2 Heuristic theory of channel-guided propagation

In this paper, the mathematical model for describing laser pulse evolution is based on the wave equation:

(1) $$\begin{align}\left({\nabla}^2-\frac{1}{c^2}\frac{\partial^2}{\partial {t}^2}\right)E={k}^2\left(1-{\eta}_{\mathrm{r}}^2\right)\widehat{E},\end{align}$$

where $E=\operatorname{Re}\left[\widehat{E}{e}^{ik\left(z- ct\right)}\right]$ is the transverse electric field of the laser, $\widehat{E}\left(r,z,t\right)$ is the slowly varying amplitude envelope, $\omega = ck=2\pi /\lambda$ is the laser frequency, $k$ is the laser wave number, $\lambda$ is the laser wavelength, $c$ is the speed of light in a vacuum, $t$ is a time independent variable, ${\eta}_{\mathrm{r}}$ is the index of refraction of the medium and $\left|1-{\eta}_{\mathrm{r}}^2\right|\ll 1$ is assumed. Here, ${\eta}_{\mathrm{r}}$ is generally a nonlinear function of the medium density profile:

(2) $$\begin{align}{\eta}_{\mathrm{r}}=1-\frac{k_{\mathrm{p}}^2(r)}{k^2},\end{align}$$

where ${k}_{\mathrm{p}}$ is the plasma wave number. In the following, it will be assumed that ${\eta}_{\mathrm{r}}$ is real, that is, dissipative effects such as collisions and absorption will be neglected. The following analysis is carried out under the axisymmetric cylindrical coordinate system, with $z$ being the axial propagation direction and $r$ the radial component:

(3) $$\begin{align}{\nabla}_{\perp}^2={\nabla}^2-\frac{\partial^2}{\partial {z}^2}\simeq \frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial }{\partial r}\right),\end{align}$$

where ${\partial}^2/\partial {z}^2$ is typically small, which will be neglected.

The propagation of an ultrashort laser pulse in a preformed plasma channel will be considered. Propagation is considered in the limits of low power $P/{P}_{\mathrm{c}}\ll 1$ and low intensity ${a}_0^2\ll 1$ , such that nonlinear effects (e.g., relativistic self-focusing) can be neglected and the density channel can be assumed to be unaffected by the laser pulse, where $P$ is the laser power, ${P}_{\mathrm{c}}$ is the critical power for relativistic self-focusing and ${a}_0$ is the laser relativistic intensity. The starting point for describing laser pulse evolution in plasma is the linear paraxial wave equation^[ Reference Esarey, Sprangle, Krall and Ting^26] for the transverse component of the laser field, which can be written in the following form:

(4) $$\begin{align}\left({\nabla}_{\perp}^2+2 ik\frac{\partial }{\partial z}\right)\widehat{E}={k}_{\mathrm{p}}^2(r)E,\end{align}$$

where ${k}_{\mathrm{p}}^2(r)={k}_{{\mathrm{p}}0}^2n(r)/{n}_0$ , ${k}_{{\mathrm{p}}0}={\omega}_{{\mathrm{p}}0}/c$ is the plasma wave number, ${\omega}_{{\mathrm{p}}0}=\sqrt{n_0{e}^2/\left({\varepsilon}_0{m}_{\mathrm{e}}\right)}$ is the plasma frequency, ${n}_0$ is the electron plasma density on the central axis ( $r=0$ ), $e$ is the elementary charge, ${\varepsilon}_0$ is the vacuum permittivity, ${m}_{\mathrm{e}}$ is the electron mass in an international system of units and $n(r)$ is the radial distribution of the plasma channel density that will be introduced below.

Figure 1 Calculated plasma electron-density profiles $n(r)$ as a function of radial position $r$ with axis density ${n}_0=1\times {10}^{16}\ {\mathrm{cm}}^{-3}$ and matched spot size ${w}_{\mathrm{m}}=100\ \unicode{x3bc} \mathrm{m}$ for a parabolic channel (black line) compared with a channel with an ${r}^4$ component with ${E}_{\varPsi}=0.3$ at the same ${w}_{\mathrm{m}}$ (blue line) and a channel at the same $\left\langle n\right\rangle$ (red line), respectively.

To analyze the behavior of the pulse radius, consider the local normalized-intensity-weighted mean-squared radius $\left\langle {r}^2\right\rangle$ defined by the following:

(5) $$\begin{align}\left\langle {r}^2\right\rangle =\frac{\int_0^{\infty }{r}^3{\left|\widehat{E}\right|}^2 \mathrm{d}r}{\int_0^{\infty }r{\left|\widehat{E}\right|}^2 \mathrm{d}r}.\end{align}$$

We say that a laser propagating in a plasma channel satisfies a quasi-matched propagation condition if its second-order ${\partial}^2\left\langle {r}^2\right\rangle /\partial {z}^2$ remains zero. Plasma channels focus laser pulses due to a radially varying refractive index. Self-steepening and depletion have been neglected to simplify the analysis of the problem. The matched spot size ${w}_{\mathrm{m}}$ can be obtained from the radial electron-density distribution according to the following^[ Reference Benedetti, Schroeder, Esarey and Leemans^64]:

(6) $$\begin{align}8\pi {r}_{\mathrm{e}}{\int}_0^{\infty }n(r)\left(\frac{2{r}^2}{w_{\mathrm{m}}^2}-1\right){e}^{-\frac{2{r}^2}{w_{\mathrm{m}}^2}} r\mathrm{d}r-1=0.\end{align}$$

Equation (6) gives the general matched spot size for applicable densities, besides the parabolic density profiles. High-order contributions should be introduced for a complete depiction of density profiles. One model provides an excellent description, as shown in Figure 1, with quartic terms of the outer edge density profile for $\mid r\mid >2{w}_{\mathrm{m}}$ as follows^[ Reference Turner, Gonsalves, Bulanov, Benedetti, Bobrova, Gasilov, Sasorov, Korn, Nakamura, van Tilborg, Geddes, Schroeder and Esarey^91]:

(7) $$\begin{align}n(r)={n}_0+\frac{1-{E}_{\varPsi}}{\pi {r}_{\mathrm{e}}{w}_{\mathrm{m}}^4}{r}^2+\frac{E_{\varPsi}}{2\pi {r}_{\mathrm{e}}{w}_{\mathrm{m}}^6}{r}^4,\end{align}$$

where ${E}_{\varPsi}$ is a scaling factor for the relative contributions of the quadratic distribution ( ${r}^2$ term) and the quartic distribution ( ${r}^4$ term). ${r}_{{\mathrm{ch}}}$ is the maximum radius of the outer edge of the channel. Considering a specific plasma evolution over time, the average density $\left\langle n\right\rangle ={\int}_0^{r_{{\mathrm{ch}}}}n(t) r\mathrm{d}r/{r}_{{\mathrm{ch}}}^2$ is approximately conserved, which will be discussed in the Section 5. The channel can guide multiple matched laser sizes at various moments with various ${E}_{\psi }$ , as the black and red lines shown in Figure 1. When it comes to a specific matched spot, multiple matched channels can be produced with various ${E}_{\psi }$ and $\left\langle n\right\rangle$ , as the black and blue lines show in Figure 1. In the description of channel-guided laser physics, the parameters ${E}_{\psi }$ , ${w}_{\mathrm{m}}$ and $\left\langle n\right\rangle$ are closely related. Equation (7) can be rewritten into an equivalent form:

(8) $$\begin{align}n(r)={n}_0+\Delta {n}_{{\mathrm{r}}2}\frac{r^2}{r_{{\mathrm{ch}}}^2}+\Delta {n}_{{\mathrm{r}}4}\frac{r^4}{r_{{\mathrm{ch}}}^4}.\end{align}$$

For ${E}_{\varPsi}<1$ and ${r}^2\ll {r}_{{\mathrm{ch}}}^2$ , a rough approximation can be taken:

(9) $$\begin{align}n(r)\simeq {n}_0+\Delta {n}_{{\mathrm{ch}}}\frac{r^2}{r_{{\mathrm{ch}}}^2}={n}_0+\Delta {n}_{{\mathrm{r}}2}\left(1+\frac{\delta \Delta {n}_{{\mathrm{r}}2}}{\Delta {n}_{{\mathrm{r}}2}}\right)\frac{r^2}{r_{{\mathrm{ch}}}^2},\end{align}$$

where $\delta \Delta {n}_{{\mathrm{r}}2}=\Delta {n}_{{\mathrm{r}}4}{r}^2/{r}_{{\mathrm{ch}}}^2\ll \Delta {n}_{{\mathrm{r}}2}.$

Approximate forms for the envelope equations describing laser pulse evolution, that is, a solution to the paraxial wave equation (Equation (4)) can be found by the method of the heuristic theory, considering a Gaussian model laser beam with the following form:

(10) $$\begin{align}E={E}_{\mathrm{s}}{{e}}^{i\theta -\left(1-i\;{\varphi}_{\mathrm{G}}\right)\frac{r^2}{w^2}},\end{align}$$

where ${E}_{\mathrm{s}}$ , ${\varphi}_{\mathrm{G}}$ , $w$ , $\theta$ represent the amplitude, curvature, spot size and phase shift, respectively. Analysis of the paraxial wave equation (Equation (4)) with an index of refraction of an ideal radially parabolic density profile form indicates that the spot size $w$ evolves according to heuristic theory^[ Reference Esarey, Sprangle, Krall and Ting^26]:

(11) $$\begin{align}\frac{\partial^2w}{\partial {z}^2}=\frac{4}{kw^2}\left(1-\frac{w^4}{w_{\mathrm{m}}^4}\right),\end{align}$$

where the matched beam spot size ${w}_{\mathrm{m}}$ is given by the following:

(12) $$\begin{align}{w}_{\mathrm{m}}^4=\frac{r_{{\mathrm{ch}}}^2}{\pi {r}_{\mathrm{e}}\Delta {n}_{{\mathrm{ch}}}}.\end{align}$$

Note that the high-order density profile component contributes to a mismatched spot as mathematical equivalence:

(13) $$\begin{align}{w}_{\mathrm{m}}={w}_{{\mathrm{mM}}}\left(1-\frac{1}{4}\frac{\delta \Delta {n}_{{\mathrm{r}}2}}{\Delta {n}_{{\mathrm{r}}2}}\right)={w}_{{\mathrm{mM}}}+\delta {w}_{\mathrm{m}}\kern1pt,\end{align}$$

where ${w}_{{\mathrm{mM}}}=\sqrt[4]{r_{{\mathrm{ch}}}^2/\left(\pi {r}_{\mathrm{e}}\Delta {n}_{{\mathrm{r}}2}\right)}$ is the matched spot size of the paraxial density profile. The laser evolution from the paraxial wave equation (Equation (4)), with the initial (at $z=0$ ) conditions ${\varphi}_{\mathrm{G}}=0,\theta =0,\partial w/\partial z=0$ , is given by the following^[ Reference Esarey and Leemans^65]:

(14) $$\begin{align}E={E}_0\left[1-\frac{\delta {w}_{\mathrm{m}}}{w_{\mathrm{m}}}\left(1-\frac{2{r}^2}{w_{\mathrm{m}}^2}\right){{e}}^{-{ik}_{\mathrm{os}}z}\right]{e}^{-\frac{r^2}{w_{\mathrm{m}}^2}}.\end{align}$$

We can get the radial intensity profile evolution as follows:

(15) $$\begin{align} I&={EE}^{\ast}\nonumber\\ &={E}_0^2\left\{1+{\left[\Psi \chi \left(1-\chi \right)\right]}^2-2\Psi \chi \left(1-\chi \right)\cos {k}_{{\mathrm{os}}}z\right\}{e}^{-\chi},\end{align}$$

where ${k}_{{\mathrm{os}}}=4/\left({kw}_{\mathrm{m}}^2\right)$ is the betatron wave number and ${\chi =2{r}^2/{w}_{\mathrm{m}}^2}$ . One coefficient can be defined as

(16) $$\begin{align}\Psi =\frac{1}{8}\frac{\Delta {n}_{{\mathrm{r}}4}}{\Delta {n}_{{\mathrm{r}}2}}\sqrt[4]{\pi {r}_{\mathrm{e}}\Delta {n}_{{\mathrm{r}}2}{r}_{{\mathrm{ch}}}^2}\end{align}$$

to denote the rate of contributions of the ${r}^4$ term to that of the ${r}^2$ term.

The above proof illustrates the impact of the ${r}^4$ term of the electron density on laser intensity distribution in accordance with Equation (15), which is obviously different from the Gaussian model within ideal parabolic density profiles. The employed centroid displacement technique^[ Reference Gonsalves, Nakamura, Lin, Osterhoff, Shiraishi, Schroeder, Geddes, Cs. Tóth and Leemans^90] can provide parabolic plasma channel shape information, that is, $\Delta {n}_{{\mathrm{r}}2}$ , intuitively. However, without an explicit expression, the measurement of $\Delta {n}_{{\mathrm{r}}4}$ generally relies on an iterative agreement between the experimental data and the simulation process. In addition, the intensity distribution measurement as a novel method can get the off-axial density $\Delta {n}_{{\mathrm{r}}4}$ at the channel outlet via fitting Equation (15) for a fixed $z$ .

3 Simulation of laser propagation in plasma channels

Actually, it is important to specify the laser mode injected into the plasma channel. In our experiments, the collimated laser used with a particularly large ${Z}_{\mathrm{R}}$ can be approximated as a plane wave due to its spot size being significantly larger than the radius of the plasma channel. However, the diffraction shaping effect on the pulse wavefront by the electrode hole and capillary wall from the electrode surface to the front end of the plasma channel is usually disregarded, because of the gas inlet position^[ Reference Spence and Hooker^69]. We implemented a cylindrical 2D-vector integral transform^[ Reference Kotlyar and Kovalev^93] calculation in cylindrical coordinates to approximately explain the infinite diffraction propagation by an ideal scattering boundary (corresponding to an electrode and the capillary wall), as shown in Figure 2(a), where the data on the negative side of the radial axis are obtained by mirroring. The method is efficient in dealing with diffraction but requires extremely high spatial resolution. Within limited computing resources, we set an initial plane wave electric field with a wavelength ${\lambda =1\ \unicode{x3bc} \mathrm{m}}$ and a spatial step of $0.1\lambda$ in a $50\ \unicode{x3bc} \mathrm{m}\times 600\ \unicode{x3bc} \mathrm{m}\left(r\times z\right)$  calculation domain. The result of the calculation offers proof of the following, for the propagation of a laser in a capillary with ${r}_{{\mathrm{ch}}}\gg \lambda$ , at only a short distance $z$ that satisfies the Fresnel condition^[ Reference Siegman^94]:

(17) $$\begin{align}\frac{z}{2{r}_{{\mathrm{ch}}}}\ge \sqrt[3]{\frac{2{r}_{{\mathrm{ch}}}}{\lambda }}.\end{align}$$

Figure 2 (a) The cylindrical 2D-vector integral transform calculation result of an input plane wave electric field $E$ with wavelength $\lambda =1\ \unicode{x3bc} \mathrm{m}$ diffraction propagation by an ideal scattering boundary in a $50\ \unicode{x3bc} \mathrm{m}\times 600\ \unicode{x3bc} \mathrm{m}\left(r\times z\right)$ domain, where the data on the negative side of the radial axis are obtained by mirroring. (b) The normalized transverse $E$ profile at $z=500\ \unicode{x3bc} \mathrm{m}$ from calculation data (black line) and fitting the profile with the theoretical function ${J}_1(x)/x$ (blue line) and the approximate Gaussian function (red line) for the channel-guided laser requirement.

The wavefront can be represented approximately as a Gaussian mode:

(18) $$\begin{align}E={E}_0\frac{2{J}_1(x)}{x}\simeq {E}_0{{e}}^{-\frac{r^2}{w^2}},\end{align}$$

as shown in Figure 2(b), where $x={x}_{10}r/{r}_{{\mathrm{ch}}}$ , ${J}_1$ is the first-order Bessel function and ${x}_{10}$ is the first zero point of ${J}_1(x)$ . In a vacuum, the edge density of a discharge capillary plasma channel drops steeply^[ Reference Garland, Tauscher, Bohlen, Boyle, D’Arcy, Goldberg, Põder, Schaper, Schmidt and Osterhoff^79], and the refractive index change of the gas outside the electrode is over two orders of magnitude smaller than that of the plasma^[ Reference Buckingham and Graham^95]. The model could be regarded as input consistent with the quasi-Gaussian mode in Equation (16) at $z=0$ .

In order to check the applicability of the aforementioned heuristic theory introduced, we have performed cylindrical particle-in-cell (PIC) simulations by the Smilei^[ Reference Benedetti, Schroeder, Esarey and Leemans^64] code. The simulated transport domain is 500 μm  $\times$  3 cm (corresponding channel radius $250\;\unicode{x3bc} \mathrm{m}$ and length 3 cm) in the y–z plane. The simulations consider a plasma channel with a density distribution via the y-direction described in Equation (7) with the ideal matched spot size ${w}_{{\mathrm{mM}}}=100$ μm and ${E}_{\psi}\,{=}\,0.3$ . According to the theory above, it can be calculated that the channel density depth $\Delta {n}_{{\mathrm{r}}2}=7.08\times {10}^{16}\ {\mathrm{cm}}^{-3}$ , quartic density contribution $\Delta {n}_{{\mathrm{r}}4}=1.13\times {10}^{17}\ {\mathrm{cm}}^{-3}$ and coefficient $\Psi =0.5002$ . Initially, at $t=0$ , a parallel laser with wavelength $\lambda =1$ μm is emitted from origin $z=0$ . The betatron wave number ${k}_{{\mathrm{os}}}=0.64\ {\mathrm{cm}}^{-1}$ can be calculated.

Figure 3 (a) The cylindrical PIC simulation result of a laser with wavelength $\lambda =1\ \unicode{x3bc} \mathrm{m}$ propagating in plasma with ideal matched spot size ${{w}_{{\mathrm{mM}}}=100\ \unicode{x3bc} \mathrm{m}}$ and ${E}_{\psi}=0.3$ , where the data on the negative side of the radial axis are obtained by mirroring. The simulation domain corresponds to plasma channel radius $r=500\;\unicode{x3bc} \mathrm{m}$ and length $L=3$ cm. (b) The coupling parameter $\Psi \cos {k}_{{\mathrm{os}}}z$ varying along the z-direction between the fitting to the simulation data via Equation (15) at every $z$ position in the solid blue line and the calculation values of Equation (15) based on the initial set simulation parameters in the broken red line. The two show good consistency after laser propagating at 1.2 cm.

One of the simulation results is given in Figure 3(a). It is obvious that the laser evolves complex models due to introducing a quartic density profile contribution. We fitted the laser intensity profiles along $z$ -direction based on the form of Equation (15). The regularity of the coupling parameter $\Psi \cos {k}_{{\mathrm{os}}}z$ from the fitting results is represented by the blue line in Figure 3(b). The numerical result calculated with the above paragraph values along the $z$ -direction is also shown in Figure 3(b) by the broken red line. It can be found that the simulation data well agree with the calculation from its parameters, when satisfying an approximate condition ${k}_{{\mathrm{os}}}z\succsim \pi /4$ . Figure 3(b) shows this condition as a demarcation by a broken black line. The limitation must be pointed out that Equation (15) cannot describe the process from a mismatched spot to a quasi-matched spot. Therefore, the broken light blue line fitting with Equation (15) shown in Figure 3 is unphysical. The condition indicates that there is a limit distance for a parallel laser pulse evolving into the model described by Equation (15), which is related to the focusing ability of the plasma. Meanwhile, it is should be considered that the betatron oscillation damps at a distance of ${kl}_{\tau }/{k}_{{\mathrm{os}}}$ ^[ Reference Esarey and Leemans^65], which is the upper limit of the capillary length, where ${l}_{\tau }$ is the laser pulse width. Under such constraints, we adopted discharged plasma in capillaries with channel depth of the order of ${10}^{17}\ {\mathrm{cm}}^{-3}$ and 3-cm length in the experiments.

4 Experimental setup and methods

Figure 4(a) provides a schematic overview of the experimental setup. We used a laser probe pulse to characterize the plasma channel properties. Laser pulses with a wavelength of 798 nm and spot radius of 4 mm were produced by the front end of the new developed terawatt laser system at Peking University as an upgrade part of the Compact LAser Plasma Accelerator (CLAPA)^[ Reference Zhu, Wu, Liao, Geng, Zhu, Li, Xu, Li, Shou, Yang, Wang, Wang, Wang, Chen, He, Zhao, Ma, Lu, Tajima, Lin and Yan^96]. Collimated laser pulses are exported to the experiment area through a 50:50 beam-splitter downstream of the regenerative amplifier. The pulse energy is less than 0.2 mJ, which is low enough to not modify the channel profile.

The capillary is placed on a motorized hexapod stage that can help to precisely adjust the alignment between a capillary and collimated laser pulses, which will affect the output laser profile. For this paper, the length of all used capillaries is 3 cm and the radii are 150, 250 and 350 μm. The off-axis deviation can be reduced to $33\ \unicode{x3bc} \mathrm{rad}$ by judging the concentricity of the continuous diffraction imaging at the exit for 3-cm long capillaries and further to $5\ \unicode{x3bc} \mathrm{rad}$ by judging the concentricity of the laser spot in the presence of the plasma channel. The capillary bulk material was either sapphire ( ${\mathrm{Al}}_2{\mathrm{O}}_3$ ) or a high-temperature laminating (HTL) resin. The HTL resin as a novel 3D-print material that is cheaper can be in a single piece and is easily mechanically machinable. Helium (He) and argon (Ar) are two types of gas that can be filled in the capillary respectively whose pressure can be controlled from 10 to 300 Torr. For the system, the scaling law between the average plasma density and pressure measured from spectroscopy is $\left\langle n\right\rangle \left[{\mathrm{cm}}^{-3}\right]=2.57\times {10}^{15}\cdot P\left[\mathrm{Torr}\right]$ . Both capillary ends were open such that gas escapes to the vacuum chamber. During injecting gas, a high voltage of 0–30 kV from the incentive source is applied to the electrodes located at both ends of the capillary, which will lead to gas discharge to generate plasma. A circuit loop is then formed and the plasma is heated to change into a channel by a current pulse with a peak of 100–400 A. Laser pulses can be guided to propagate through the channels at probable time windows. Figure 4(c) shows a photo of plasma channel luminescence during discharge of the gas-filled capillary.

Figure 4 (a) Schematic overview of the experimental setup. Laser pulses propagate from right to left along a capillary as the axis in the z-direction. The capillary is carried by a six-axis mover. Gas is injected from pipes to fill the capillary. Electrodes connected to the incentive source are located at both ends. The imaging system consisting of a single object lens and a CCD is downstream. (b) Typical experimentally measured transverse laser spot intensity distributions at the capillary exit plane during plasma discharge for plasma channels with $\Psi <0.2$ (the upper subfigure) and $\Psi >0.5$ (the lower one). (c) Discharge phenomenon of a gas-filled capillary viewed from the side.

A simple lens imaging system is set for this detection. The distance between the lens with a focal length of 25 cm and the capillary outlet is about 30 cm. The transverse, time-integrated laser pulse profile was imaged with a charge-coupled device (CCD) camera, which has $1600\times 1200$ pixels and a resolution of $0.517\;\unicode{x3bc} \mathrm{m}/\mathrm{pixel}$ . Figure 4(b) shows examples of measured (single shot, background subtracted) transverse laser pulse intensity distributions at the location of the capillary exit aperture. The upper one represents typical laser spots, which are almost transversely Gaussian, downstream across plasma channel density profiles with $\Psi <0.2$ , where the quartic density contributions can be negligible. The lower one represents those that are no longer fundamental model Gaussian with $\Psi >0.5$ , where the quartic density contributions are obvious in the initial process of plasma channel evolutions, generally.

A DG645 device was used to achieve laser pulses with a repetition rate of 1 kHz propagating through discharged plasma channels at various times, by allocating delay times of the thyratron switch trigger signal referring to the regenerative amplifier trigger signal. The thyratron switch in the incentive source is triggered to close with several ns time jitter. Helmholtz coils are set around current transform cables to get the discharge current time-evolving waveforms and a photodiode was used to get signals when the laser pulses arrived. Both can be measured by an oscilloscope at the same time. The relative time difference between the timing moments of two waveforms is the basis for the time relationship between a laser pulse and a plasma channel.

All the experimental parameter groups are listed in Table 1. As an example, the result of the No. 9 experiment will be analyzed as follows, and is shown in Figure 5. The technique adopted in our example is the simple RLC circuit. The selectable ranges of the resistance ( $R$ ), inductance ( $L$ ) and capacitance ( $C$ ) are 0– $500\;\Omega$ , 0– $18\;\unicode{x3bc} \mathrm{H}$ and 0.5–30 nF, respectively. A principle ${R}^2\approx 2L/C$ followed makes the current waveform, shown in Figure 5 by the solid red line, approximate to a single sine pulse to suppress electromagnetic impacts from the discharge on the electronic equipment, which can be described as follows:

(19) $$\begin{align}I={I}_0{e}^{-\alpha t}\sin \omega t,\quad t>0,\end{align}$$

Table 1 Parameters for all experimental groups.

Figure 5 Current waveform measured in the No. 9 experiment shown with a solid red line and fitting results with the waveform based on Equation (19) shown with a broken red line, reconstructed evolution of $\Psi$ from experimental data shown with blue triangle marks and an exponential decline tendency of $\Psi$ evolution shown with a dotted blue line. Both of the red curves above have the $y$ -axis on the right in red. Both of the blue curves have the $y$ -axis on the left in blue. Subfigure groups made up of a lower one, a measured profile and an upper one, a reconstructed profile via fitting results, in the same column from left to right, correspond to a laser pulse propagating through plasma channels at 245, 343, 458, 537 and 644 ns, respectively. All subfigures are under the unified ruler and color bar placed in the upper right corner inside.

where $\alpha$ and $\omega$ are the decline and oscillation time scale, respectively. The fitting curve shown in Figure 5 based on Equation (19) of the current waveform can provide the time origin of the discharge. The CCD is synchronously triggered by the DG645 device and is able to acquire the transverse, time-integrated intensity profile of the laser pulses. The quasi-Newton algorithm^[ Reference Fletcher^97] and the model identification technique^[ Reference Yuan, Xu, Zhao, Hong, Lu, Feng, Chen, Zou, Zhang, Qin and Zhu^98] can be studied to fit the intensity images with Equation (15). The two rows of subfigures are respectively the experiential profiles (lower ones) and the reconstructed profiles via fitting results (upper ones) in Figure 5, where the intensity is normalized. Notice that the pulse is no longer transversely fundamental Gaussian. The subfigure groups (both in every column) from left to right correspond to a laser pulse propagating through plasma channels at 245, 343, 458, 537 and 644 ns, respectively. The unified ruler and color bar are placed in the upper right corner inside Figure 5. The blue triangle marks are coefficients $\Psi$ defined in Section 2 reconstructed from the fitting results with the experimental measurements. It is almost certain that there is an exponential decline tendency ${e}^{-\varsigma t}$ of $\Psi$ , which indicates that the ${r}^4$ density contribution decreases in the whole current heating process. It is related to the current drop from the peak over time. The coefficient $\varsigma$ can be defined as a time-dependent decline rate to denote the decrease rate of $\Psi$ , that is, the contribution of a quartic density profile. For the case in Figure 5, $\varsigma =5.72\ {\mathrm{s}}^{-1}$ from the fitted dotted blue line.

5 Experimental results and discussion

In previous work, the impact of discharge voltages and avalanche processes^[ Reference Druyvesteyn and Penning^99] on the development of subsequent plasma channels was rarely considered. Here, we evaluated the influence of the discharge voltages on plasma density evolution with quartic distributions by three styles of waveforms with different current pulse widths keeping the peak currents the same as much as possible, as shown in Figure 6(a). The three different styles of waveforms are obtained by adjusting circuit parameters $R$ , $L$ and $C$ simultaneously. It is difficult to ensure the same current waveform. Only if the high voltages $V$ of Styles 1–3 are respectively 20, 18 and 15 kV, the peaks are almost equal, that is, those of the No. 9, No. 10 and No. 1 experimental groups. The waveform oscillation time scale $\omega$ can be reconstructed as $6.985$ , $2.764$ and $0.017\;{\unicode{x3bc} \mathrm{s}}^{-1}$ , respectively, which is a highly discriminative parameter. However, for such short current pulses, the characteristic parameters of the current transmission loop strongly depend on the current itself. Gas-filled capillaries are small enough loads to neglect. The plasma channel-guided effect starts almost between 200 and 300 ns after the current time origin.

Figure 6 (a) Measured (solid lines) and fitting (broken lines) current waveforms of Style 1 in red, Style 2 in blue and Style 3 in black with $\omega =6.985$ , $2.764$ and $0.017\;{\unicode{x3bc} \mathrm{s}}^{-1}$ respectively in the No. 9, No. 10 and No. 1 experiments correspondingly. (b) Reconstructed evolutions of density profile coefficient $\Psi$ shown with triangle marks and exponential decline tendencies shown with broken lines with various styles of waveforms in No. 9 (red), No. 10 (blue) and No. 1 (black) experiments.

At the rising edge of the current, the gas is rapidly broken down and transformed into almost fully ionized plasma. Discharge voltages positively push the process via more electron emissions^[ Reference Druyvesteyn and Penning^99]. From around the current peak moment, the non-uniform plasma profile is formed and begins to evolve. The evolution process is positively correlated with the current, resulting in the heating effect. In order to evaluate the impact of the breakdown on the evolution, we manipulated the incentive source to generate plasma in a waveform with a higher voltage but a smaller heating current, that is, a steeper falling edge of the current. The decline $\varsigma$ of $\Psi$ still decreases with the current, as shown in Figure 6(b), which infers that the gain from voltages is not enough to offset the reduction in current, and the impact of the breakdown can be negligible compared to that of current heating.

Figure 7 Reconstructed evolutions of density profile coefficient $\Psi$ shown with triangle marks and exponential decline tendencies shown with broken lines (a) with various high voltages $V=15$ , 20, 25 and 28 kV for He plasma in the No. 1 (red), No. 2 (blue), No. 3 (black) and No. 4 (green) experiments, (b) with various high voltages $V=15$ , 20, 25 and 28 kV for Ar plasma in the No. 5 (red), No. 6 (blue), No. 7 (black) and No. 8 (green) experiments. (c) Scaling of the exponential decline rate $\varsigma$ with high voltages for He plasma shown with the light-blue line and Ar plasma shown with the light-red line. The circle marks are in colors corresponding to those in (a) and (b).

For the same style of waveform with almost the same oscillation scale $\omega$ , currents are almost positively linearly related to voltages. In the experiments with the same style waveform, the circuit parameters $R$ , $L$ and $C$ are fixed and adjusting voltages would make the peak currents change. In the following analysis of the influences, it is clear that the effect from currents is dominant compared to that of the voltages.

In order to clarify the influence of the experimental conditions on the profile evolutions, we adopt the control variable method for group comparisons listed in Table 1. Figure 7(a) shows reconstructed profile evolutions with triangle marks and exponential decline tendencies with broken lines with various high voltages $V=15$ , 20, 25 and 28 kV respectively in red, blue, black and green corresponding to the No. 1, No. 2, No. 3 and No. 4 experiments for He plasma. Figure 7(b) shows those in the No. 5, No. 6, No. 7 and No. 8 experiments for Ar plasma. Both Figures 7(a) and 7(b) evidence that $\varsigma$ increases with voltage increasing, essentially that is an increase in the heating effect of the current, which is shown in Figure 7(c). Meanwhile, they also evidence that $\varsigma$ of Ar plasma is larger than that of He plasma, as shown in Figure 7(c). It can be interpreted that the discharge of Ar plasma is accompanied by weaker electron-ion heat conduction^[ Reference Lindstrøm, Adli, Boyle, Corsini, Dyson, Farabolini, Hooker, Meisel, Osterhoff, Röckemann, Schaper and Sjobak^45].

Figure 8(a) shows reconstructed profile evolutions with triangle marks and exponential decline tendencies with broken lines with various capillary diameters $d=300$ , $500$ and $700\;\unicode{x3bc} \mathrm{m}$ , respectively in red, blue and black corresponding to the No. 11, No. 9 and No. 12 experiments, which evidences that $\varsigma$ increases with diameters decreasing, as shown in Figure 8(c). Figure 8(b) shows reconstructed profile evolutions with triangle marks and exponential decline tendencies with broken lines with various capillary pressures $P=80$ , 90 and 100 Torr, respectively in red, blue and black corresponding to the No. 13, No. 14 and No. 9 experiments, which evidences that $\varsigma$ increases with pressures decreasing, as shown in Figure 8(d). The capillary load is small enough that the difference in the parameters of the capillaries themselves hardly affects the circuit characteristics. It is a widely held view that current densities depending on diameters and plasma densities depending on pressures are important factors for local plasma heating. When the current is the same, less plasma will be heated to a higher temperature in less time. Besides, more advanced techniques are needed to address the starting moment of the channel-guided effect experimentally. The problem brings some disturbance to the analysis.

Figure 8 Reconstructed evolutions of the density profile coefficient $\Psi$ shown with triangle marks and exponential decline tendencies shown with broken lines (a) with various capillary diameters $d=300$ , $500$ and $700\ \unicode{x3bc} \mathrm{m}$ in the No. 11 (red), No. 9 (blue) and No. 12 (black) experiments, and (b) with various pressures $p=80$ , 90 and 100 Torr in the No. 13 (red), No. 14 (blue) and No. 9 (black) experiments. (c) Scaling of the exponential decline rate $\varsigma$ with diameters. The circle marks are in colors corresponding to those in (a). (d) Scaling of the exponential decline rate $\varsigma$ with pressures. The circle marks are in colors corresponding to those in (b).

From the experimental results, we can roughly describe the phenomenon of the plasma channel evolution. Previous research work^[ Reference Bobrova, Esaulov, Sakai, Sasorov, Spence, Butler, Hooker and Bulanov^67, Reference Broks, Garloff and van der Mullen^68, Reference Broks, van Dijk, Bastiaens, Boller and van der Mullen^92] has described the ohmic heating model of plasmas under quasi-thermal equilibrium conditions. It is the basis on which the $J{-}T$ model^[ Reference Bobrova, Esaulov, Sakai, Sasorov, Spence, Butler, Hooker and Bulanov^67] is well suited for parabolic density distributions. We could consider that the parabolic density distribution indicated by the model $n(r)/{n}_0=1+{ar}^2/{r}_{{\mathrm{ch}}}^2$ is approximately valid when $\Psi \to 0$ , where $a$ is a main current-related parameter that needs to be measured^[ Reference Broks, Garloff and van der Mullen^68]. A rough evolution hypothesis can be described by the conservation of the number of particles $\partial \left\langle n\right\rangle /\partial t=0$ in the plasma channel. The evolution is from a significant quartic density distribution to a parabolic density distribution holding average constant. However, the approximation is limited as the sub-microsecond timescale is close to the characteristic timescale of acoustic velocity.

So far, the quartic density profile evolution at the outer edge of the channel has not been succinctly described as a brief model. We suspect that this has a great deal to do with heat conduction, nonlocal thermal equilibrium, potential balance and particle collisions and diffusion with the wall material. Probably, it is a reasonable explanation to associate quartic density profile contributions with the plasma presheath of arc discharges^[ Reference Mancinelli, Prevosto, Chamorro, Minotti and Kelly^100]. Due to the conservation of particle numbers, the density at the outer edge of channels is positively correlated with the plasma temperature in the central channel region. Obviously, the same goes for the temperature at the edge. Thus, the thickness of the presheath is negatively correlated with temperature. In this way, the profile evolution process is linked to the initial temperature rise of the central plasma. To a certain extent, $\Psi$ and $\varsigma$ would physically express the temperature and temperature rate, respectively. It is possible to introduce why voltages and currents are positive factors to quicken plasma evolutions, and electron-ion heat conduction, radii and pressures are negative. We consider that quickening plasma evolutions is equivalent to increasing the exponential decline tendency coefficient $\varsigma$ of the rate $\Psi$ of the quartic density contribution.

The density profile evolution offers additional ideas for optimizing the plasma channel. For applications of capillaries such as guiding ultra-intense laser pulses and laser–plasma acceleration, ideal parabolic density profiles seem to require sufficient plasma heating but within larger temperature gradients. The laser–plasma relativistic interactions considering quartic profile contributions remain unstudied. For charged particle beam transports with plasma channels as APLs, the current density profile is a crucial factor. Plasma with almost entirely parabolic distributions and large temperature gradients is not conducive to maintaining beam emittance. Far from quasi-thermal equilibrium conditions, the impact of quartic profile contributions on the current density profile is unknown. A balance should be found on whether to change circuit parameters or to choose different plasma synchronous moments when searching for the desired currents.

One more experiment was carried out with the same contributions but with laser wavelength $\lambda =532$ nm in the No. 9 experiment. We deployed a Q-switched laser. Equations (12)–(16) do not exhibit a significant dependence on wavelength. Figure 9 confirms this. The reconstructed profile evolutions measured by the laser pulses from the front end shown with red triangle marks and the Q-switched laser shown with blue ones have a good consistency. The trigger of the latter is subjected to severe electromagnetic shock from the plasma discharges, resulting in a shorter time window for measurements. The slight difference between the two tendencies comes from the sensitivities of different photodiodes, possibly.

Figure 9 Reconstructed profile evolutions shown with triangle marks and exponential decline tendencies shown with broken lines from measurements with laser wavelength $\lambda =786$ nm of the front (red) end and $\lambda =532$ nm of a Q-switched laser (blue), respectively.