1018 cm⫺3 the 6-eV discharge electrons were calculated to increase the Stark width to approximately 1280 GHz, and the increased ion temperature increased the Doppler width to 118 GHz. Consequently the linewidth and the optical gain cross section of the laser transition are expected to be increased and decreased, respectively, by a factor of approximately 20. We can see that for this example the greatly increased gain length produced by driving the laser within a doped plasma channel is predicted to be almost exactly balanced by the increase in the linewidth of the laser transition. Similar considerations are likely to be important for other collisionally excited OFI laser systems driven within the plasma channel of doped hydrogen-filled capillary discharge waveguides. However, the small-signal gain coefficient depends critically on the conditions of the plasma channel and on the properties of the energy levels of the lasant ion, and ascertaining whether operating within waveguides of this type can increase the net single-pass gain that can be achieved will require more-detailed modeling. It may be possible to develop gas-filled capillary discharge waveguides for which the ratio ⌬N e /N e (0) is significantly greater than has hitherto been demonstrated, by using capillaries of smaller diameter or by employing magnetic fields to shape the radial plasma profile.48 This development would allow the partial pressure of hydrogen relative to that of the lasant ions to be decreased, reducing the associated additional contribution to the Stark broadening of the laser transition.

2. Recombination Optical Field Ionization Lasers Doped hydrogen-filled capillary discharge waveguides may be well suited to driving recombination OFI lasers within the plasma channel. Grout et al. have shown theoretically40 that one can dramatically increase the gain of recombination OFI lasers by doping the lasant gas into a buffer gas of hydrogen. In their scheme, OFI of hydrogen generates a dense pool of cold electrons, which decreases the average electron temperature T e significantly from that achieved by OFI of the target laser gas alone. Because the rate of three-body recombination scales49 as T e ⫺9/2, use of a buffer gas can greatly increase the gain that is generated. For example, Grout et al. calculated that for pure-argon targets the small-signal gain of the recombination laser transition at 23.2 nm in Ar7⫹ does not exceed 10 cm⫺1. However, for targets that comprised 4 mbars of argon doped into 68 mbars of hydrogen the elec-

Spence et al.

tron temperature was calculated to be reduced to approximately 7 eV, leading to a small-signal gain coefficient as high as 350 cm⫺1. It would seem that this recombination laser scheme could be applied to doped hydrogen-filled capillary discharge waveguides. In this case the pool of cold electrons would be produced by the discharge ionization of the hydrogen, and the lasant ion stage before recombination would be generated by OFI with linearly polarized laser radiation. As indicated above, inverse bremsstrahlung heating of the discharge electrons by femtosecond driving laser pulses is not expected to be efficient; it will leave a high density of discharge electrons with a temperature of ⬃6 eV. The energy of the OFI electrons will be higher than this value, but their density will be low. Consequently it would seem that immediately after the passage of the driving laser pulse the plasma conditions will be similar to those for which Grout et al. calculated high gain. The advantage of operating in a doped hydrogenfilled capillary discharge waveguide is that the gain length may be extended to long lengths. Figure 11 shows the calculated distribution of argon ion stages as a function of radius and length for a 250-nm, 100-fs laser pulse focused to a waist of 30 ␮m and a peak intensity of 5 ⫻ 1016 W cm⫺2 at the entrance to a hydrogen-filled capillary discharge waveguide doped with 4 mbars of argon. It can be seen that the required ion stage before recombination, Ar8⫹, is generated over a length of more than 300 mm. If these conditions are achieved, they should lead to extreme saturation of the laser transition and extraction of a high proportion of the energy stored in the population inversion.

Fig. 11. Calculated argon ion stage as a function of radius and propagation distance z after passage of a 250-nm, 100-fs laser pulse focused to a peak intensity of 5 ⫻ 1016 W cm⫺2 and a waist of 30 ␮m at the entrance to a hydrogen-filled capillary discharge waveguide doped with 4 mbars of argon. The matched spot size of the un-ionized channel was assumed to be 22 ␮m.


147

3. Discussion In summary, for collisionally excited OFI lasers the large increase in gain length that could be generated by operation within a doped hydrogen-filled capillary discharge waveguide will, to some extent, be offset by increases in the linewidth of the laser transition and the consequent decrease in the gain coefficient. Detailed modeling, or experiment, will be required in each case for determining whether there will be an advantage to pumping within this type of waveguide. However, there would appear to be a distinct advantage to pumping recombination OFI lasers in doped hydrogen-filled capillary discharge waveguides. This interesting possibility warrants further investigation theoretically and experimentally.

B. Laser Wakefield Acceleration The acceleration of charged particles by the large electric fields generated within laser-produced plasma waves has been of great scientific interest since the concept was proposed50 by Tajima and Dawson in 1979. The main reason for this interest is the large acceleration gradients that can be achieved in principle, offering the prospect of compact accelerators for acceleration to GeV energies and beyond. The accelerating electric field within a plasma wave can be as much as the order of the wave-breaking field E 0 ⫽ m e ␻ p c/e, where ␻ p is the plasma frequency.51 For electron densities of the order of 1018 cm⫺3 we find that E 0 ⬃ 100 GV m⫺1 , some 3 orders of magnitude greater than the accelerating field of order 100 MV m⫺1 that is achievable with conventional rf-driven accelerators. A wide variety of laser-driven plasma accelerator schemes has been investigated to date,52–56 as recently reviewed by Bingham.57 For a number of applications the laser wakefield acceleration scheme is particularly promising because it offers the possibility of generating pulses of nearly monoenergetic electrons with pulse durations of only a few femtoseconds. In this approach the ponderomotive force at the front and the back of an intense laser pulse pushes plasma electrons away from the laser pulse in the forward and backward directions to excite a trailing plasma wave that travels at the group velocity of the driving laser pulse. The plasma wave contains both accelerating and deaccelerating phases, each of which is divided into radially focusing and defocusing phases. For laser wakefield acceleration, electrons must be injected into the plasma wave from an external source. In order that the range of accelerating fields experienced by the electrons within the bunch be small, the duration of the bunch must be short compared with the plasma period, ␶ p ⫽ 2 ␲ / ␻ p . The length over which electrons may be accelerated is known as the dephasing length, L D ⫽ (1/2)␭ p 3 /␭ 2 , where ␭ p and ␭ are the wavelengths of the plasma wave and the driving laser, respectively,58,59 and the factor of (1/2) allows for the fact that the radial electric field is focusing for only half of the accelerating phase of the plasma wave. To provide some indication of the plasma conditions that are suitable for laser wakefield acceleration it is use-

148


Spence et al.

ful to employ a nonrelativistic analytical model.59,60 Within this model, for a laser pulse that is Gaussian in space and time the peak longitudinal electric field on axis is given by E zpeak

⫽

4I max冑␲ ln 2 ec 2 N c ␶

冉冊冋冉冊册 ␻ p␶

4 冑ln 2

2

exp ⫺

␻ p␶

4 冑ln 2

2

, (2)

where N c ⫽ 4 ␲ c m e ␧ 0 /␭ e is the critical plasma density and I max and ␶ are the peak intensity and duration (full width at half-maximum) of the laser pulse. It can be shown straightforwardly from Eq. (2) that for Gaussian laser pulses the accelerating electric field is a maximum when the resonance condition ␻ p ␶ ⫽ 4 冑ln 2 ⬇ 3.33 is met. Provided that the intensity of the driving laser can be maintained over the dephasing length, electrons injected into the correct phase of the plasma wave may be accelerated to an energy of W ⫽ E zpeakL D . For resonant excitation E zpeak ⬀ ␻ p , and because L D ⬀ ␻ p ⫺3 , the maximum electron energy is proportional to ␻ p ⫺2 , or N e ⫺1 . Therefore, provided that the driving laser pulse can be guided over the dephasing length, the electron energy produced by laser wakefield acceleration increases as the electron density is reduced. Operation at lower plasma densities also has the advantage that the plasma period is longer, making it easier to inject a pulse of electrons with a duration short compared to ␶ p . We now employ the results of the linear theory outlined above to estimate the electron energies that might be generated in a gas-filled capillary discharge waveguide. The lowest electron density for which guiding in hydrogenfilled capillary discharge waveguides has been demonstrated27,29 to date is approximately 1.5 ⫻ 1018 cm⫺3 , which correspondes to a dephasing length of 16 mm for laser pulses with ␭ ⫽ 800 nm. We note that this dephasing length is much greater than the interaction length that would be achieved in the absence of a waveguide. The resonant pulse duration for this plasma density is 48 fs, and the plasma period is 91 fs. The hydrogen-filled capillary discharge waveguide has been demonstrated to guide optical pulses with an approximately constant spot size of 30 ␮m. Assuming this value, we predict that 1-J, 48-fs optical pulses guided over the dephasing length will generate a plasma wave with a peak longitudinal electric field of 25 GV m⫺1, which will be capable of accelerating electrons over the dephasing length to an energy of 0.4 GeV. The peak axial intensity of the driving pulses considered above is 1.5 ⫻ 1018 W cm⫺2 , approximately an order of magnitude greater than the intensity of the pulses guided to date in hydrogen-filled capillary discharge waveguides, and it is appropriate to consider whether guiding will still be effective. For a fully ionized plasma channel, such as that produced by a hydrogen-filled capillary discharge waveguide, the guiding properties are independent of intensity until relativistic and ponderomotive effects become important. For the plasma conditions considered, the critical power for relativistic self-focusing is 20 TW, approximately equal to the peak power of the 1-J, 48-fs laser pulses assumed here. However, Sprangle et al. have 2 2

2 2

shown that for pulses with a duration shorter than the plasma period, as is the case for the laser wakefield scheme, relativistic and ponderomotive focusing are ineffective. For example, numerical simulations61 by those authors have shown that a fully ionized plasma channel can guide pulses with a mean intensity of 1 ⫻ 1018 W cm⫺2 over more than 10 Rayleigh ranges with little temporal or spatial distortion. The plasma channel was found to be effective at guiding the laser pulse even though the laser pulse drove a strong plasma wave that was suitable for accelerating electrons to high energies. We note that the spectral blueshift observed in the present experiments is unlikely to have a significantly deleterious effect on the generation of the plasma wave. As discussed in Subsection 3.B, the observed blueshift is caused by ionization of low levels of impurity species from the capillary wall. This ionization occurs predominantly on the leading edge of the laser pulse, causing it to steepen slightly, but the peak and trailing edge of the pulse are not affected greatly. The properties of the plasma wave generated by an intense laser pulse depend primarily on the peak intensity of the pulse and the temporal profile of the trailing edge and not on the laser wavelength. Thus this plasma wave should not be affected significantly by the presence of low levels of partially ionized impurity species in the plasma channel. Further numerical and experimental work will be necessary to ascertain whether in fact there is no such effect in practice. This preliminary analysis suggests that the use of hydrogen-filled capillary discharge waveguides is a promising guiding technique for laser wakefield acceleration. Guiding of intense pulses over the lengths that correspond to the dephasing length has already been demonstrated for hydrogen-filled capillary discharge waveguides. Acceleration to electron energies above 1 GeV could be achieved if waveguides able to operate at lower plasma pressures were developed. Further, it was recently demonstrated62 theoretically that, for plasma channels in which there is a suitable longitudinal increase in the axial electron density, there exists a point on the trailing plasma wave that propagates at c. This propagation would permit acceleration beyond the dephasing limit. It may be possible to create the desired longitudinal variation of the plasma density in gas-filled capillary discharge waveguides by flowing the initial gas fill along tapered capillaries. This approach could be extended to developing plasma channels with periodic variations in the plasma density, with potential applications in phase-matched high-harmonic generation and the development of plasma-based free-electron lasers.

5. CONCLUSIONS We have presented the results of recent investigations of a new type of waveguide for high-intensity laser pulses: the gas-filled capillary discharge waveguide. Interferometric measurements and magnetohydrodynamic simulations showed that hydrogen-filled capillary discharges are able to form stable, long-lived, and essentially fully ionized parabolic plasma channels suitable for guiding intense laser pulses.

Spence et al.

We have presented the results of recent experiments to demonstrate guiding of laser pulses with a peak input intensity of 1.2 ⫻ 1017 W cm⫺2 in hydrogen-filled capillary discharge waveguides as much as 50 mm long. Under good guiding conditions the coupling and propagation losses of the waveguide were determined to be ⬍4% and (7 ⫾ 1) m⫺1 , respectively. Measurements of the spectra of pulses transmitted by the waveguide showed negligible spectral broadening, although a small blueshift was observed. The blueshift was found to be consistent with laser-induced ionization of low levels of impurity species from the capillary wall. It is likely that this spectral shift can be reduced significantly by improved design and by the use of alternative materials for the capillary. Gas-filled capillary discharge waveguides have a number of important features. Producing the plasma channel requires no auxiliary laser systems. The discharge circuit is simple and does not require fast switching of high voltages. The plasma channel is long-lived, permitting straightforward synchronization with the laser system, and is essentially fully ionized, thereby minimizing temporal or spatial distortion of the propagating laser pulse. Finally, of great practical importance, the lifetime of the capillary is long: Approximately 103 laser pulses have been guided through a single capillary with no degradation of the guiding performance.27,29 We have considered two potential applications of gasfilled capillary discharge waveguides: short-wavelength lasers and particle acceleration. We found that for collisionally excited optical field ionization lasers the large increase in gain length that can be generated by operation within a doped hydrogen-filled capillary discharge waveguide would be offset to some extent by increases in the linewidth of the laser transition. However, it was suggested that high single-pass gains could be produced by pumping of recombination OFI lasers in doped hydrogenfilled capillary discharge waveguides. We presented a simple analysis of the prospects of applying hydrogen-filled capillary discharge waveguides to laser wakefield acceleration. It was shown that, for the plasma channels that have already been demonstrated to guide intense laser pulses, acceleration of electrons to energies of the order of 0.4 GeV should be possible. The extension of this approach to lower electron densities should permit acceleration to energies of 1 GeV and above. Further, the development of waveguides with longitudinal structure may allow the dephasing limit to be overcome, and such waveguides could find a wide variety of applications in laser–plasma physics.


Hooker is very grateful to the Royal Society for the support of a Royal Society University Research Fellowship. D. J. Spence’s @physics.ox.ac.uk.

The authors acknowledge invaluable technical assistance from the Astra laser and target area support staff at the Rutherford Appleton Laboratory. They also thank the Engineering and Physical Sciences Research Council (EPSRC, UK) for financial support through grant GR/ M88969. D. J. Spence and A. Butler appreciate the support of studentships from the EPSRC, and D. J. Spence also thanks the Rutherford Appleton Laboratory for a Cooperative Award in Science and Engineering. S. M.

e-mail

address

is

d.spence1

REFERENCES 1. 2. 3. 4. 5.

6.

7.

8.

9.

10.

11. 12.

13.

ACKNOWLEDGMENTS

149

14. 15. 16.

17.

H. M. Milchberg, C. G. Durfee, and T. J. McIlrath, ‘‘Highorder frequency conversion in the plasma waveguide,’’ Phys. Rev. Lett. 75, 2494–2497 (1995). E. Esarey, P. Sprangle, J. Krall, A. Ting, and G. Joyce, ‘‘Optically guided laser wakefield acceleration,’’ Phys. Fluids B 5, 2690–2697 (1993). E. Esarey, P. Sprangle, J. Krall, and A. Ting, ‘‘Overview of plasma-based accelerator concepts,’’ IEEE Trans. Plasma Sci. 24, 252–288 (1996). D. V. Korobkin, C. H. Nam, S. Suckewer, and A. Goltsov, ‘‘Demonstration of soft x-ray lasing to ground state in Li III,’’ Phys. Rev. Lett. 77, 5206–5209 (1996). B. E. Lemoff, G. Y. Yin, C. L. Gordon, C. P. J. Barty, and S. E. Harris, ‘‘Demonstration of a 10-Hz femtosecond pulsedriven XUV laser at 41.8 nm in Xe IX,’’ Phys. Rev. Lett. 74, 1574–1577 (1995). S. Jackel, R. Burris, J. Grun, A. Ting, C. Manka, K. Evans, and J. Kosakowskii, ‘‘Channeling of terawatt laser pulses by use of hollow waveguides,’’ Opt. Lett. 20, 1086–1088 (1995). B. Cros, C. Courtois, G. Matthieussent, A. Di Bernardo, D. Batani, N. Andreev, and S. Kuznetsov, ‘‘Eigenmodes for capillary tubes with dielectric walls and ultraintense laser pulse guiding,’’ Phys. Rev. E 65, 026405 (2002). F. Dorchies, J. R. Marques, B. Cros, G. Matthieussent, C. Courtois, T. Velikoroussov, P. Audebert, J. P. Geindre, S. Rebibo, G. Hamoniaux, and F. Amiranoff, ‘‘Monomode guiding of 1016 W/cm2 laser pulses over 100 Rayleigh lengths in hollow capillary dielectric tubes,’’ Phys. Rev. Lett. 82, 4655– 4658 (1999). C. Courtois, B. Cros, G. Malka, G. Matthieussent, J. R. Marques, N. Blanchot, and J. L. Miquel, ‘‘Experimental study of short high-intensity laser-pulse monomode propagation in centimeter-long capillary tubes,’’ J. Opt. Soc. Am. B 17, 864–867 (2000). P. Monot, T. Auguste, P. Gibbon, F. Jakober, G. Mainfray, A. Dulieu, M. Louisjacquet, G. Malka, and J. L. Miquel, ‘‘Experimental demonstration of relativistic self-channeling of a multiterawatt laser pulse in an underdense plasma,’’ Phys. Rev. Lett. 74, 2953–2956 (1995). G. Z. Sun, E. Ott, Y. C. Lee, and P. Guzdar, ‘‘Self-focusing of short intense pulses in plasmas,’’ Phys. Fluids 30, 526–532 (1987). A. B. Borisov, A. V. Borovskiy, V. V. Korobkin, A. M. Prokhorov, O. B. Shiryaev, X. M. Shi, T. S. Luk, A. McPherson, J. C. Solem, K. Boyer, and C. K. Rhodes, ‘‘Observation of relativistic and charge-displacement self-channeling of intense subpicosecond ultraviolet (248-nm) radiation in plasmas,’’ Phys. Rev. Lett. 68, 2309–2312 (1992). P. Sprangle, E. Esarey, and A. Ting, ‘‘Nonlinear interaction of intense laser pulses in plasmas,’’ Phys. Rev. A 41, 4463– 4467 (1990). E. Esarey, P. Sprangle, J. Krall, and A. Ting, ‘‘Self-focusing and guiding of short laser pulses in ionizing gases and plasmas,’’ IEEE J. Quantum Electron. 33, 1879–1914 (1997). C. G. Durfee and H. M. Milchberg, ‘‘Light pipe for highintensity laser pulses,’’ Phys. Rev. Lett. 71, 2409–2412 (1993). S. P. Nikitin, I. Alexeev, J. Fan, and H. M. Milchberg, ‘‘High efficiency coupling and guiding of intense femtosecond laser pulses in preformed plasma channels in an elongated gas jet,’’ Phys. Rev. E 59, R3839–R3842 (1999). P. Volfbeyn, E. Esarey, and W. P. Leemans, ‘‘Guiding of laser pulses in plasma channels created by the ignitor–heater technique,’’ Phys. Plasmas 6, 2269–2277 (1999).

150 18.

19.

20.

21.

22.

23.

24.

25. 26. 27. 28.

29.

30. 31.

32.

33.

34.

35. 36.

J. Opt. Soc. Am. B / Vol. 20, No. 1 / January 2003 E. W. Gaul, S. P. Le Blanc, A. R. Rundquist, R. Zgadzaj, H. Langhoff, and M. C. Downer, ‘‘Production and characterization of a fully ionized He plasma channel,’’ Appl. Phys. Lett. 77, 4112–4114 (2000). J. Fan, E. Parra, I. Alexeev, K. Y. Kim, H. M. Milchberg, L. Y. Margolin, and L. N. Pyatnitskii, ‘‘Tubular plasma generation with a high-power hollow Bessel,’’ Phys. Rev. E 62, R7603–R7606 (2000). Y. Ehrlich, C. Cohen, A. Zigler, J. Krall, P. Sprangle, and E. Esarey, ‘‘Guiding of high intensity laser pulses in straight and curved plasma channel experiments,’’ Phys. Rev. Lett. 77, 4186–4189 (1996). D. Kaganovich, A. Ting, C. I. Moore, A. Zigler, H. R. Burris, Y. Ehrlich, R. Hubbard, and P. Sprangle, ‘‘High efficiency guiding of terawatt subpicosecond laser pulses in a capillary discharge plasma channel,’’ Phys. Rev. E 59, R4769– R4772 (1999). S. M. Hooker, D. J. Spence, and R. A. Smith, ‘‘Guiding of high-intensity picosecond laser pulses in a dischargeablated capillary waveguide,’’ J. Opt. Soc. Am. B 17, 90–98 (2000). D. J. Spence and S. M. Hooker, ‘‘Simulations of the propagation of high-intensity laser pulses in discharge-ablated capillary waveguides,’’ J. Opt. Soc. Am. B 17, 1565–1570 (2000). T. Hosokai, M. Kando, H. Dewa, H. Kotaki, S. Kondo, N. Hasegawa, K. Nakajima, and K. Horioka, ‘‘Optical guidance of terrawatt laser pulses by the implosion phase of a fast Z-pinch discharge in a gas-filled capillary,’’ Opt. Lett. 25, 10–12 (2000). C. Fauser and H. Langhoff, ‘‘Focussing of laser beams by means of a z-pinch formed plasma guiding system,’’ Appl. Phys. B 71, 607–609 (2000). D. J. Spence and S. M. Hooker, ‘‘Investigation of a hydrogen plasma waveguide,’’ Phys. Rev. E 63, 015401(R) (2001). A. Butler, D. J. Spence, and S. M. Hooker, ‘‘Guiding of highintensity laser pulses with a hydrogen-filled capillary discharge waveguide,’’ Phys. Rev. Lett. 89, 185003 (2002). N. A. Bobrova, A. A. Esaulov, J. I. Sakai, P. V. Sasorov, D. J. Spence, A. Butler, S. M. Hooker, and S. V. Bulanov, ‘‘Simulations of a hydrogen-filled capillary discharge waveguide,’’ Phys. Rev. E 65, 016407 (2002). D. J. Spence, A. Butler, and S. M. Hooker, ‘‘First demonstration of guiding of high-intensity laser pulses in a hydrogenfilled capillary discharge waveguide,’’ J. Phys. B 34, 4103– 4112 (2001). E. Yablonovitch, ‘‘Energy conservation in the picosecond and subpicosecond photoelectric effect,’’ Phys. Rev. Lett. 60, 795–796 (1988). D. L. Matthews, P. L. Hagelstein, M. D. Rosen, M. J. Eckart, N. M. Ceglio, A. U. Hazi, H. Medecki, B. J. Macgowan, J. E. Trebes, B. L. Whitten, E. M. Campbell, C. W. Hatcher, A. M. Hawryluk, R. L. Kauffman, L. D. Pleasance, G. Rambach, J. H. Scofield, G. Stone, and T. A. Weaver, ‘‘Demonstration of a soft-x-ray amplifier,’’ Phys. Rev. Lett. 54, 110– 113 (1985). S. Suckewer, C. H. Skinner, H. Milchberg, C. Keane, and D. Voorhees, ‘‘Amplification of stimulated soft-x-ray emission in a confined plasma column,’’ Phys. Rev. Lett. 55, 1753– 1756 (1985). H. Fiedorowicz, A. Bartnik, J. Dunn, R. F. Smith, J. Hunter, J. Nilsen, A. L. Osterheld, and V. N. Shlyaptsev, ‘‘Demonstration of a neonlike argon soft-x-ray laser with a picosecond-laser-irradiated gas puff target,’’ Opt. Lett. 26, 1403–1405 (2001). H. Daido, R. Kodama, K. Murai, G. Yuan, M. Takagi, Y. Kato, I. W. Choi, and C. H. Nam, ‘‘Significant improvement in the efficiency and brightness of the J ⫽ 0 – 1 19.6-nm line of the germanium laser by use of double-pulse pumping,’’ Opt. Lett. 20, 61–63 (1995). J. Nilsen, B. J. Macgowan, L. B. Dasilva, and J. C. Moreno, ‘‘Prepulse technique for producing low-Z Ne-like x-ray lasers,’’ Phys. Rev. A 48, 4682–4685 (1993). J. J. Rocca, V. Shlyaptsev, F. G. Tomasel, O. D. Cortazar, D. Hartshorn, and J. L. A. Chilla, ‘‘Demonstration of a dis-

Spence et al.

37.

38. 39. 40. 41.

42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52.

53.

54.

55.

56.

charge pumped table-top soft-x-ray laser,’’ Phys. Rev. Lett. 73, 2192–2195 (1994). K. A. Janulewicz, J. J. Rocca, F. Bortolotto, M. P. Kalachnikov, V. N. Shlyaptsev, W. Sandner, and P. V. Nickles, ‘‘Demonstration of a hybrid collisional soft-x-ray laser,’’ Phys. Rev. A 6303, 033803 (2001). P. B. Corkum, N. H. Burnett, and F. Brunel, ‘‘Abovethreshold ionization in the long-wavelength limit,’’ Phys. Rev. Lett. 62, 1259–1262 (1989). B. E. Lemoff, C. P. J. Barty, and S. E. Harris, ‘‘Femtosecondpulse-driven, electron-excited XUV lasers in 8-timesionized noble gases,’’ Opt. Lett. 19, 569–571 (1994). M. J. Grout, K. A. Janulewicz, S. B. Healy, and G. J. Pert, ‘‘Optical-field induced gas mixture breakdown for recombination x-ray lasers,’’ Opt. Commun. 141, 213–220 (1997). S. Sebban, R. Haroutunian, P. Balcou, G. Grillon, A. Rousse, S. Kazamias, T. Marin, J. P. Rousseau, L. Notebaert, M. Pittman, J. P. Chambaret, A. Antonetti, D. Hulin, D. Ros, A. Klisnick, A. Carillon, P. Jaegle, G. Jamelot, and J. F. Wyart, ‘‘Saturated amplification of a collisionally pumped opticalfield-ionization soft x-ray laser at 41.8 nm,’’ Phys. Rev. Lett. 86, 3004–3007 (2001). G. J. Pert, ‘‘Inverse bremsstrahlung in strong radiation fields at low temperatures,’’ Phys. Rev. E 51, 4778–4789 (1995). L. Spitzer, Physics of Fully Ionized Gases (Interscience, Princeton, N.J., 1962). R. D. Cowan, The Theory of Atomic Structure and Spectra (University of California Press, Berkeley, Calif., 1981). J. D. Hey and P. Breger, ‘‘The classical Kramers–Gaunt factor in Stark-broadening calculations,’’ J. Phys. B 22, L79– L82 (1989). J. D. Hey and P. Breger, ‘‘Calculated Stark widths of oxygen ion lines,’’, J. Quant. Spectrosc. Radiat. Transfer 24, 349– 364 (1980). S. M. Hooker, ‘‘Inner-shell soft x-ray lasers in Ne-like ions driven by optical field ionization,’’ Opt. Commun. 182, 209– 219 (2000). V. V. Ivanov, K. N. Koshelev, E. S. Toma, and F. Bijkerk, Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, Russia (personal communication, 2002). G. J. Pert, ‘‘Models of collisional radiative recombination,’’ J. Phys. B 23, 619–650 (1990). T. Tajima and J. M. Dawson, ‘‘Laser electron accelerator,’’ Phys. Rev. Lett. 43, 267–270 (1979). A. Ogata and K. Nakajima, ‘‘Recent progress and perspectives of laser-plasma accelerators,’’ Laser Part. Beams 16, 381–396 (1998). F. Amiranoff, J. Ardonceau, M. Bercher, D. Bernard, B. Cros, A. Debraine, J. M. Dieulot, J. Fusellier, F. Jacquet, J. M. Joly, M. Juillard, G. Matthieussent, P. Matricon, P. Mine, B. Montes, P. Mora, R. Morano, J. Morillo, F. Moulin, P. Poilleux, A. E. Specka, and C. Stenz, ‘‘The plasma beatwave acceleration experiment at Ecole Polytechnique,’’ Nucl. Instrum. Methods Phys. Res. A 363, 497–510 (1995). F. Amiranoff, S. Baton, D. Bernard, B. Cros, D. Descamps, F. Dorchies, F. Jacquet, V. Malka, J. R. Marques, G. Matthieussent, P. Mine, A. Modena, P. Mora, J. Morillo, and Z. Najmudin, ‘‘Observation of laser wakefield acceleration of electrons,’’ Phys. Rev. Lett. 81, 995–998 (1998). R. Wagner, S. Y. Chen, A. Maksimchuk, and D. Umstadter, ‘‘Electron acceleration by a laser wakefield in a relativistically self-guided channel,’’ Phys. Rev. Lett. 78, 3125–3128 (1997). D. Gordon, K. C. Tzeng, C. E. Clayton, A. E. Dangor, V. Malka, K. A. Marsh, A. Modena, W. B. Mori, P. Muggli, Z. Najmudin, D. Neely, C. Danson, and C. Joshi, ‘‘Observation of electron energies beyond the linear dephasing limit from a laser-excited relativistic plasma wave,’’ Phys. Rev. Lett. 80, 2133–2136 (1998). C. E. Clayton, K. A. Marsh, A. Dyson, M. Everett, A. Lal, W. P. Leemans, R. Williams, and C. Joshi, ‘‘Ultrahigh-gradient acceleration of injected electrons by laser-excited relativistic electron-plasma waves,’’ Phys. Rev. Lett. 70, 37–40 (1993).

Spence et al. 57. 58. 59.

R. Bingham, ‘‘Plasma accelerators: part 1,’’ submitted to Plasma Phys. Controlled Fusion. P. Mora and F. Amiranoff, ‘‘Electron acceleration in a relativistic electron-plasma wave,’’ J. Appl. Phys. 66, 3476– 3481 (1989). F. Dorchies, F. Amiranoff, V. Malka, J. R. Marques, A. Modena, D. Bernard, F. Jacquet, P. Mine, B. Cros, G. Matthieussent, P. Mora, A. Solodov, J. Morillo, and Z. Najmudin, ‘‘Acceleration of injected electrons in a laser wakefield experiment,’’ Phys. Plasmas 6, 2903–2913 (1999).

Vol. 20, No. 1 / January 2003 / J. Opt. Soc. Am. B 60. 61. 62.

151

L. M. Gorbunov and V. I. Kirsanov, ‘‘Excitation of plasma waves by short electromagnetic wave packets,’’ Zh. Eksp. Teor. Fiz. 93, 509–518 (1987). P. Sprangle, E. Esarey, J. Krall, and G. Joyce, ‘‘Propagation and guiding of intense laser pulses in plasmas,’’ Phys. Rev. Lett. 69, 2200–2203 (1992). P. Sprangle, B. Hafizi, J. R. Penano, R. F. Hubbard, A. Ting, A. Zigler, and T. M. Antonsen, ‘‘Stable laser-pulse propagation in plasma channels for GeV electron acceleration,’’ Phys. Rev. Lett. 85, 5110–5113 (2000).