Present address: ]Lawrence Berkeley National Laboratory, Berkeley, California
94720, USA

# All-optical GeV electron bunch generation in a laser-plasma accelerator via
truncated-channel injection

A. Picksley<EMAIL_ADDRESS>[  John Adams Institute for Accelerator Science
and Department of Physics, University of Oxford, Denys Wilkinson Building,
Keble Road, Oxford OX1 3RH, United Kingdom  J. Chappell  John Adams Institute
for Accelerator Science and Department of Physics, University of Oxford, Denys
Wilkinson Building, Keble Road, Oxford OX1 3RH, United Kingdom  E. Archer
John Adams Institute for Accelerator Science and Department of Physics,
University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH,
United Kingdom  N. Bourgeois  Central Laser Facility, STFC Rutherford Appleton
Laboratory, Didcot OX11 0QX, United Kingdom  J. Cowley  John Adams Institute
for Accelerator Science and Department of Physics, University of Oxford, Denys
Wilkinson Building, Keble Road, Oxford OX1 3RH, United Kingdom  D. R. Emerson
Scientific Computing Department, STFC Daresbury Laboratory, Warrington WA4
4AD, United Kingdom  L. Feder  John Adams Institute for Accelerator Science
and Department of Physics, University of Oxford, Denys Wilkinson Building,
Keble Road, Oxford OX1 3RH, United Kingdom  X. J. Gu  Scientific Computing
Department, STFC Daresbury Laboratory, Warrington WA4 4AD, United Kingdom  O.
Jakobsson  John Adams Institute for Accelerator Science and Department of
Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford
OX1 3RH, United Kingdom  A. J. Ross  John Adams Institute for Accelerator
Science and Department of Physics, University of Oxford, Denys Wilkinson
Building, Keble Road, Oxford OX1 3RH, United Kingdom  W. Wang  John Adams
Institute for Accelerator Science and Department of Physics, University of
Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, United Kingdom
R. Walczak  John Adams Institute for Accelerator Science and Department of
Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford
OX1 3RH, United Kingdom  Somerville College, Woodstock Road, Oxford OX2 6HD,
United Kingdom  S. M. Hooker  John Adams Institute for Accelerator Science and
Department of Physics, University of Oxford, Denys Wilkinson Building, Keble
Road, Oxford OX1 3RH, United Kingdom

(August 30, 2024)

###### Abstract

We describe a simple scheme, truncated-channel injection, to inject electrons
directly into the wakefield driven by a drive pulse guided by an all-optical
plasma channel. We use this approach to generate dark-current-free
$1.2\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$, 4.5% relative energy spread
electron bunches with $120\text{\,}\mathrm{T}\mathrm{W}$ laser pulses guided
in a $110\text{\,}\mathrm{m}\mathrm{m}$-long hydrodynamic optical-field-
ionized (HOFI) plasma channel. Our experiments and particle-in-cell
simulations show that high-quality electron bunches were only obtained when
the drive pulse was closely aligned with the channel axis, and was focused
close to the density down-ramp formed at the channel entrance. Start-to-end
simulations of the channel formation, and electron injection and acceleration
show that increasing the channel length to $410\text{\,}\mathrm{m}\mathrm{m}$
would yield $3.65\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ bunches, with a
slice energy spread $\sim$5\text{\times}{10}^{-4}\text{\,}$$.

††preprint: APS/123-QED

In the last 15 years it has become well established that laser-driven plasma
accelerators Tajima and Dawson (1979) can accelerate few-femtosecond Van
Tilborg  _et al._ (2006); Ohkubo  _et al._ (7 03); Debus  _et al._ (2010);
Lundh  _et al._ (2011); Heigoldt  _et al._ (2015) electron bunches to GeV-
scale energies in accelerator stages only a few centimetres long Leemans  _et
al._ (2006); Karsch  _et al._ (2007); Wang  _et al._ (2013); Leemans  _et al._
(2014); Shin  _et al._ (2018); Gonsalves  _et al._ (2019); Ke  _et al._
(2021). These highly desirable features make laser-plasma accelerators
attractive for driving compact, femtosecond-duration light sources Corde  _et
al._ (2013); Albert and Thomas (2016), including free-electron lasers (FELs).
Indeed, FEL gain has recently been reported in experiments utilizing laser-
Wang  _et al._ (2021); Labat  _et al._ (2022) and beam-driven Pompili  _et
al._ (2022); Galletti  _et al._ (2022) plasma accelerators. Other work has
demonstrated generation of incoherent soft x-radiation in an undulator Fuchs
_et al._ (2009), and at photon energies in the keV range Kneip  _et al._
(2010) from betatron oscillations in the plasma accelerator itself, and at MeV
energies from Thomson scattering Phuoc  _et al._ (2012); Powers  _et al._
(2013); Khrennikov  _et al._ (2015).

The most demanding applications of laser-plasma accelerators, such as driving
short-wavelength FELs, require the generation of multi-GeV electron bunches
with high peak current, low transverse emittance, and small shot-to-shot
jitter of the bunch properties. For these applications it is preferable to
operate in the linear, or quasi-linear, regime Esarey  _et al._ (9 08) in
order to prevent uncontrolled self-injection at one or more points along the
length of the accelerator. However, this regime brings two significant
challenges. First, relativistic self-focusing does not occur and hence the
drive laser pulse must be guided over the tens to hundreds of Rayleigh ranges
required to reach multi-GeV energies. Second, trapping of electrons in the
plasma wave is more difficult since the electric fields of the plasma wave are
lower than in the highly nonlinear regime Esarey  _et al._ (9 08).

In order to meet the first of these challenges, we have developed low-density
hydrodynamic optical-field-ionized (HOFI) plasma channels Gonsalves  _et al._
(2017); Shalloo (2018); Shalloo  _et al._ (2018, 2019), building on pioneering
work by Milchberg et al. Durfee and Milchberg (1993); Durfee  _et al._ (1995);
Clark and Milchberg (1997); Kumarappan  _et al._ (2005). These channels have
low losses, and could in principle be several metres long Picksley  _et al._
(2020a). Experiments have demonstrated guiding of relativistically-intense
laser pulses through $200\text{\,}\mathrm{m}\mathrm{m}$-long plasma channels
with axial densities as low as
$1\text{\times}{10}^{17}\text{\,}\mathrm{c}\mathrm{m}^{-3}$ Picksley  _et al._
(2020b, a); Miao  _et al._ (2020); Feder  _et al._ (2020), and operation at
kHz pulse repetition rates Alejo  _et al._ (2022).

Figure 1: Schematic of truncated-channel injection scheme. The main figure
shows how the driving (orange) and channel-forming (red) laser pulses were
coupled into a hybrid gas jet-cell, and the configuration of the far-field
input, exit mode, and electron diagnostics. Left inset: (a) Measured axicon
longitudinal intensity profile, (b) calculated initial electron temperature
profile, and (c) calculated density profile of the truncated HOFI plasma
channel $3.5\text{\,}\mathrm{n}\mathrm{s}$ after arrival of the channel-
forming pulse. In each panel, the black curve shows the relative magnitude of
each variable along the optical axis. (d) Results of fluid simulations of the
longitudinal variation of the total density $n=n_{\mathrm{e}}+n_{\mathrm{H}}$
for three simulations with $T_{\mathrm{e}}(r,z-\delta z)$ with $\delta
z=0,-100,$-200\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$. Right inset:
Measured transverse fluence profiles of the drive laser at focus and at the
exit of the HOFI channel.

In this paper we demonstrate a new approach that can addresses both challenges
in a single stage. We show that the density down-ramp formed at the start of a
HOFI channel can promote electron injection directly into the quasi-linear
wakefield driven by a channel-guided drive pulse. We demonstrate
experimentally that this truncated-channel injection (TCI) scheme produces
electron bunches with an energy up to
$1.2\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$, and a root-mean-square (rms)
energy spread of $\sigma_{\mathrm{E}}/\mu_{\mathrm{E}}=$4.5\text{\,}\%$$ from
$120\text{\,}\mathrm{T}\mathrm{W}$ ($5.8\text{\,}\mathrm{J}$,
$50\text{\,}\mathrm{f}\mathrm{s}$) laser pulses guided in 110-mm-long HOFI
channels. We show that these high-quality TCI bunches are only generated when
the drive laser is: (i) focused close to the down-ramp at the channel
entrance; and (ii) well-aligned with, and hence guided by, the HOFI channel.
In contrast, we find that bunches produced by ionization injection have much
larger energy spread and are generated preferentially when the drive pulse is
mis-aligned with the channel. The experimental results are found to be in
excellent agreement with start-to-end simulations of the complete TCI scheme,
which includes hydrodynamic simulations of the formation of the HOFI channel,
high-resolution particle-in-cell (PIC) simulations of the electron injection,
and boosted-frame PIC simulations of the accelerator stage. These simulations
show that dephasing was not reached in our experiments, and that extending the
channel length to the dephasing length
$L_{\mathrm{d}}\approx$410\text{\,}\mathrm{m}\mathrm{m}$$ would yield electron
bunches of energy $3.65\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$, a slice
energy spread below the per-mille level, a peak current of
$0.8\text{\,}\mathrm{k}\mathrm{A}$, and a normalized transverse emittance of
$\epsilon_{\perp}<$5\text{\,}\mathrm{m}\mathrm{m}\,\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$$.
These properties appear to be well matched to the requirements of soft X-ray
FELs driven by $100\text{\,}\mathrm{T}\mathrm{W}$ class lasers.

Before describing this work in detail we note that Scott et al. Scott  _et
al._ (2020) have investigated electron injection in plasma structures
generated by a laser prepulse; these had a similar transverse shape to those
we investigate here, but were much shorter
($\sim$100\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$) and were not
coupled to a guiding structure. Several groups have recently investigated GeV-
scale electron acceleration in all-optical plasma channels. Oubrerie et al.
used $40\text{\,}\mathrm{T}\mathrm{W}$ laser pulses to generate
$\sim$50\text{\,}\mathrm{p}\mathrm{C}$$,
$1.1\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ bunches with relative energy
spread $4\text{\,}\%$, following blade-induced shock injection in a 15-mm-long
all-optical channel generated with an axiparabola Oubrerie  _et al._ (2022).
Miao et al. used $300\text{\,}\mathrm{T}\mathrm{W}$ laser pulses to generate
electron beams with large energy spreads, and a central energy peaking at
$\sim$5\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ following ionization
injection in nitrogen-doped, 20-cm-long HOFI channels Miao  _et al._ (2022).
Very recent work investigating electron injection via density transitions
generated by hydrodynamically-expanding shocks has demonstrated it as a
feasible injection scheme for plasma-based Foerster  _et al._ (2022) and
laser-based v. Grafenstein  _et al._ (2023) acceleration.

Figure 1 outlines the TCI scheme, and the experimental arrangement employed.
Our experiments were undertaken with the Astra-Gemini TA3 Ti:sapphire laser at
the Rutherford Appleton Laboratory, which delivered two independent beams with
central wavelength $\lambda_{0}=$800\text{\,}\mathrm{n}\mathrm{m}$$ and full-
width at half-maximum (FWHM) durations of $47\pm
3.\text{\,}\mathrm{f}\mathrm{s}$.

Figure 2: (a) Left axis: Measured injection probability, $p_{\mathrm{inj}}$,
as a function of $\Delta z=z_{\mathrm{ch}}-z_{\mathrm{f}}$ for all (grey
circles) and well guided (green squares) shots. Right axis: Variation with
$\Delta z$ of the trapped electron bunch charge, $Q_{b}$, at the end of the
HOFI channel calculated from PIC simulations. Inset (grey): PIC calculation of
the longitudinal phase space distribution of the injected electron bunch at
the channel exit for $\Delta z=$0.8\text{\,}\mathrm{m}\mathrm{m}$$, indicated
by the grey data point. This simulation is used in comparison to experimental
measurements throughout the study. (b) Two examples of the measured,
angularly-resolved, electron energy distributions (left), and the
corresponding exit modes (right) of the guided drive beam for $\Delta
z=$0.8\text{\,}\mathrm{m}\mathrm{m}$$. The black cross shows the drive input
position. Below the upper plot is shown a comparison between the measured and
simulated angularly-integrated electron energy spectra. (c) Plots as in (b),
but for $\Delta z=$-11.2\text{\,}\mathrm{m}\mathrm{m}$$: The upper plot shows
an example of the high charge, but large energy spread beams that are observed
with a large input offset ($\delta
r=$50\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$). The lower plot shows
an example of the high-quality guiding, but lack of electron injection, that
is observed for well-aligned beams ($\delta
r=$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$).

The channel-forming beam, of pulse energy $86\pm
17.\text{\,}\mathrm{m}\mathrm{J}$, was focused by an axicon lens to form an
initial plasma column within the 110-mm-long gas target. The drive beam, of
pulse energy $5.8(0.2)\text{\,}\mathrm{J}$, was focused into the gas target by
an off-axis paraboloid, used at $f$/40, to form a focus of spot size
($1/\mathrm{e}^{2}$ intensity radius)
$w_{0}\approx$40\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$. The Rayleigh
range and peak intensity of the drive pulse were determined to be
$z_{\mathrm{R}}\approx$6\text{\,}\mathrm{m}\mathrm{m}$$ and
$I_{\mathrm{pk}}\approx$2.4\text{\times}{10}^{18}\text{\,}\mathrm{W}\,\mathrm{c}\mathrm{m}^{-2}$$
($a_{0}\approx 1.0$) respectively. The delay between the arrival of the two
pulses was set to $\Delta\tau=$3.5\text{\,}\mathrm{n}\mathrm{s}$$. A small
fraction ($<$1\text{\,}\%$$) of the beam was transmitted through the final
turning mirror and brought to focus in order to measure the transverse offset,
$\delta r$, of the drive pulse focus relative to the axis of the plasma
channel on-shot.

The target comprised a $110\text{\,}\mathrm{m}\mathrm{m}$ long hybrid gas jet-
cell Aniculaesei  _et al._ (2018). The drive and channel-forming beams were
coupled in and out of this via $6\text{\,}\mathrm{m}\mathrm{m}$ diameter
ceramic pinholes at the entrance and exit of the cell. A $2\text{\,}\%$ mix of
nitrogen in hydrogen was pulsed into the cell. Fluid simulations sup
indicated that the rms variation of the gas density along the cell axis was
$<$3\text{\,}\%$$ inside the cell, and that beyond the pinholes the atomic
density decreased by a factor of e in a distance of
$4.2\text{\,}\mathrm{m}\mathrm{m}$ (i.e. $<z_{\mathrm{R}}$).

After leaving the cell the drive laser pulse was reduced in intensity by
reflection from an optical wedge and imaged by an achromatic Keplerian
telescope onto a 16-bit CMOS detector and two fiber-based optical
spectrometers. The electron energy spectrum was measured by an electron
spectrometer comprising two $30\text{\,}\mathrm{c}\mathrm{m}$-long,
$1.01(0.01)\text{\,}\mathrm{T}$ dipole magnets and two scintillating screens
imaged onto 16-bit CMOS detectors sup .

Truncating the plasma channel was achieved by translating the axicon along its
optical axis to vary the longitudinal position of the start of the HOFI plasma
channel. In principle the start of the channel can be calculated from the
diameter ($26\text{\,}\mathrm{m}\mathrm{m}$) of the central hole in the axicon
and the approach angle of the axicon ($1.6^{\circ}$). In practice, however,
the characteristic length over which the intensity of the line focus increases
from zero is determined by the wavefront uniformity of the collimated beam,
diffraction, and alignment of the axicon to the beam. Figure 1(a) shows the
measured intensity distribution, $I_{\mathrm{ax}}(r,z)$, near the start of the
axicon line focus, and Fig. 1(b) shows the calculated 2D electron temperature
profile $T_{\mathrm{e}}(r,z)$ of the resulting initial plasma column sup . It
can be seen that the strong intensity dependence of optical field ionization
causes the onset of the plasma column to be significantly sharper than that of
$I_{\mathrm{ax}}(r,z)$. The body of this initial plasma column expands
radially, driving a cylindrical shock into the surrounding gas, to form a HOFI
channel. During this expansion a collar of neutral gas builds up on the outer
surface of the shock front Picksley  _et al._ (2020a); Feder  _et al._ (2020),
which is ionized by the drive pulse as it propagates, to form a deep, low-loss
conditioned HOFI (CHOFI) channel Picksley  _et al._ (2020a); Miao  _et al._
(2020).

As shown in Fig. 1(c), hydrodynamic simulations show that the front edge of
the initial plasma column expands longitudinally to form a hemispherical
acoustic shock wave; the expansion of the electron density and neutral density
($n=n_{\mathrm{e}}+n_{\mathrm{H}}$) along the laser axis resembles a Sedov-
Taylor-like spherical expansion for the times simulated
($0\text{\,}\mathrm{n}\mathrm{s}5\text{\,}\mathrm{n}\mathrm{s}$) Hutchens
(2000); sup , leading to a lower shock height than that formed by the _radial_
expansion that occurs in the main part of the channel. We note that the
leading edge of the drive laser will ionize any neutral atoms, to give a new
electron density equal to $n$. The scale length of the transition,
$\hat{L}_{\mathrm{tr}}=n_{\mathrm{e0}}/(dn_{\mathrm{e}}/dz)$, where
$n_{\mathrm{e0}}$ is the axial density in the main part of the HOFI channel
can be varied by adjusting the relative position of the start of the axicon
focus $z_{\mathrm{ch}}$ and the plume of gas exiting the cell entrance. Fig.
1(d) shows the longitudinal density profile $n(z)$ calculated for a fixed
initial gas density distribution and $T_{\mathrm{e}}(r,z-\delta z)$, with
$\delta z=0,-100,$-200\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$ where
$T_{\mathrm{e}}(r,z)$ is shown in Fig. 1(b). The black dashed line indicates
$n_{\mathrm{gas}}$ at $\tau=0$. It can be seen that the ramp length is
$\approx$70\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$ and the ratio of
the peak shock density to the on-axis density in the main part of the channel,
$n_{\mathrm{shock}}/n_{\mathrm{e0}}$ can be varied from approximately 2 to 6,
corresponding to a variation of
$\hat{L}_{\mathrm{tr}}=$70\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}15\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$.
These scale lengths are short compared to the plasma wavelength
$\lambda_{p}=2\pi
c(n_{e0}e^{2}/m_{e}\epsilon_{0})^{-1/2}\approx$100\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$,
and hence injection is expected to occur as a result of the abrupt change in
wakefield phase as the plasma electrons cross the density transition Suk  _et
al._ (2001); Schmid  _et al._ (2010); Buck  _et al._ (2013), rather than a
result of the decrease in wake velocity that occurs for long
($\hat{L}_{\mathrm{tr}}\gg\lambda_{p}$) density transitions Bulanov  _et al._
(1998).

We examined the TCI scheme by setting the longitudinal position,
$z_{\mathrm{f}}$, of the drive pulse focus to coincide with the position of
the front pinhole, and varying $z_{\mathrm{ch}}$. Their separation, $\Delta
z=z_{\mathrm{ch}}-z_{\mathrm{f}}$, was varied between
$-11.2\text{\,}\mathrm{m}\mathrm{m}$ and $0.8\text{\,}\mathrm{m}\mathrm{m}$.
We note that in a standard guiding experiment $\Delta
z\ll$0\text{\,}\mathrm{m}\mathrm{m}$$, so that the entrance to the channel is
far upstream of the drive pulse focus. The on-axis plasma density of the
channel was set to a low value,
$\sim$1.3(0.1)e17\text{\,}\mathrm{c}\mathrm{m}^{-3}$$, so that ionisation
injection was not observed for well-guided pulses in a standard guiding
geometry ($\Delta z=$-11.2\text{\,}\mathrm{m}\mathrm{m}$$), defined using
limits on the measured spot size and average fluence of the light imaged at
the exit of the plasma channel sup .

Figure 2(a) shows the recorded injection probability, $p_{\mathrm{inj}}$, as a
function of $\Delta z$, where injection is considered to have occurred if the
total charge recorded by the electron spectrometer was above
$0.05\text{\,}\mathrm{p}\mathrm{C}$. For well-guided shots, electron bunches
were observed for $$-3.2\text{\,}\mathrm{m}\mathrm{m}$\leq\Delta
z\leq$0.8\text{\,}\mathrm{m}\mathrm{m}$$, i.e. when the the drive focus was
well within $z_{R}$ of the leading edge of the plasma channel. The injection
success rate for 124 well-guided shots with $\Delta
z\geq$-3.2\text{\,}\mathrm{m}\mathrm{m}$$ was $74\%$. In contrast, no electron
injection was observed in 73 well-guided shots when $\Delta
z\leq$-7.2\text{\,}\mathrm{m}\mathrm{m}$$.

With the channel entrance at its most downstream position, $\Delta
z=$0.8\text{\,}\mathrm{m}\mathrm{m}$$, high-quality guiding and electron
acceleration could be observed simultaneously when the input beam was well
aligned with the HOFI channel, as shown in Fig. 2(b). Electron bunches of pC-
scale charge with mean energies in excess of
$1\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ and few-percent energy spreads were
consistently observed for input offsets $\delta
r\lesssim$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$. In contrast, as
shown in the lower panel of Fig. 2(c), electron injection was not observed for
well guided shots when the channel entrance was located upstream of the drive
pulse focus ($\Delta z\leq$-7.2\text{\,}\mathrm{m}\mathrm{m}$$). In this
configuration, injection was only observed when the drive pulse was offset
transversely from the channel axis, an example of which is shown in the upper
panel of Fig. 2(c).

Figure 3: Variation with $\delta r$ of the spectra of the output drive and
electron beams. (a) Intensity-weighted average wavelength,
$\langle\lambda\rangle$, of the drive pulses for $\Delta
z=$0.8\text{\,}\mathrm{m}\mathrm{m}$$. The size of data points corresponds to
the relative fluence of the measured guided mode; grey crosses represent shots
that were not guided. Black squares show binned averages of guided shots (see
below). The grey triangle shows the result obtained from the PIC simulation
for $\Delta z=$0.8\text{\,}\mathrm{m}\mathrm{m}$$. (b) Mean bunch energy
$\mu_{E}$ for guided shots sup  for TCI (purple squares) and ionisation
injection (blue circles); bins containing only a single event are shown by
‘$+$’ markers. For the ionisation injection dataset, $\Delta
z=$-11.2\text{\,}\mathrm{m}\mathrm{m}$$ and
$n_{e0}=$2.2(0.1)e17\text{\,}\mathrm{c}\mathrm{m}^{-3}$$. (c) Ratio of the
average rms energy spread $\sigma_{E}$ of electron bunches to $\mu_{E}$. For
all plots the data points with error bars are averages over
$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$-wide bins, with the error
bars showing one standard error.

The spectral shift of the drive pulses provides insight into wakefield
excitation along the plasma channel. Figure 3(a) shows, as a function of
$\delta r$, the intensity-weighted average wavelength,
$\langle\lambda\rangle$, of the transmitted drive pulse, together with a
measure of the wavelength shift,
$\delta\langle\lambda\rangle=\langle\lambda\rangle/\lambda_{0}$. It can be
seen that well-aligned shots ($\delta
r\lesssim$30\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$) are strongly
correlated with high transmission and significant red-shifting, corresponding
to good guiding and strong wakefield excitation. In contrast, shots with large
$\delta r$ are strongly associated with low transmission and significant blue-
shifting, consistent with significant ionization by the drive pulse.

Figures 3(b) and (c) show corresponding effects in the measured electron
energies, with larger mean energies $\mu_{E}$ measured for smaller $\delta r$.
As shown in Fig. 3(c), very well aligned shots ($\delta
r\leq$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$) have dramatically
lower relative rms energy spread — $\sigma_{E}/\mu_{E}\approx 7\%$ on average,
with a best of $4.5\%$ — than less well aligned shots. Furthermore, these well
aligned shots exhibited percent-level energy stability, with a mean energy of
$\mu_{\mathrm{E}}=$1.033(0.010)\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$.

For larger input offsets ($10<\delta
r\leq$50\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$), electron bunches of
higher charge ($\sim$10\text{\,}\mathrm{p}\mathrm{C}$$) were obtained, but
their spectra typically comprised multiple peaks superposed on a continuous
spectrum extending up to $\sim$1.0\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$,
with a mean energy
$\mu_{\mathrm{E}}\sim$0.8\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$. For very
large offsets ($\delta r>$50\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$),
strong blue-shifting of the drive pulse was observed, and the electron spectra
were effectively continuous, with
$\mu_{\mathrm{E}}\sim$0.5\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$.

Figures 3(b) and (c) also show results for the case of ionization injection
from the nitrogen dopant, which was studied by increasing the on-axis density
to $n_{\mathrm{e0}}=$2.2(0.1)e17\text{\,}\mathrm{c}\mathrm{m}^{-3}$$ and
shifting the axicon to $\Delta z=$-11.2\text{\,}\mathrm{m}\mathrm{m}$$, to
suppress injection via TCI. It is noticeable that $\mu_{E}$ is lower, and much
less sensitive to $\delta r$, for ionisation injection. It was also observed
that, with ionization injection, electrons were preferentially injected when
the drive pulse was slightly mis-aligned with respect to the axis of the
channel; injection occurred only twice in 26 shots with $\delta
r\leq$20\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$ ($\sim 7\%)$ but
occured in $\sim 30\%$ of well-guided events for $20<\delta
r\leq$50\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$.

Figure 4: (a) Laser spot size (top) and TCI bunch charge (bottom) during
propagation for input modes of different sizes. (b)-(e) Longitudinal phase
space of the beam injected via TCI at $z=$1\text{\,}\mathrm{m}\mathrm{m}$$ ,
$110\text{\,}\mathrm{m}\mathrm{m}$ (exit of cell),
$200\text{\,}\mathrm{m}\mathrm{m}$, and $410\text{\,}\mathrm{m}\mathrm{m}$
(one dephasing length) respectively. The electron density profile is shown
underlaid grey. Also shown is the beam energy distribution (red) and current
profile (blue). (f) Evolution of the properties of the TCI-injected bunch
during propagation, showing the relative energy spread of the beam
($\sigma_{E}/\mu_{E}$, blue), the mean relative slice energy spread (red), and
bunch chirp ($h$, purple).

The data presented in Figs 2 and 3 may be interpreted as follows. At the lower
density investigated in this work, high quality electron injection only
occurred when the drive pulses were: (i) well aligned with the channel
($\delta r<$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$); and (ii)
focused close to the channel entrance ($\Delta
z=$0.8\text{\,}\mathrm{m}\mathrm{m}$$). For these conditions, the drive pulse
was strongly red-shifted, consistent with strong wake excitation, and
$\sim$1\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ electron bunches with few-
percent energy spread were injected and accelerated with an injection
probability of $\sim 80\%$. When the channel entrance was located far upstream
of the drive focus, electron bunches were only generated if the drive pulse
was mis-aligned; in this case, the electron spectra were broad-band, and the
transmitted drive pulses had low energy transmission and were less strongly
red-shifted, or even blue-shifted. These observations are consistent with
poorly-aligned drive pulses interacting with the higher-density channel walls
and surrounding neutral gas, triggering uncontrolled electron injection,
potentially at several points along the channel. For the case of ionization
injection, broad-band electron bunches were generated for a wide range of
$\delta r$, suggesting that high-quality guiding was not necessary, and that
injection occurred at several points along the plasma channel.

To gain further insight we performed start-to-end modelling of the channel
formation and electron acceleration process. The chain of codes employed is
described in detail in sup , but, briefly was as follows. First, the particle-
in-cell (PIC) code EPOCH Arber  _et al._ (2015) was used to calculate the
initial electron temperature distribution $T_{\mathrm{e}}(r,z)$ [see Fig.
1(b)] from $I_{\mathrm{ax}}(r,z)$. Second, the Eulerian code FLASH Fryxell
_et al._ (2000) was used to perform three-dimensional hydrodynamic simulations
of the evolution of the truncated plasma column up to the arrival time of the
drive pulse, using $T_{\mathrm{e}}(r,z)$ and the initial longitudinal gas
density distribution obtained from fluid modelling of the hybrid cell-jet.
Third, propagation of the drive pulse through the plasma channel was
calculated with the PIC code FBPIC Lehe  _et al._ (2016) in two stages: (i)
electron bunch generation and trapping at the down-ramp was calculated by a
high-resolution simulation in the lab-frame; then (ii) this bunch was input
into a boosted-frame simulation to calculate acceleration over the remainder
of the channel. An Airy-like drive laser focus with $\delta r=0$ and other
parameters similar to the experiment was assumed.

The right-hand axis of Fig. 2(a) shows the calculated electron bunch charge
$Q_{b}$ at the channel exit for several values of $\Delta z$. It can be seen
that the variation of $Q_{b}$ with $\Delta z$ is qualitatively similar to that
of the measured $p_{\mathrm{inj}}$, although the range of $\Delta z$ for which
injection is observed in the simulations is smaller. The inset to this figure
shows the calculated longitudinal phase space of the bunch at the exit of the
channel for $\Delta z=$0.8\text{\,}\mathrm{m}\mathrm{m}$$. The properties of
this bunch are:
$\mu_{\mathrm{E}}=$1.175\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$,
$\sigma_{E}=$38\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$ ($3.2\text{\,}\%$),
and $Q_{b}=$1.8\text{\,}\mathrm{p}\mathrm{C}$$. The calculated FWHM bunch
duration was $9.4\text{\,}\mathrm{f}\mathrm{s}$, and the mean slice energy
spread was $1.0(0.4)\text{\,}\%$. The average accelerating gradient
experienced by the bunch was approximately
$11\text{\,}\mathrm{G}\mathrm{V}\,\mathrm{m}^{-1}$. The spectra of the
electron bunch and the transmitted drive pulse are found to be in good
agreement with the measurements, as evident in Figs 2(b) and 3(a).

Simulations showed that efficient coupling of the drive pulse and electron
bunch into the HOFI channel depended on $w_{0}$ and $z_{f}$. For the
conditions of Fig. 2, approximately $20\text{\,}\%$ of the trapped charge was
coupled into the channel, the rest being immediately dephased and ejected from
the wakefield. Improper matching of the drive into the HOFI channel was found
to have a deleterious effect on the bunch properties since poor matching
results in rapid spot-size oscillations, which in turn leads to transverse
ejection of particles from the wakefield with each oscillation. These effects
are evident in Fig. 4(a), which shows the evolution with $z$ of the mode size
and trapped charge for the same channel parameters as Fig. 2 and several Airy
input modes mismatched to the channel by $\Delta=w_{\mathrm{m}}-w_{0}$. It can
also be seen that for a matched Gaussian mode $>$40\text{\,}\%$$ of injected
bunch charge is transported into the channel, increasing the peak current by
an order of magnitude compared to the Airy mode.

Figure 4 also shows the evolution of the bunch characteristics for longer
channels than investigated experimentally, i.e. up to the dephasing length
$L_{\mathrm{d}}\approx$410\text{\,}\mathrm{m}\mathrm{m}$$. At
$z=L_{\mathrm{d}}$ the properties of the bunch are found to be:
$\mu_{\mathrm{E}}=$3.65\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$;
$\sigma_{\mathrm{E}}=$487\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$
($\sigma_{\mathrm{E}}/\mu_{E}=$13.3\text{\,}\%$$); an rms duration of
$\sim$26\text{\,}\mathrm{f}\mathrm{s}$$, corresponding to a peak current
$I_{\mathrm{peak}}\approx$0.8\text{\,}\mathrm{k}\mathrm{A}$$; and a normalized
projected transverse emittance of
$\epsilon_{n,\perp}\approx$5.8\text{\,}\mathrm{m}\mathrm{m}\,\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$$
(with a slice-average of
$\epsilon_{n,\perp}^{\mathrm{slice}}\approx$3.6\text{\,}\mathrm{m}\mathrm{m}\,\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$$).

The bunch exhibits negative chirp
$h=$-1.2\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\,\mathrm{f}\mathrm{s}^{-1}$$
immediately after injection (as in Döpp  _et al._ (2018)), but the chirp
increases with acceleration to positive values, becoming
$h=$49\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\,\mathrm{f}\mathrm{s}^{-1}$$ at
$z=L_{\mathrm{d}}$. We note that, thanks to the linear longitudinal phase-
space, a plasma-based dechirper D’Arcy  _et al._ (2019) could be used to
reduce $\sigma_{\mathrm{E}}$ to the level of the slice-averaged energy spread,
$\sigma_{\mathrm{E}}^{\mathrm{slice}}/\mu_{\mathrm{E}}\sim$5.0\text{\times}{10}^{-4}\text{\,}$$,
resulting in electron bunches that appear to be well-suited to driving a
compact FEL. In the 1D limit, three conditions must be satisfied Huang and Kim
(2007): (i) $\sigma_{E}/\mu_{E}<\rho$; (ii)
$L_{g}=\lambda_{u}/4\pi\sqrt{3}\pi\rho<z_{R}\approx\pi\sigma_{r}^{2}/\lambda_{R}$;
and (iii)
$\epsilon_{\perp}=\epsilon_{n,\perp}^{\mathrm{slice}}/\gamma<\lambda_{R}/4\pi$.
Here $\rho$ is the Pierce parameter, $L_{g}$ the power gain length,
$\lambda_{u}$ the undulator period, $\sigma_{r}$ the transverse bunch size,
and $\lambda_{R}$ is the FEL wavelength. For the bunches in Fig. 4, the most
stringent of these conditions is found to be (iii), and hence the range of
possible FEL wavelengths satisfies $\lambda_{R}\gtrsim
4\pi\epsilon_{\perp}\approx$6\text{\,}\mathrm{n}\mathrm{m}$$.

Alternatively, the large energy chirp generated during acceleration could
instead be used to compress the bunch to sub-fs duration, resulting in high
($\gtrsim$10\text{\,}\mathrm{k}\mathrm{A}$$) peak currents — we note that
similar electron bunch parameters were recently assumed in a proposed scheme
for generating high peak power, attosecond-duration soft x-rays Emma  _et al._
(2021).

In summary, we have demonstrated a simple scheme to inject electrons directly
into the wakefield driven by a drive pulse guided by an all-optical plasma
channel. We used this approach to generate dark-current-free bunches of energy
$1.2\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ and 4.5% relative energy spread
with $120\text{\,}\mathrm{T}\mathrm{W}$ laser pulses guided in a
$110\text{\,}\mathrm{m}\mathrm{m}$-long HOFI plasma channel. Our measurements,
confirmed by PIC simulations, showed that high-quality electron bunches were
only obtained when the drive pulse was well aligned with the channel axis and
focused close to the density down-ramp formed at the channel entrance. In
contrast, bunches injected via ionization had much higher energy spread, and
were found to occur preferentially when the drive was misaligned with the
channel. Start-to-end simulations of the channel formation, and electron
injection and acceleration showed that increasing the channel length to
$L_{\mathrm{d}}\approx$410\text{\,}\mathrm{m}\mathrm{m}$$ would yield
$3.65\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ bunches, with a slice relative
energy spread below the per-mille level.

This experiment is the first to exploit sculpting of the longitudinal and
transverse density profile of all-optical plasma channels to control electron
injection into a plasma channel accelerator stage. The use of additional laser
pulses, plus further tailoring of the gas density prior to formation of the
HOFI channel, can be expected to provide additional control. We note that HOFI
channels have been operated at repetition rates in the kilohertz range Alejo
_et al._ (2022), and new techniques have been developed for driving GeV-scale
plasma accelerator stages with few-joule, few-kHz thin-disc lasers Jakobsson
_et al._ (2021); van de Wetering  _et al._ (2023). These advances, and those
reported here, show promise for the development of high-repetition-rate
compact radiation sources, including FELs.

The authors would like to acknowledge useful discussions with Rémi Lehe. This
work was supported by the UK Science and Technology Facilities Council (STFC
UK) [grant numbers ST/R505006/1, ST/S505833/1 & ST/V001655/1]; the Engineering
and Physical Sciences Research Council [grant numbers EP/R513295/1 &
EP/V006797/1]; and the Central Laser Facility of the United Kingdom. This
material is based upon work supported by the Air Force Office of Scientific
Research under award number FA9550-18-1-7005. Computing resources provided by
STFC Scientific Computing Department’s SCARF cluster. The CFD work used in
this study was supported by funding from the CLF/EPAC (Extreme Photonics
Applications Centre). The software used in this work was developed in part by
the DOE NNSA- and DOE Office of Science- supported Flash Center for
Computational Science at the University of Chicago and the University of
Rochester.

This research was funded in whole, or in part, by EPSRC and STFC, which are
Plan S funders. For the purpose of Open Access, the author has applied a CC BY
public copyright licence to any Author Accepted Manuscript version arising
from this submission.

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##  I Supplemental Material

In this Supplemental Information we provide further details of the laser
guiding and electron energy diagnostics, the hybrid jet-cell target, and the
start-to-end simulations.

##  II Diagnosing guiding

###  II.1 On-shot far-field monitor calibration

Fig. S1: Correlation between the centroid position of the focal spot measured
by a camera placed at the focus (Focal camera, FC) and via a small portion of
the beam transmitted through the final turning mirror (Far-field monitor, FFM)
in the (a) horizontal and (b) vertical directions. Pearson’s correlation
coefficients of ${\rho=-0.90}$ (horizontal) and ${\rho=-0.93}$ (vertical) were
measured, indicating strong correlations between the two cameras and hence
validity of this diagnostic as a proxy for the position of the drive pulse at
focus. (c) Example images of the same focus recorded on (i) FC and (ii) FFM
looking at a $\pm$120\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$ region
around the identified centroid.

Calibration of the on-shot far-field monitor (FFM) was performed by
simultaneously recording images of the foci of both the leakage beam and the
main drive pulse on two separate cameras; the “FFM” and “Focal” cameras
respectively. The FFM utilized the $<1\%$ transmitted component from the final
turning mirror before the interaction (after the focusing optic) and hence
shared a common path with the main beam up until this point, minimising
possible sources of uncorrelated fluctuations. The calibration dataset
consisted of 192 shots taken in a low-power mode at
$0.05\text{\,}\mathrm{H}\mathrm{z}$, matching the repetition-rate of the full-
power beam. Figure S1 shows the correlations between the (a) horizontal and
(b) vertical focal spot centroids on the two cameras. The leakage-to-main beam
Pearson’s correlation coefficient was calculated to be $-0.90$ and $-0.93$ in
the horizontal and vertical directions respectively, indicating strong
(negative) correlation between the two (the sign of the correlation arising
from the relative orientation of the CCDs). Example images of the same focus
measured on both cameras are shown in Fig. S1(c).

Ellipses can be fitted to the distributions shown in Fig. S1(a) and (b), with
the length of the minor axes representing an average measure of the
uncertainty of the true focus position as measured by the FFM. Therefore, an
estimate of the uncertainty of a single measurement of the radial offset with
respect to the axis of the guiding channel is a summation in quadrature of
these values combined with the measured spatial jitter of the axicon focus,
corresponding to an overall uncertainty of
$9.1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$. For this reason,
$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$-wide bins are used
throughout this work when studying the interaction as a function of the radial
input offset.

###  II.2 Characterisation of well-guided shots

(a) Spot-size distribution on exit mode image.

(b) Distribution of average fluence on exit mode image.

Fig. S2: Measured distributions of (a) spot-size and (b) average fluence
within the automatically-identified ROI on exit mode images. Each distribution
is fitted with a Gaussian Mixture Model (grey curve), with the dominant
contribution represented by the shaded region. The thresholds used to identify
well-guided events are represented by the dashed lines.

The distribution of the drive beam energy at the exit of the guiding channel
was imaged on every shot by the exit mode imaging system described in the main
text. A region of interest (ROI) in each image is automatically identified by
finding the largest contour within the image that contained pixels with values
above $(1/\mathrm{e})\;\times$ the $99^{\mathrm{th}}$-percentile of the pixel
counts. The distribution of light belonging to the guided mode is identified
by fitting an ellipse to this contour, and applying a mask to the image around
its centre-of-mass of size three times the fitted ellipse in order to isolate
the signal from the guided mode. The $D4\sigma$-method is then used to
calculate the minor and major spot-sizes ($w_{\mathrm{\min}}$ and
$w_{\mathrm{maj}}$) of the guided mode within the masked image. The average
pixel count contained within the identified spot ($\mu_{\mathrm{fluence}}$) is
also recorded.

Well-guided events are characterised by guided modes with a combination of
small spot-sizes and high fluence on the exit mode imager, since this
indicates minimal evolution during propagation with high throughput from the
channel. The thresholds for these requirements were found by applying Gaussian
Mixture Models (GMMs) to the measured distributions of $w_{\mathrm{min}}$,
$w_{\mathrm{maj}}$ and $\mu_{\mathrm{fluence}}$ to estimate the dominant
contribution, as demonstrated by the hatched, shaded regions in the
distributions in Fig. S2. The number of Gaussian peaks within the GMM, $n$,
was allowed to vary between $1$ and $6$ for each distribution, with the best
fit value for $n$ selected using the Akaike information criterion that
includes both a goodness-of-fit measure and a preference for fewer components
(smaller $n$) to avoid over-fitting. For the spot-size measurements [Fig.
S2(a)], this dominant contribution — centred around the lowest measured values
of $w_{\mathrm{min}}$ and $w_{\mathrm{maj}}$ — corresponds to well-guided
shots, with spot-sizes comparable to the input drive beam spot-size
($\sim$40\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$). It should be noted
that images of the guided mode on the exit mode imager included a non-constant
background contribution from scattered axicon light, leading to measured spot-
sizes that often exceeded direct measurements of the input drive beam focus
where this effect was not present. The dominant contribution to the
distribution of the average exit mode fluence [Fig. S2(b)] instead corresponds
to poorly-guided events, with minimal intensity within the identified region
of interest mainly as a result of scattered light from the axicon beam.
Thresholds for well-guided shots were defined using the fitted mean and width
of these dominant contributions, resulting in the following requirements for a
shot to be considered “well-guided”:

  * •

$w_{\mathrm{min}}\leq$72\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$,

  * •

$w_{\mathrm{maj}}\leq$110\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$,

  * •

$\mu_{\mathrm{fluence}}\geq$410\text{\,}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{s}$$.

These requirements typically led to approximately $50\%$ of all events being
characterised as well-guided.

##  III Gas target longitudinal uniformity

###  III.1 Simulations

Fig. S3: Left: Normalised steady-state pressure profile from CFD simulations
of the gas target. The simulated pressure profile includes a localised
pressure spike around the centre of target along its top surface (“transducer
axis” — red, dashed) which was observed via measurements with pressure
transducers. Right: Histogram of the simulated normalised gas pressure along
the central axis of the target, indicating relative longitudinal pressure
variations of $3.0\text{\,}\%$ RMS.

Computational fluid dynamics (CFD) simulations of hydrogen gas flow within the
gas target were performed using the open source software code_saturne Fournier
_et al._ (2011); cod . The simulations were performed by members of the
Scientific Computing Department, STFC Daresbury Laboratory. The gas inlet,
central chamber of the target and entrance and exit pinholes were reproduced
in simulation, with hydrogen gas flowed into the target at a backing pressure
of $1.0\text{\,}\mathrm{b}\mathrm{a}\mathrm{r}$. The simulations indicated
that a steady state was reached approximately
$22\text{\,}\mathrm{m}\mathrm{s}$ after the valve opened. As shown in Fig. S3,
during steady-state the simulations predicted that along the central axis of
the target — the laser propagation axis — the gas pressure is uniform to
within $3\%$ RMS between $z=4-106$ mm. Experimental verification of the
longitudinal pressure profile along the central axis could not be undertaken
in-situ at RAL, and hence a separate measurement campaign was performed, as
discussed in the following section.

In the experiments performed at RAL, two independently-calibrated gas pressure
transducers (type: RS-PRO 797-4983) were connected to the upper surface of the
target volume to measure the absolute gas pressure, and its evolution, within
the target on every shot. One transducer was located near the centre of the
target ($z=$59\text{\,}\mathrm{m}\mathrm{m}$$), while the other was located
towards the front ($z=$29\text{\,}\mathrm{m}\mathrm{m}$$). The CFD simulations
predicted that the gas pressure distribution at the top of the target — along
the axis measured by the pressure transducers — featured a small
($\sim$5\text{\,}\mathrm{m}\mathrm{m}$$) region around the centre, directly
above the gas inlet, where the pressure increased by approximately $30\%$ [see
Fig. S3]. This pressure bump was observed in the experiment, since the centre
transducer measured pressures that were $27\pm 1.\text{\,}\%$ higher than the
transducer towards the front, consistent over fill pressures ranging from
$20\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}170\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$.

###  III.2 Measurement of gas pressure uniformity along laser axis

The uniformity of the longitudinal gas pressure profile along the central axis
of the gas target was measured at the Oxford Plasma Accelerator Laboratory
(OPAL) using the plasma fluorescence technique discussed in Ref. Miao  _et
al._ (2022). To do this, an extended hydrogen plasma column was formed by
focusing a $120\pm 4.\text{\,}\mathrm{m}\mathrm{J}$,
$45\text{\,}\mathrm{f}\mathrm{s}$ laser pulse with the same axicon used in the
experiments at RAL (approach angle $1.6\text{\,}{}^{\circ}$). The RMS spatial
jitter of the axicon focus along the length of the target was measured to be
$$0.45\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$\times$0.44\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$
$(x\times y)$. Fluorescence light emitted from this plasma was imaged
transversely on to a 16-bit CMOS sensor (Andor Xyla 5.5,
$10\text{\,}\mathrm{m}\mathrm{s}$ exposure time) through a window on top of
the gas target. This window allowed viewing access to the longitudinal region
extending from $z=17-105$ mm within the
$110\text{\,}\mathrm{m}\mathrm{m}$-long target, as shown in Fig. S4(b). A
bandpass filter centered on $656\pm 10.\text{\,}\mathrm{n}\mathrm{m}$ was
placed in front of the sensor to isolate the hydrogen-alpha fluorescence.

Fig. S4: (a) Calibration dataset demonstrating the measured plasma
fluorescence along the gas target as a function of backfill pressure. (b)
Example (false-colour) raw fluorescence image of the axicon-generated hydrogen
plasma at OPAL. (c) An example of a calibration curve,
$F(z=$40\text{\,}\mathrm{m}\mathrm{m}$,P)$, demonstrating the polynomial fit
used to extract the relationship between measured fluorescence intensity and
gas pressure.

Firstly, a calibration dataset was taken that allowed the relation between the
measured plasma fluorescence along the target and the gas pressure to be
determined. This involved uniformly filling the gas target (and entire target
chamber) with hydrogen gas over pressures ranging from
$P=$0.3\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}34\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$$.
The fluorescence produced by the plasma column generated by the axicon focus
within the hydrogen backfill was then captured by the imaging setup as a
function of the longitudinal position along the target, $z$, as demonstrated
in Fig. S4(a). The average measured fluorescence of ten shots, integrated over
the transverse axis, was then fit to a seventh-order polynomial to produce a
relation between the light intensity and the gas pressure, $F(z,P)$, at each
longitudinal position within the cell. The uncertainty on the fit includes
contributions from, and is dominated by, the shot-to-shot fluctuations in the
measured plasma fluorescence during backfill operation, which were typically
on the order of $15\%$. An example of this fit for
$z=$40\text{\,}\mathrm{m}\mathrm{m}$$ is shown in Fig. S4(c). These
calibration curves could then be used to infer the gas pressure along the
target by measuring the plasma fluorescence when pulsing gas into the cell.

The target chamber was then evacuated and the gas target was operated in a
pulsed mode, using the same valve opening time
($150\text{\,}\mathrm{m}\mathrm{s}$), backing pressure
($1.0\text{\,}\mathrm{b}\mathrm{a}\mathrm{r}$) and relative laser arrival time
($\Delta t=$146\text{\,}\mathrm{m}\mathrm{s}$$) as used in the main experiment
at RAL. Again, ten events were recorded, with the measured fluorescence signal
averaged. The gas pressure along the central axis of the target could then be
extracted by inverting the calibration curves and relating the measured
fluorescence in pulsed operation, $f(z)$, to the longitudinal gas pressure,
$P(z)$. The gas target was kept in place in both the backfill and pulsed
operation measurements to ensure that any modification of the axicon focus by
the pinhole at the front of the target was consistent for both datasets.

Fig. S5: Upper: Comparison of the normalised simulated longitudinal gas
pressure profile (black, solid) and that measured experimentally (red) along
the propagation axis of the laser. Lower: average plasma fluorescence counts
measured during pulsed operation. Right: histogram of the measured normalised
gas pressure, indicating relative longitudinal pressure variations of
$4.1\text{\,}\%$ RMS.

The results of such a measurement are shown in Fig. S5. The lower axis shows
the average fluorescence counts measured in pulsed operation mode. The upper,
central plot compares the (normalised) measured longitudinal pressure along
the central axis, reconstructed via inversion of the calibration functions, to
that predicted by the CFD simulation. Good agreement between the two datasets
can be seen, with a reduced chi-squared statistic of $\chi^{2}_{\nu}=1.16$
indicating that the measurement was consistent with simulations of the
longitudinal pressure along the central axis of the target. The measured
relative RMS gas pressure variation along the target was found to be only
$4.1\%$ within the region accessible to the measurement ($z=17-105\,$mm,
approximately $80\%$ of the length of the target), and very similar to the
$3\%$ RMS variations predicted by CFD simulations.

The total uncertainty on the reconstructed pressure at each longitudinal
position includes contributions from the uncertainty on the polynomial fit to
the calibration curves combined with the shot-to-shot fluctuations of the
plasma fluorescence measured in pulsed operation mode. The relative
uncertainty on the measurement grew along $z$ as the focus generated at the
far end of the target corresponds to light at the outer edges of the near-
field laser profile on the axicon, and hence was more susceptible to laser
energy and pointing jitters. In addition, the fluorescence signal in this
region was significantly dimmer, reducing the signal-to-noise ratio and
increasing the fit uncertainty in the calibration data, as can be seen in Fig.
S4. The relative fluctuation in the measured fluorescence profile in pulsed
operation mode was typically on the order of $15\%$, similar to backfill
operation, and indicates that no further significant fluctuations were
introduced in pulsed operation mode. The dominant contribution to the shot-to-
shot fluctuations observed during these measurements was therefore consistent
for both datasets and was likely a result of the laser energy jitter, measured
to be $3.5\%$ RMS.

##  IV Electron energy uncertainty

The electron bunch spectral distribution was measured by dispersing the beam
using a series of two $30\text{\,}\mathrm{c}\mathrm{m}$-long permanent dipole
magnets located $0.59\text{\,}\mathrm{m}$ downstream of the exit pinhole, each
with magnetic field strength of $1.01(0.01)\text{\,}\mathrm{T}$ and
longitudinally separated by $5\text{\,}\mathrm{c}\mathrm{m}$. The beams were
vertically dispersed on to two scintillating screens (type: Lanex Regular),
the surfaces of which were imaged onto two 16-bit CMOS detectors (type: Andor
Neo 5.5). The combined height of the scintillating screens enabled measurement
of electron energies ranging from $350\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$
to $3\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ while simultaneously imaging the
guided mode of the drive pulse. The scintillating screens provided a charge-
dependent scintillation signal, calibrated via consideration of the collection
efficiency of the imaging setup in combination with well-known properties of
scintillating screens, resulting in an overall charge uncertainty of $26\%$.

A vertical array of eleven
$145\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$-thick fiducial tungsten
wires was placed $1.86\text{\,}\mathrm{m}$ downstream of the exit of the
second dipole to cast identifying shadows in the scintillation signal. The
positions of these shadows could be used to reconstruct the trajectories of
accelerated electrons and minimise the electron-beam-pointing uncertainty Wang
_et al._ (2013). The main additional contribution to the energy uncertainty
arises from uncertainties in the positions of the electron spectrometer
components and the accuracy of the measured dipole field maps.

To quantify the relative contributions of these uncertainties, the electron
propagation through the electron spectrometer was solved numerically for
different electron energies (e.g. Fig. S6(a)), allowing comparison between the
reconstructed energy of electrons on the scintillating screen and their true
energy. The relative positions of all beamline components after the gas target
were allowed to vary in both $y$ and $z$ within the experimentally-determined
uncertainty ($\pm\,$1\text{\,}\mathrm{m}\mathrm{m}$$) with the field map
amplitudes also randomly varied according to their measurement accuracy
($\pm\,1\%$). The resulting contributions to the energy uncertainty were then
combined in quadrature to estimate the total energy-dependent systematic
uncertainty, shown in Fig. S6(b).

Fig. S6: (a) Measured electron spectrometer magnet field maps (blue) with
field contours in the range $0.8\leq B_{x}\leq 1.1$ T ($\Delta B_{x}=0.04$ T)
marked. Example electron trajectories are overlaid for energies between
$0.3\leq E\leq 2.0$ GeV ($\Delta
E=$0.1\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$). (b) Relative uncertainty as
a function of electron energy for one (blue, dashed) and two (red, solid)
magnets. The increase in systematic uncertainty for two magnets without
trajectory reconstruction is shown by the red, dotted line.

The advantage of employing two consectutive dipoles, rather than a single
magnet, is clear from Fig. S6(b). For a single magnet, the relative
uncertainty scaled linearly with the electron energy, increasing from $\sim
2\%$ at $0.3\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ — the minimum energy that
could be measured on the scintillating screen — to approximately $11\%$ at
$2.0\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$. In contrast, the systematic
energy uncertainty was decreased when using two magnets as a result of the
increased dispersion of the beam. In this case, the relative energy
uncertainty at $1.0\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ was approximately
$1\%$, with a discontinuity at
$\sim$1.2\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ created by electron
trajectories exiting either the upper side of the second magnet or its far
end. In both cases, the dominant contribution to the systematic uncertainties
was the vertical position of the scintillating screen. As a result of the much
reduced systematic energy uncertainty over the energies of interest, the two
magnet setup was preferred.

For high-quality electron beams with low energy spread, as was typically
observed in the TCI setup, no shadows from the fiducial wires were present
within the electron spectra, which increases the systematic energy
uncertainty. The additional energy uncertainty is dominated by uncertainties
in the (vertical) beam pointing angle at the plasma exit as this results in
changes of the position and angle of the beam as it enters the first dipole
magnet.

Fig. S7: Histogram of $e^{-}$-beam transverse position on ESPEC screen for
well-guided shots. Shown in pink is the distribution for TCI ($\Delta
z=$0.8\text{\,}\mathrm{m}\mathrm{m}$$) and in blue is the distribution for II
($\Delta z=$-11.2\text{\,}\mathrm{m}\mathrm{m}$$). The width of the positional
distributions (and their uncertainties) are converted into horizontal beam-
pointing distributions via $\sigma_{x^{\prime}}=\sigma_{x}/z(E)$, where $z(E)$
represents the energy-dependent distance between the exit of the gas target
and the position of the signal on the scintillating screen.

The pointing angle of the electron beam as it exits the plasma can be affected
by its relative betatron phase advance or the pulse-front tilt of the drive
pulse Popp  _et al._ (2010). While this could not be reconstructed on-shot for
electron bunches that did not interact with the fiducial wires, the
distribution of the pointing angle of all shots in the dataset can be
estimated from the position of the transverse centroid of the electron bunches
on the spectrometer scintillating screen. The results of this analysis are
shown in Fig. S7 for the two injection schemes investigated in this study —
truncated-channel injection (TCI) in pink, and ionisation injection (II) in
blue. Shown is a histogram of the transverse centroid position of electron
bunches on the screen with respect to the left-edge of the screen (total width
of $80\text{\,}\mathrm{m}\mathrm{m}$) for all well-guided shots in the
datasets. The width of these centroid position distributions represents a
measure of the shot-to-shot pointing angle fluctuations of the electron beam,
multiplied by the energy-dependent distance between the exit of the plasma
channel and the front surface of the scintillator screen. Gaussian fits to the
distributions give pointing fluctuations of
$4.01(0.54)\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$ and
$1.52(0.05)\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$ for TCI and II
respectively. This represents a direct measure of the horizontal electron beam
pointing fluctuations which, on their own, do not significantly contribute to
energy uncertainties as the electrons are dispersed vertically. However,
laser-plasma accelerators are typically assumed to be cylindrically symmetric
for many-cycle pulses, and hence the transverse beam-pointing fluctuations
measured here would be expected to be representative of the magnitude of the
vertical pointing fluctuations. It is hypothesised that the difference in
horizontal pointing fluctuations between the two injection schemes are a
result of the difference in injection location. For TCI, injection occurs at
the channel entrance and therefore injected electrons will be more sensitive
to the effect of slight transverse offsets and mis-matching of the laser pulse
on subsequent propagation and acceleration within the channel.

The resulting increase in relative systematic energy uncertainty for the two
magnet setup without trajectory reconstruction — assuming a vertical pointing
fluctuation of $4\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$ — is shown
in Fig. S6(b), represented by the dotted red line. At an energy of
$1.0\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$, the relative uncertainty
increases from approximately $1\%$ to close to $3\%$.

##  V Start-to-end Simulations

The timescales associated with the TCI scheme range from the femtosecond-scale
for channel ionisation and electron injection, to multiple nanosecond scale
for the truncated HOFI plasma channel expansion, and range from the micron
scale for the density transition to the meter-scale for low-density
propagation. A simulation that included all of the required physics over these
time- and length-scales is beyond the capabilities of current particle-in-cell
(PIC) codes. We therefore employed a chain of codes, as described below:

  1. 1.

The fraction of electrons ionised, and initial electron momentum (through the
process of OFI) was calculated using several separate particle-in-cell (PIC)
simulations using the code EPOCH Arber  _et al._ (2015) in the two-dimensional
geometry. The initial axicon line focus $I_{\mathrm{ax}}(r,z)$ was converted
to an electron temperature distribution $T_{\mathrm{e}}(r,z)$ using these
simulations as a look-up table. We have previously outlined details of the
electron species thermalisation Shalloo  _et al._ (2018) in HOFI plasma
channel expansion.

  2. 2.

As described above, the initial gas density distribution $n_{\mathrm{gas}}(z)$
was separately determined via fluid simulations using the open source,
computational fluid dynamics software code_saturne cod . The steady state
solution provided details of the length of the gas density plumes at the gas
jet-cell entrance.

  3. 3.

$T_{\mathrm{e}}(r,z)$ and $n_{\mathrm{gas}}(z)$ were input into the Eulerian
magnetohydrodynamic code FLASH (v4.6.2) Fryxell  _et al._ (2000) on a three-
dimensional $(400\times 400\times 1000)$
$\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{3}$ grid, with adaptive-mesh-
refinement used to dynamically resolve the fine-scale structure at the shock
fronts. The simulation used a three-temperature model such that the electron,
ion and radiation species could be independently evolved, with the electron
and ion species also able to exchange heat. The energy diffusion and thermal
conductivity modules were activated as this has previously been found to
improve modelling of species propagation Picksley  _et al._ (2020a). Tabulated
values were used for the equation of state (EoS) and ionisation fraction of
atomic $H$, along with the Rosseland and Planck opacities, with tables
obtained from the PROPACEOS package. Radiation transport was modelled using
multi-group diffusion, with a total of 6 energy groups distributed between
$10\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ and
$12\text{\,}\mathrm{e}\mathrm{V}$.

  4. 4.

The calculated electron density $n_{\mathrm{e}}(r,z)$ and neutral density
$n_{\mathrm{H}}(r,z)$ profiles after a time $\Delta\tau$ were extracted from
the FLASH simulation and input into FBPIC Lehe  _et al._ (2016) — a PIC code
that operates in the $r$-$z$ geometry. For the trapping section, simulations
were run utilising the multi-GPU capabilities of FBPIC in the laboratory frame
Jalas  _et al._ (2017), and were propagated over
$4\text{\,}\mathrm{m}\mathrm{m}$. The simulation window was $600\times 3500$
cells on the $(r,z)$ cylindrical grid and co-propagated with the laser pulse.
Electron and neutral species was initialised with $3\times 3\times 8$ macro-
particles in the $r\times z\times\theta$ directions.

  5. 5.

Since injection does not occur in the low-density HOFI plasma channel after
the truncation section, it was possible to capture the physics of bunch
acceleration using a boosted frame simulation with fewer macroparticles. This
was necassary due to the length scales involved. Macroparticles from the
laboratory simulation that made up the trapped bunch were isolated, and input
into a second FBPIC simulation carried out with $\gamma_{\mathrm{boost}}=12$
Kirchen  _et al._ (2016, 2020), and $1\times 1\times 8$ macro-particles in the
$r\times z\times\theta$ directions. Matched boundary layers with $128$ damping
cells were included transversely to account for light that leaked out of the
plasma channel at an oblique angle. Simulation outputs were vizualized using
the open-PMD framework Huebl  _et al._ (2015). It should be noted that a
single, long propagation, laboratory frame simulation was also carried out to
confirm there were no additional contributions from ionisation injection
during the meter-scale propagation section of the HOFI plasma channel.