The role of the gas/plasma plume and self-focusing in a gas-filled capillary discharge waveguide for high-power laser-plasma applications

The gas-filled capillary discharge waveguide (CDW) is a useful medium for investigating high-power laser-plasma interactions over extended lengths because guiding can increase the interaction length to many Rayleigh lengths.¹ Major applications of these waveguides include the laser-plasma wakefield accelerator (LWFA),^2–7 where terawatt-scale femtosecond laser pulses are used to generate MeV,^8–10 and GeV¹¹ electron beams over typical lengths of 30–40 mm, and the plasma-based Raman amplifier,^12,13 where two counter-propagating laser pulses in a CDW allow resonant transfer of energy from a “pump” laser pulse to a “probe” pulse. In both of these applications, there is an increasing trend towards using very high incident laser powers, where relativistic effects can become important.^14,15

The gas-filled CDW is formed when a potential difference of 10–20 keV is applied between two electrodes located at either end of a capillary to initiate a discharge in readily ionized gas such as hydrogen. The gas is pre-injected into the capillary and underdense plasma is formed via avalanche breakdown and collisional ionization.¹⁶ Thermal conduction to the capillary walls results in a high on-axis plasma temperature and a temperature gradient between the axis and the wall. In equilibrium, the pressure across the capillary is uniform, which results in a parabolic radial density distribution and a refractive index profile suitable for waveguiding.

For a low-intensity laser pulse propagating in the channel, diffraction can be exactly compensated by the focusing effect of the plasma lens and the laser pulse can be transported over several centimetres with excellent mode structure and little energy loss for a matched spot size.^1,17 Laser pulse propagation at very high intensity is modified significantly by the intensity-dependent refractive index due to relativistic effects,¹⁸ and ponderomotive focusing,¹⁹ which are highly nonlinear. LWFAs typically operate in the so-called bubble regime²⁰ characterized by a normalized laser vector potential a₀ = eA₀/m_ec > 1, where A₀ is the vector potential, e and m_e are the charge and mass of the electron, respectively, and c is the speed of light in vacuum. Relativistic self-focusing overcomes vacuum diffraction when the power P₀ exceeds a critical power, $Pc=17.4(ω0/ωp)2 [GW]$⁠, where ω₀ and ω_p are the laser and plasma frequencies, respectively. For lower power, it increases the effective Rayleigh length by a factor of $(1−P0/Pc)−1/2$⁠.²¹ The Rayleigh length in vacuum is Z_R = πw₀²/λ, where w₀ is the (1/e²) radius of the laser beam and λ its wavelength. Ponderomotive self-focusing occurs when the laser pulse duration exceeds a plasma period, so that the rear of the pulse propagates in a density channel formed by the ponderomotive pressure of the pulse front. Both processes lead to an increase in a₀, which enables strong density wake formation, electron injection, and subsequent acceleration.²² Non-linear self-focusing along with diffraction and plasma guiding determine the laser pulse evolution in the CDW and the relative strengths of these processes are particularly important in the entrance region of the CDW because the laser focal plane typically coincides with the capillary entrance plane.

To fully describe the laser pulse evolution, the entrance region outside the capillary must be included in any study, which forms the topic of this paper. The plume regions, which in the case of the CDW are the regions beyond the capillary itself, are well known for material processing applications using plasma jets²³ or plasma guns,²⁴ but have been neglected in previous laser-plasma studies.^9,25 A gas-filled CDW plume has two distinct components arising from initial gas pre-injection and later discharge breakdown, i.e., there is an expanding neutral gas plume followed by an expanding plasma plume. Here, we show that the density of the gas/plasma plume is sufficiently high for laser self-focusing to occur at a significant distance in front of the entrance of the CDW. The vacuum laser focus is chosen to coincide with the entrance plane. Numerical and analytical methods are used to determine the self-focusing in the plume region and this is compared with laser evolution when the plume is neglected.

The schematic of our CDW system is shown in Fig. 1(a). The alumina capillary under investigation here is manufactured by femtosecond laser micromachining,²⁶ and has a length of 40 mm and diameter of 300 μm. Gas inlets, with 0.2 mm × 0.8 mm rectangular apertures, are located 10 mm from each end of the capillary. Injection of hydrogen gas is controlled by a pulsed solenoid valve that is open for a duration of 200 ms. A high voltage electrical pulse (20 kV peak voltage, 900 ns full-width duration at half-maximum)²⁷ is applied 100 μs after valve closure to initiate breakdown. The copper electrode geometry [Fig. 1(b)] shows the on-axis aperture extending from the capillary edge with a diameter of 700 μm and widening to 2 mm at a distance of −5 mm (negative distances along the laser propagation axis refer to the region outside the capillary). The high-power laser pulse is usually synchronized with the plasma current pulse on a sub-microsecond timescale. Therefore, the laser initially encounters the neutral gas plume far from the entrance plane and then, closer to the entrance, the expanding hot plasma plume that has a typical temperature of ≈5 eV.¹⁶ The precise rate of plasma expansion (with a velocity of ∼10⁴–10⁵ m/s)^28,29 depends on discharge characteristics and is discussed further in Sec. IV. In regions, where the intensity is greater than 10¹⁴ W/cm² non pre-ionized gas is ionized by the intense laser field.

The plume of gas/plasma extends beyond the Perspex housing and is clearly visible to the naked eye, as shown in Fig. 1(a). However, the main region of interest is in the vicinity of the entrance plane, where the laser beam is converging. The expected focusing system in vacuum of a typical high-power laser used in LWFA experiments^7,30 utilizes a large F/number focusing element (F/16) to focus the beam of wavelength λ = 800 nm to a (1/e²) radius w₀ = 20 μm at the entrance of the CDW, which would give an intensity of 2.2 × 10¹⁸ W/cm² and a₀ = 1.0 at the focus in vacuum. However, in the case, where a gas/plasma plume exists in this laser focusing region, significant self-focusing enhances the rate of focusing. For our conditions, this occurs when a₀ reaches a value of ≈0.5, at Z = −3 mm. This is equivalent to approximately −2Z_R, where Z_R = 1.6 mm, and defines the region where laser self-focusing is important.

As a first step into gaining insight into the structure of our capillary plume, we have modelled the neutral gas density distribution under steady-state conditions using the 2D FLUENT fluid code.³¹ Simulations have been carried out for two cases: one including the electrode geometry and the other without it, to illustrate the influence of the geometry. As shown in Fig. 2, for the steady-state case, the hydrogen molecular density between the inlets is 3.6 × 10¹⁸ cm⁻³, reducing to 1.1 × 10¹⁸ cm⁻³ at the entrance (for a backing pressure of 150 mbar), when the valve is open. Upon valve closure, when the pressure reaches equilibrium, the on-axis electron plasma density is uniform when the gas is ionized along the entire capillary and n_e is estimated to be 2.9 × 10¹⁸ cm⁻³ (from Ref. 32). The electrode geometry has a significant effect on the gas density profile in these simulations. In the absence of the electrodes [upper contour map of Fig. 2(a)], the gas density would decay rapidly with distance beyond the capillary entrance plane and the density at Z = −3 mm is negligible (<10¹⁷ cm⁻³). The presence of the electrodes leads to confinement of the gas [lower contour map of Fig. 2(a)], thus maintaining a relatively high density with a rapid fall-off only beyond the extent of the electrodes. In this case, n_e is estimated to be ≈0.7 × 10¹⁸ cm⁻³ at Z = −3 mm.

The extent of self-focusing in the plume region, assuming it is wholly comprised of plasma, has been calculated analytically and compared with simulations using the 3D axisymmetric quasi-static simulation code WAKE.³³ A study has been made of plasma densities in the range of 1.5−3.5 × 10¹⁸ cm⁻³, which are comparable with that obtained at the entrance plane in the FLUENT simulation (Fig. 2) and are also typical for LWFA experiments.^8–11 An intense laser pulse propagating through plasma undergoes self-focusing²¹ due to the relativistic mass nonlinearity and electron density depletion associated with the ponderomotive pressure.¹⁹ Relativistic effects change the effective dielectric constant of the medium to $ε=1−ωp2ne/(ω02γn)$⁠, where $γ=(1+a02/2)1/2$ is the Lorentz factor associated with the quiver motion, n_e is the plasma density, and n is the unperturbed background plasma density. In our model, we consider the field amplitude of a Gaussian transverse laser pulse profile varying as $a2(Z,r)=[a02(Z)/f2(Z)]exp[−r2/w02f2(Z)]$⁠, where $f(Z)=w(Z)/w0$ is the normalized pulse radius and r is the radial coordinate. With the aid of the Poisson equation and at quasi-steady state, we obtain the electron density as a function of the laser vector potential. This density results in the dielectric function $ε=ε0−ε2r2/w02$⁠, where the second order term ε₂ is due to the relativistic mass nonlinearity and electron density depletion associated with the ponderomotive pressure of the laser. Following Refs. 34 and 35, we obtain the governing equation for the spot size evolution in a dispersive medium as

where ξ = Z/Z_R. The second term of (1) is due to diffraction, the third term is due to plasma inhomogeneity and the fourth term is responsible for self-focusing. Equation (1) is a second-order differential equation, which excludes pulse depletion and can be solved upon defining the plasma density profile n(Z,r) as

where $g(Z)=(1+erf[Z+|L0|/δZ])/2$ describes the plume region, L₀ = −3.2 mm is the initial longitudinal position, δZ = 3.25 mm is the density ramp length, R₁ = 95 μm is the parabolic channel radius, R₂ = 150 μm is the capillary radius, n₀ is the on-axis density, and Δn = 0.3n₀ is the increase in density at r = R₁. The description of the entrance region density closely matches that inferred from the FLUENT simulation for the electrode geometry. We then apply the boundary condition f(−L₀) = w(−L₀)/w₀ and the initial radius of curvature of the wavefront is equal to the initial curvature in vacuum at Z = −L₀.

In both the analytic model and the WAKE simulation (Fig. 3), a curved phase-front has been chosen as an initial condition to account for the focusing laser pulse. Focusing in vacuum leads to $a0=1.0$ at the capillary entrance plane, after which $a0$ diminishes as a consequence of diffraction (channel guiding is not included here). Inclusion of plasma in the plume region results in additional self-focusing and, therefore, a higher peak a₀ close to, or at, the entrance plane. Increasing the density results in stronger self-focusing that can increase a₀ significantly. Similar behaviour is obtained in the two methods and the respective peak a₀ values agree to within 6%. As an example: a₀ reaches 1.83 (analytical) and 1.93 (WAKE) for a density of 3.49 × 10¹⁸ cm⁻³. The small discrepancy between the two methods could be due to finite pulse length and pulse depletion effects that are accounted for in the numerical simulations and absent in the analytical calculations. The depletion length L_dep = τc(ω₀/ω_p)², where τ is the pulse duration [see Fig. 4(b) later], in the numerical simulations is appreciably short (only ∼4 mm at the higher density).

To illustrate the influence of the plasma plume region on the laser pulse evolution in the CDW, WAKE simulations of the waveguide have been carried out both without and with the plume, as shown in Fig. 4, for an on-axis density n_e = 3.0 × 10¹⁸ cm⁻³. In the former case, the laser is focused in vacuum onto the entrance plane [peak a₀ = 1.0], while for the latter case, self-focusing in the plume is included, as in Fig. 3(b) [peak a₀ ≈ 1.8]. Both cases show that variation of the spot size due to self-focusing leads to a mismatch of the waveguide and high order transverse modes are excited, which result in scalloping. The matched waist R_m[μm] ≈ 1.48 × 10⁵ R_w^1/2[μm]/n_e^1/4[cm⁻³] = 36 μm, for a R_w ≈ 100 μm parabolic channel radius¹⁶ is larger than the self-focused waist and results in peaks in a₀ separated by a period ≈12 mm (close to the low intensity scallop period λ_C = π²R_m²/λ = 16 mm). In both cases (with and without plume), a₀ ≈ 2.0 after one scallop period. However, the evolution of a₀ around the capillary entrance is very different.

Without the plume a₀ increases from 1.0 to only 1.41 at Z = 1.4 mm, while inclusion of the plume leads to a₀ = 1.85 at Z = −0.3 mm, a 30% increase, which corresponds to a 70% increase in the intensity. As shown in Fig. 5, this drives a strongly non-linear plasma wakefield at the beginning of the waveguide. The initial growth of a₀ arises primarily from the laser waist size reduction [Fig. 4(a)] with subsequent pulse length compression [Fig. 4(b)] and photon deceleration contributing significantly to the growth of a₀ at one scallop period. This later growth in a₀ is driven by the spatio-temporal density profile of the wakefield bubble. In the LWFA, the position along the capillary where electron self-injection occurs and acceleration begins will depend on particular experimental parameters but our results show that the laser evolution in the plume region may, in many cases, lead to self-injection within the first mm of the waveguide. This subject will be discussed elsewhere.

To investigate the case, where a very long plasma plume is ejected following the generation and expansion of the hot plasma, we have carried out time-averaged measurements (Fig. 6) by imaging fluorescence from the plume in the transverse plane onto either a charge-coupled device (CCD) camera (pixel size 4.65 μm), to obtain a 2D image, or onto the narrow entrance slit (width 10 μm) of an imaging optical spectrometer (resolution 1.4 nm) to measure the spectra of Stark-broadened hydrogen Balmer line emission³⁶ from the background gas neutral atoms. The latter method enables the spatially resolved electron plasma density to be measured. Self-absorption leads to an over-estimation of the absolute density³⁷ compared with interferometry techniques.³² The longitudinal variation of the fluorescence has been studied at intervals of 0.5 mm by translating the relay lens and detectors together, where the step size is chosen to be larger than the depth-of-focus (∼0.1 mm). Measurements are time-averaged because a 2 ms shutter opening time has been used for both diagnostic methods employed here (CCD imaging and Stark spectroscopy). However, the peak density is approximately double the time-averaged value.³⁸ Furthermore, the similarity between the transverse profiles of CCD and the spectral data, as illustrated in Fig. 7, show that the CCD images can be used as a reliable qualitative measurement of plasma density variations. Both of our diagnostic methods resolve the characteristic parabolic profile of the channel.¹⁶ 

The on-axis density of this plasma plume in the 3 mm region of interest is found to be almost equal to that inside the capillary. Figure 8 compares two transverse density profiles obtained by measuring Stark-broadening using the spectrometer. The parabola formed inside the capillary quickly flattens at ≈1 mm resulting in a uniform transverse profile and remains flat until Z = −3 mm and beyond. Only beyond Z = −3 mm does the on-axis density begin to decrease slightly, as shown in the CCD camera data of Fig. 9. Light scattered off the inner surface of the electrode aperture [Fig. 9(a)] highlights the electrode edge and helps to illustrate the tight confinement of the plume even up to Z = −8 mm (i.e., that a “plasma plume jet” is formed). These measurements show that the on-axis plasma density remains high, in contrast to the steady-state FLUENT simulations of neutral cold gas that show a density decrease around the entrance (Z = 0 mm) and between the inlet and the entrance (Z = 10 mm).

Our measurements show that the on-axis density does not decrease by an order of magnitude until Z = −30 mm, i.e., beyond the Perspex housing, and a linear drop-off is maintained up to Z = −60 mm. A weak dependence on the gas backing pressure (in the range of 40–160 mbar) has been found for the density drop-off. Reduced gas flow at lower pressure reduces the pressure of the expanding plume (the plasma temperature is also lower¹⁶), marginally increasing the rate of density drop-off. However, the relative difference is negligible in the region of interest up to Z = −3 mm. At any given pressure, therefore, the on-axis density along the region of interest can be taken to be constant, to within experimental uncertainty. The steady-state neutral gas simulations (Fig. 2) predict a much more rapid drop-off of the density than that measured experimentally. It is therefore concluded that the steady-state simulations exaggerate the role of the electrode geometry. The measurements show that the electrode/housing regions have negligible influence on the on-axis density and hence self-focusing in the plume must always be accounted for. This would apply even for very short electrode/housing regions.

Consideration must now be given to how these time-averaged measurements apply to gas-filled CDW laser wakefield experiments. Gas pre-injection on the ms timescale ensures that the neutral gas plume is always extremely long and of high density, however, the precise extent of the expanding plasma plume as encountered by the incident laser pulse is specific to each experiment. The density profile of the plasma plume drops off exponentially away from the capillary entrance with a leading edge velocity (at n_e/e) of ∼1 × 10⁴ m/s.²⁸ Establishing a parabolic channel can take as little as 50 ns after the onset of the discharge¹⁶ however electron injection typically occurs after a further ∼100 ns^8–10,17 such that the plasma plume leading edge has propagated ∼2 mm upon laser arrival. The time-averaged measurements would, in such a case, overestimate the plasma plume density in the region of interest and laser self-ionization of the neutral gas would be required to drive the strong self-focusing, most of which is predicted to occur within one just Rayleigh length of the capillary entrance (Fig. 3). A denser expanding plasma plume in the region of interest requires a larger discharge/laser delay time and/or a hotter, more rapidly expanding plume. Clearly, a time-resolved study of the expanding plasma plume on the sub-microsecond timescale would be of interest although it must be reiterated that the long neutral gas plume would support the requisite laser self-ionization.

It is shown that the gas/plasma plume of a CDW can have an on-axis density that is comparable with that in the capillary for a significant distance beyond its entrance (and exit) plane, and therefore must be taken into account for high-intensity laser pulses, where self-ionization of the gas plume and subsequent self-focusing leads to a rapid increase in intensity and a plasma density wake. Self-focusing in the plume leads to an increase in a₀ at the beginning of the waveguide, which may lead to early electron injection and acceleration, in contrast with middle-to-late⁹ injection and acceleration. This has implications for the length of capillary required for electron beam generation, and also for the position of the focus of the laser beam. It is well known that the capillary length must exceed the dephasing length, where electrons outrun the plasma wave, and, thus, has a bearing on the electron beam parameters including beam quality.⁶ The scallop period λ_C scales inversely with n_e. However, if injection is guaranteed to occur early, the required length would only be of the order of 3–8 mm for our parameters. Shorter CDWs would have the advantage in that lower discharge voltages are sufficient for breakdown, and laser beam alignment is easier because the capillary is only two or three Rayleigh lengths long and much shorter than λ_C.

The extent of the plume to be considered depends on the laser parameters and focusing conditions. For example, matching to the capillary studied here (n_e = 3.0 × 10¹⁸ cm⁻³, R_m = 36 μm) considerably lengthens the Rayleigh length (Z_R ≈ 5 mm in this case compared with 1.6 mm considered above). The larger F/number will lead to earlier self-focusing if the laser power is sufficient and, therefore, a longer plume region will need to be considered. Given that the on-axis plume density decreases with distance beyond −3 mm, the longitudinal variation in density must be properly described and, hence, the specific geometric design of the capillary electrode/housing will become important if the radially expanding plasma plume jet impinges on it.

We acknowledge support of the U.K. EPSRC, the EC's LASERLAB-EUROPE/LAPTECH (Grant Agreement No. 284464), EuCARD-2 (Grant Agreement No. 312453), and the Extreme Light Infrastructure (ELI) European Project. We thank S. Kalmykov and N. Bourgeois for discussions on WAKE and D. Clark and T. McCanny for technical support.