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Linear wakefield generation: plots of $E/E_{0}$, $v/c$, and $n_\mathrm{e}/n_{0}$ (in blue) for laser pulse $a=a_{0}\sin(\pi \zeta/L)$ with pulse length $L= {\rm (a)}\, \frac{1}{2}, {\rm (b)}\, 1, {\rm (c)}\, 2 \, \lambda_\mathrm{p}$ and $a_{0} = 0.2$ (in red).

Linear wakefield generation: plots of $E/E_{0}$, $v/c$, and $n_\mathrm{e}/n_{0}$ (in blue) for laser pulse $a=a_{0}\sin(\pi \zeta/L)$ with pulse length $L= {\rm (a)}\, \frac{1}{2}, {\rm (b)}\, 1, {\rm (c)}\, 2 \, \lambda_\mathrm{p}$ and $a_{0} = 0.2$ (in red).

Linear wakefield generation: plots of $E/E_{0}$, $v/c$, and $n_\mathrm{e}/n_{0}$ (in blue) for laser pulse $a=a_{0}\sin(\pi \zeta/L)$ with pulse length $L= {\rm (a)}\, \frac{1}{2}, {\rm (b)}\, 1, {\rm (c)}\, 2 \, \lambda_\mathrm{p}$ and $a_{0} = 0.2$ (in red).

Non-linear wakefield generation: plots of $E/E_{0}$, $v/c$, and $n_\mathrm{e}/n_{0}$ (in blue) for laser pulse $a=a_{0}\sin(\pi \zeta/L)$ with $a_{0}= {\rm (a)}\, 0.5, {\rm (b)}\, 1.0$ and ${\rm (c)}\, 1.5$ (in red).

Non-linear wakefield generation: plots of $E/E_{0}$, $v/c$, and $n_\mathrm{e}/n_{0}$ (in blue) for laser pulse $a=a_{0}\sin(\pi \zeta/L)$ with $a_{0}= {\rm (a)}\, 0.5, {\rm (b)}\, 1.0$ and ${\rm (c)}\, 1.5$ (in red).

Non-linear wakefield generation: plots of $E/E_{0}$, $v/c$, and $n_\mathrm{e}/n_{0}$ (in blue) for laser pulse $a=a_{0}\sin(\pi \zeta/L)$ with $a_{0}= {\rm (a)}\, 0.5, {\rm (b)}\, 1.0$ and ${\rm (c)}\, 1.5$ (in red).

Wake amplitude for (a) $a_{0}=1$ with varying pulse length for a linear (blue) and non-linear (red) wake, and (b) $L = \lambda_\mathrm{p}$ and varying $a_{0}$.

Wake amplitude for (a) $a_{0}=1$ with varying pulse length for a linear (blue) and non-linear (red) wake, and (b) $L = \lambda_\mathrm{p}$ and varying $a_{0}$.

Evolution of beam waist (red line) of an ideal Gaussian beam, with curvature of phase fronts (blue lines) and Rayleigh range, $z_\mathrm{R}$, (shaded grey) also shown.

Behaviour of laser beam waist focussed on boundary of sharp density plasma transition for $P/P_\mathrm{cr} =$ (a) 0, (b) 0.5, (c) 0.9, (d) 1.0 and (e) 1.1.

Behaviour of laser beam waist focussed on boundary of sharp density plasma transition for $P/P_\mathrm{cr} =$ (a) 0, (b) 0.5, (c) 0.9, (d) 1.0 and (e) 1.1.

Behaviour of laser beam waist focussed on boundary of sharp density plasma transition for $P/P_\mathrm{cr} =$ (a) 0, (b) 0.5, (c) 0.9, (d) 1.0 and (e) 1.1.

Behaviour of laser beam waist focussed on boundary of sharp density plasma transition for $P/P_\mathrm{cr} =$ (a) 0, (b) 0.5, (c) 0.9, (d) 1.0 and (e) 1.1.

Behaviour of laser beam waist focussed on boundary of sharp density plasma transition for $P/P_\mathrm{cr} =$ (a) 0, (b) 0.5, (c) 0.9, (d) 1.0 and (e) 1.1.

Behaviour of the waist of the leading edge of a laser pulse focussed on boundary of sharp density transition on a plasma with radial guiding channel with $\delta n/\delta n_\mathrm{cr} =$ (a) 0, (b) 0.1, (c) 0.5, (d) 1.0 and (e) 1.4.

Behaviour of the waist of the leading edge of a laser pulse focussed on boundary of sharp density transition on a plasma with radial guiding channel with $\delta n/\delta n_\mathrm{cr} =$ (a) 0, (b) 0.1, (c) 0.5, (d) 1.0 and (e) 1.4.

Behaviour of the waist of the leading edge of a laser pulse focussed on boundary of sharp density transition on a plasma with radial guiding channel with $\delta n/\delta n_\mathrm{cr} =$ (a) 0, (b) 0.1, (c) 0.5, (d) 1.0 and (e) 1.4.

Behaviour of the waist of the leading edge of a laser pulse focussed on boundary of sharp density transition on a plasma with radial guiding channel with $\delta n/\delta n_\mathrm{cr} =$ (a) 0, (b) 0.1, (c) 0.5, (d) 1.0 and (e) 1.4.

Behaviour of the waist of the leading edge of a laser pulse focussed on boundary of sharp density transition on a plasma with radial guiding channel with $\delta n/\delta n_\mathrm{cr} =$ (a) 0, (b) 0.1, (c) 0.5, (d) 1.0 and (e) 1.4.

Compression and photon acceleration of a wakefield driving laser pulse for $a_{0} =1$, $L=\lambda_\mathrm{p}$. (a) $n_\mathrm{e}$ (top), \emph{group} velocity (middle) and pulse shape (bottom) showing regions of speeding and slowing of the pulse. (b) $n_\mathrm{e}$ (top), \emph{phase} velocity (middle) and pulse shape (bottom) showing regions of stretching and compressing of phase.

Compression and photon acceleration of a wakefield driving laser pulse for $a_{0} =1$, $L=\lambda_\mathrm{p}$. (a) $n_\mathrm{e}$ (top), \emph{group} velocity (middle) and pulse shape (bottom) showing regions of speeding and slowing of the pulse. (b) $n_\mathrm{e}$ (top), \emph{phase} velocity (middle) and pulse shape (bottom) showing regions of stretching and compressing of phase.