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Laser-machined two-stage nozzle optimised for laser wakefield acceleration

Part of: Laser and Plasma Accelerators 2023

Published online by Cambridge University Press:  08 February 2024

V. Tomkus Open the ORCID record for V. Tomkus [Opens in a new window] ,
M. Mackevičiūtė ,
J. Dudutis Open the ORCID record for J. Dudutis [Opens in a new window] ,
V. Girdauskas Open the ORCID record for V. Girdauskas [Opens in a new window] ,
M. Abedi-Varaki Open the ORCID record for M. Abedi-Varaki [Opens in a new window] ,
P. Gečys Open the ORCID record for P. Gečys [Opens in a new window]  and
G. Račiukaitis Open the ORCID record for G. Račiukaitis [Opens in a new window]
V. Tomkus*
Affiliation:
FTMC - Center for Physical Sciences and Technology, Savanoriu Ave. 231, LT-02300 Vilnius, Lithuania
M. Mackevičiūtė
Affiliation:
FTMC - Center for Physical Sciences and Technology, Savanoriu Ave. 231, LT-02300 Vilnius, Lithuania
J. Dudutis
Affiliation:
FTMC - Center for Physical Sciences and Technology, Savanoriu Ave. 231, LT-02300 Vilnius, Lithuania
V. Girdauskas
Affiliation:
FTMC - Center for Physical Sciences and Technology, Savanoriu Ave. 231, LT-02300 Vilnius, Lithuania Vytautas Magnus University, K. Donelaicio St. 58, LT-44248 Kaunas, Lithuania
M. Abedi-Varaki
Affiliation:
FTMC - Center for Physical Sciences and Technology, Savanoriu Ave. 231, LT-02300 Vilnius, Lithuania
P. Gečys
Affiliation:
FTMC - Center for Physical Sciences and Technology, Savanoriu Ave. 231, LT-02300 Vilnius, Lithuania
G. Račiukaitis
Affiliation:
FTMC - Center for Physical Sciences and Technology, Savanoriu Ave. 231, LT-02300 Vilnius, Lithuania
*
† Email address for correspondence: vidmantas.tomkus@ftmc.lt
Rights & Permissions [Opens in a new window]

Abstract

In this paper, the modelling and manufacturing of a two-stage supersonic gas jet nozzle enabling the formation of adaptive plasma concentration profiles for injection and acceleration of electrons using few-cycle laser beams are presented. The stages are modelled using the rhoSimpleFoam algorithm of the OpenFOAM computational fluid dynamics software. The first 200–300 ${\rm \mu}$m diameter nozzle stage is dedicated to 1 % N2 + He gas jet formation and electron injection. By varying the pressure between the first and second stages of the injectors, the electron injection location could be adjusted, and the maximum acceleration distance could be ensured. By changing the concentration of the nitrogen in the gas mixture, the charge of the accelerated electrons could be controlled. The second nozzle stage is designed for acceleration in fully ionised He or hydrogen gas and forms the optimal plasma concentration for bubble formation depending on the laser pulse energy, duration and focused beam diameter. In order to reduce the diameter of the plasma profile formed by the first nozzle and the concentration drop gap between the two nozzles, a one-side straight section was introduced in the first nozzle. The shock wave reflected from the straight section of the wall propagates parallel to the shock wave of the intersecting supersonic jets and ensures a minimal gap between the jets. The second-stage longitudinal plasma concentration profile could have an increasing gas density gradient to compensate for dephasing between the electron bunch and the plasma wave due to wave shortening with increasing plasma concentration.

Keywords

laser wakefield accelerationelectron accelerationtwo-stage gas nozzleplasma applicationsplasma devices
Type
Research Article
Information
Journal of Plasma Physics , Volume 90 , Issue 1 , February 2024 , 965900102
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Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press

1. Introduction

In laser wakefield acceleration (LWFA), the ponderomotive force of the intense laser pulse drives plasma electrons outward, forming plasma waves that travel at relativistic speeds along with the laser pulse. The electrons trapped in the plasma wave can be accelerated up to GeV energies in an efficient and compact way (Tajima & Malka Reference Tajima and Malka2020). Numerical and experimental studies have been conducted on a wide variety of injection methods to produce quasi-monoenergetic electron beams, such as self-injection (Kalmykov et al. Reference Kalmykov, Yi, Khudik and Shvets2009; Mangles et al. Reference Mangles, Genoud, Bloom, Burza, Najmudin, Persson, Svensson, Thomas and Wahlström2012; Wan et al. Reference Wan, Seemann, Tata, Andriyash, Smartsev, Kroupp and Malka2022), ionisation injection (McGuffey et al. Reference McGuffey, Thomas, Schumaker, Matsuoka, Chvykov, Dollar, Kalintchenko, Yanovsky, Maksimchuk and Krushelnick2010; Pak et al. Reference Pak, Marsh, Martins, Lu, Mori and Joshi2010) and density down-ramp injection (Schmid et al. Reference Schmid, Buck, Sears, Mikhailova, Tautz, Herrmann, Geissler, Krausz and Veisz2010; Buck et al. Reference Buck, Wenz, Xu, Khrennikov, Schmid, Heigoldt, Mikhailova, Geissler, Shen and Krausz2013; Foerster et al. Reference Foerster, Döpp, Haberstroh, Grafenstein, Campbell, Chang, Corde, Cabadağ, Debus and Gilljohann2022). The electron injection based on a down-ramp plasma density transition profile has proved to be an effective technique for high-quality LWFA electron beam acceleration (Schmid et al. Reference Schmid, Buck, Sears, Mikhailova, Tautz, Herrmann, Geissler, Krausz and Veisz2010; Buck et al. Reference Buck, Wenz, Xu, Khrennikov, Schmid, Heigoldt, Mikhailova, Geissler, Shen and Krausz2013; Wang et al. Reference Wang, Feng, Ke, Yu, Xu, Qi, Chen, Qin, Zhang and Fang2021). A lot of research has been done on density down-ramp injection in beam-driven plasma wakefield acceleration (Suk et al. Reference Suk, Barov, Rosenzweig and Esarey2001; De La Ossa et al. Reference De La Ossa, Hu, Streeter, Mehrling, Kononenko, Sheeran and Osterhoff2017; Xu et al. Reference Xu, Li, An, Dalichaouch, Yu, Lu, Joshi and Mori2017). This technique has been employed recently for free-electron lasing using a high-quality LWFA electron beam of 10–50 pc (Schmid et al. Reference Schmid, Buck, Sears, Mikhailova, Tautz, Herrmann, Geissler, Krausz and Veisz2010; Buck et al. Reference Buck, Wenz, Xu, Khrennikov, Schmid, Heigoldt, Mikhailova, Geissler, Shen and Krausz2013; Wang et al. Reference Wang, Feng, Ke, Yu, Xu, Qi, Chen, Qin, Zhang and Fang2021). Ionisation injection schemes with nitrogen-mixed light gases such as He or H2 and pure He or H2 in a second section were investigated by Golovin et al. (Reference Golovin, Banerjee, Chen, Powers, Liu, Yan, Zhang, Zhang, Zhao and Umstadter2016). Nevertheless, due to the nonlinear laser propagation in LWFA plasma, it is challenging to analyse the injection mechanisms (Vieira et al. Reference Vieira, Fiúza, Silva, Tzoufras and Mori2010).

One of the most essential features of LWFA is the capability to create a suitable gas density profile for the controlled injection of electrons into the accelerator phase while maintaining the optimal plasma concentration for acceleration. One common method of creating desired density profiles is by producing a supersonic gas nozzle. The demand for stable, energetic e-beams with a wide energy range, low energy spread and high charge is prevalent in a variety of applications. Due to the differences in injection and acceleration requirements, those characteristics have been difficult to achieve simultaneously in single-stage LWFA. In electron acceleration, for instance, low-density plasma is needed since it has longer dephasing $({L_{\textrm{deph}}} \propto {n^{ - 3/2}})$ and depletion lengths $({L_{\textrm{depl}}} \propto {n^{ - 1}})$. However, self-trapping in low-density plasma is ineffective at injecting charge and provides little control over generated electron beams. To overcome this issue, several approaches have been introduced. Instead of self-trapping, these methods deterministically force background plasma electrons to become locally dephased and then become trapped and eventually accelerated by the plasma wave. There are several experimental studies (Gonsalves et al. Reference Gonsalves, Nakamura, Lin, Panasenko, Shiraishi, Sokollik, Benedetti, Schroeder, Geddes and Van Tilborg2011; Liu et al. Reference Liu, Xia, Wang, Lu, Wang, Deng, Li, Zhang, Liang and Leng2011; Pollock et al. Reference Pollock, Clayton, Ralph, Albert, Davidson, Divol, Filip, Glenzer, Herpoldt and Lu2011; Kim et al. Reference Kim, Pae, Cha, Kim, Yu, Sung, Lee, Jeong and Lee2013; Wang et al. Reference Wang, Li, Liu, Wang, Chen, Zhang, Qi, Leng, Liang and Liu2013; Vargas et al. Reference Vargas, Schumaker, He, Zhao, Behm, Chvykov, Hou, Krushelnick, Maksimchuk and Yanovsky2014; Golovin et al. Reference Golovin, Banerjee, Chen, Powers, Liu, Yan, Zhang, Zhang, Zhao and Umstadter2016; Steinke et al. Reference Steinke, Van Tilborg, Benedetti, Geddes, Schroeder, Daniels, Swanson, Gonsalves, Nakamura and Matlis2016) that physically separate the injector and accelerator stages, but in most cases, these supersonic nozzles or gas cells were designed for LWFA using multi-terawatt joule class lasers with pulse duration of 30 fs. The lower plasma concentration and relatively long pulse duration result in a millimetre-scale acceleration distance. Such gas targets are manufactured using regular computer numerical control or additive printing technology. In the case of kilohertz lasers with few-cycle pulse duration and limited energy, the acceleration distance shrinks to a few hundred microns, and tailoring of plasma targets becomes challenging. Stable LWFA acceleration using a near one-cycle laser and a one-side-shock nozzle manufactured using hybrid fused silica machining technology was demonstrated in Rovige et al. (Reference Rovige, Huijts, Andriyash, Vernier, Tomkus, Girdauskas, Raciukaitis, Dudutis, Stankevic and Gecys2020). An alternative design of an LWFA target is presented in this paper, which consists of two-stage gas micrometric nozzles: the injector, which uses mixed gas (helium with a small percentage of nitrogen), and the accelerator (with pure helium). This approach offers the advantage of independent control of the plasma concentration of the injection and acceleration stage. A novel focusing tool – an axiparabola proposed for broadband high-intensity lasers – produces Bessel–Gauss beams and allows for tuning of the phase of the plasma wave (Smartsev et al. Reference Smartsev, Caizergues, Oubrerie, Gautier, Goddet, Tafzi, Phuoc, Malka and Thaury2019; Oubrerie et al. Reference Oubrerie, Andriyash, Lahaye, Smartsev, Malka and Thaury2022). Therefore, a short injector zone with a longer acceleration distance could be required relative to the gas target formed by a circular supersonic nozzle. Several scholars have also investigated density injection parameters numerically (Samant, Upadhyay, & Krishnagopal Reference Samant, Upadhyay and Krishnagopal2014; Ekerfelt et al. Reference Ekerfelt, Hansson, Gallardo González, Davoine and Lundh2017; Massimo et al. Reference Massimo, Lifschitz, Thaury and Malka2017).

Nowadays, emerging kilohertz laser systems, generating few-cycle laser pulses with tens of mJ of energy, provide peak intensities in the range of 1018–1019 W cm−2 and can be efficiently used for particle acceleration (He Reference He2014; Zhang et al. Reference Zhang, Chen, Zou, Zhu, Li, Yang, Liu, Yu, Ma and Sheng2022; Tóth et al. Reference Tóth, Nagymihály, Seres, Lehotai, Csontos, Tóth, Geetha, Somoskői, Kajla and Abt2023). To drive the charged particles in the LWFA self-guiding bubble regime, relatively high plasma concentrations of n = 3–5 × 1019 cm−3 and tighter focusing of the laser beam to the diameter of 3–5 ${\rm \mu}$m are required. This leads to shorter acceleration distances of tens to hundreds of micrometres, resulting in a relatively low energy of the accelerated electrons and high energy spread.

This study aims to simulate and manufacture a two-stage nozzle from a single block of fused silica for injection and LWFA of electrons using few-cycle laser pulses of the Bessel–Gauss beam. The extended focal line of the Bessel–Gauss beam ensures a low diffraction and longer acceleration distance. A one-sided shock nozzle was manufactured using three-dimensional laser machining of fused silica blocks, enabling injection control with a precision of 20–40 ${\rm \mu}$m (Rovige et al. Reference Rovige, Huijts, Andriyash, Vernier, Tomkus, Girdauskas, Raciukaitis, Dudutis, Stankevic and Gecys2020). The longitudinal plasma concentration profile of the second stage can include the rising gas density gradient to reach the rephasing thanks to the shifting of the centre of the plasma bubble with a decrease of the plasma wavelength (Sprangle et al. Reference Sprangle, Hafizi, Penano, Hubbard, Ting, Moore, Gordon, Zigler, Kaganovich and Antonsen2001; Pukhov & Kostyukov Reference Pukhov and Kostyukov2008; Guillaume et al. Reference Guillaume, Döpp, Thaury, Phuoc, Lifschitz, Grittani, Goddet, Tafzi, Chou and Veisz2015; Oubrerie et al. Reference Oubrerie, Andriyash, Lahaye, Smartsev, Malka and Thaury2022).

Different machining techniques are applied for manufacturing converging–diverging channels. Conventional mechanical drilling enables fabrication of channels with diameters usually down to 1 mm (Cai et al. Reference Cai, Liu, Shi, Song and Wan2015, Reference Cai, Liu, Shi, Song and Wan2016; Cai, Liu, & Shi Reference Cai, Liu and Shi2017). However, the converging and diverging parts are manufactured in separate steps (Cai et al. Reference Cai, Liu, Shi, Song and Wan2015, Reference Cai, Liu and Shi2017). Although additive manufacturing processes like stereolithography and selective laser sintering are also limited to throat diameters down to 1 mm, complex channels are manufactured in a single step (Döpp et al. Reference Döpp, Guillaume, Thaury, Gautier, Ta Phuoc and Malka2016; Andrianaki et al. Reference Andrianaki, Grigoriadis, Skoulakis, Tazes, Mancelli, Fitilis, Dimitriou, Benis, Papadogiannis and Tatarakis2023). Femtosecond laser-induced chemical etching (FLICE), laser trepanning and electrical discharge machining (EDM) can be used to produce diameters smaller than 100 ${\rm \mu}$m (Takahashi et al. Reference Takahashi, Horiuchi, Mori, Tatsukoshi, Ono, Mikayama, Imajo and Mobley2013; Tomkus et al. Reference Tomkus, Girdauskas, Dudutis, Gečys, Stankevič and Račiukaitis2019; Chiomento et al. Reference Chiomento, Zuffi, Vieira, Tabacow, Maldonado and Samad2021; Zuffi et al. Reference Zuffi, Tabacow, Vieira and Samad2022). However, de Laval-shaped channel formation is typically a two-step process for laser trepanning and EDM techniques (Li et al. Reference Li, Wang, Wang and Zhao2018; Chiomento et al. Reference Chiomento, Zuffi, Vieira, Tabacow, Maldonado and Samad2021; Zuffi et al. Reference Zuffi, Tabacow, Vieira and Samad2022). The FLICE technique is more flexible as converging–diverging channels can be formed from one side, and additional alignment and coupling steps can be avoided (Rovige et al. Reference Rovige, Huijts, Vernier, Andriyash, Sylla, Tomkus, Girdauskas, Raciukaitis, Dudutis and Stankevic2021). However, FLICE, laser trepanning or EDM of micrometre-scale diameters are limited to low millimetre-scale depths (Takahashi et al. Reference Takahashi, Horiuchi, Mori, Tatsukoshi, Ono, Mikayama, Imajo and Mobley2013; Tomkus et al. Reference Tomkus, Girdauskas, Dudutis, Gečys, Stankevič and Račiukaitis2019; Chiomento et al. Reference Chiomento, Zuffi, Vieira, Tabacow, Maldonado and Samad2021). For larger nozzles, nanosecond rear-side milling is an appealing approach, as centimetre-sized nozzles can be formed with channel diameters down to 100 ${\rm \mu}$m (Tomkus et al. Reference Tomkus, Girdauskas, Dudutis, Gečys, Stankevič and Račiukaitis2018; Chaulagain et al. Reference Chaulagain, Karatodorov, Raclavský, Lorenz, Lamač, Albrecht, Tomkus, Dudutis, Mackevičiūtė and Gečys2021). Additionally, this technique can form converging–diverging channels up to some angle in a single step (Chaulagain et al. Reference Chaulagain, Karatodorov, Raclavský, Lorenz, Lamač, Albrecht, Tomkus, Dudutis, Mackevičiūtė and Gečys2021). In this paper, we demonstrate fabrication of a two-stage nozzle from a single block of fused silica using the rear-side milling technique only.

2. Numerical modelling of gas propagation

In this article, we propose implementing a two-stage supersonic nozzle to optimise the injection and acceleration of electrons using a Bessel–Gauss driving beam. The two-stage nozzle was simulated using the OpenFOAM computational fluid dynamics (CFD) software (figures 1 and 2) (Chen et al. Reference Chen, Xiong, Morris, Paterson, Sergeev and Wang2014; OpenFOAM, 2022). The first stage of the nozzle with a diameter of 200–300 ${\rm \mu}$m is used for the formation of a 1 % N2 + He gas jet and ionisation injection of electrons. The balance of the backing pressure between the first and second nozzle stages allows the injection of electrons into the rear part of the plasma bubble and ensures the maximal acceleration distance. The second stage of the nozzle is dedicated to the LWFA acceleration of electrons in the pure He gas and defines the optimal plasma concentration for the formation of the bubble depending on the pulse energy, duration and diameter of the focused beam. To minimise the gap of the concentration drop between the two jets, a one-side straight section (Rovige et al. Reference Rovige, Huijts, Andriyash, Vernier, Tomkus, Girdauskas, Raciukaitis, Dudutis, Stankevic and Gecys2020) was implemented to cause a shock wave reflected from the wall to be colinear to the shock wave from the intersecting supersonic jets. The longitudinal plasma concentration profile of the second stage includes the rising gas density gradient to reach the rephasing and extend the LWFA acceleration distance. It should be noted that helium gas at room temperature 293 K was used in simulations, and the backing pressure was changed from 10 to 50 bar. The corresponding maximal atomic gas concentration at the 500 ${\rm \mu}$m maximum from the nozzle outlet was correspondingly between 1.0 × 1019  and 5.2 × 1019 cm−3. Also, the gas concentration in figures 1 and 2 corresponds to the backing pressure of 27 bar.

Figure 1. Schematic of a two-stage nozzle (a,b), consisting of a first electron injection stage (1), a second acceleration stage (2) and a one-half straight wall section (3) that reduces the gas concentration drop between the two nozzles. Longitudinal profile of gas concentration (c) to form an increasing gas density gradient.

Figure 2. Helium gas density diagram of the two-stage supersonic nozzle (a) and longitudinal gas concentration profile at the backing pressure of 27 bar along the laser propagation path at a 0.5 mm distance above the outlet of the injection stage of the nozzle (b) simulated using the OpenFOAM CFD software.

3. Methods of laser processing and characterisation

Nozzles with converging–diverging channels have already been manufactured from fused silica using nanosecond laser rear-side milling (Chaulagain et al. Reference Chaulagain, Karatodorov, Raclavský, Lorenz, Lamač, Albrecht, Tomkus, Dudutis, Mackevičiūtė and Gečys2021). However, the most complicated parts of this design are the diverging part of the acceleration channel and the inclined injection channel with an angle of 30°. In this geometry, there is unaffected material below at least part of the scanning contour, increasing the chance of the channel clogging with processing debris. Therefore, it is essential to remove ablated particles from the interaction zone. For this, pulses of high air pressure were blown toward the channel after each layer.

In this study, the nozzle was fabricated with a 532 nm wavelength and 4.5 ns (FWHM) pulse duration (Atlantic 60, Ekspla) three-dimensional laser system using a nanosecond bottom-up machining method (Gečys, Dudutis, & Račiukaitis Reference Gečys, Dudutis and Račiukaitis2015; Tomkus et al. Reference Tomkus, Girdauskas, Dudutis, Gečys, Stankevič and Račiukaitis2018, Reference Tomkus, Girdauskas, Dudutis, Gečys, Stankevič, Račiukaitis, González, Guénot, Svensson and Persson2020). The laser beam in the X and Y axes was controlled by a galvanometric scanner (excelliSCAN 14, Scanlab), and the positioning in the Z axis was performed by a motorised translation table (8MT167-100, Standa). The beam was focused using an 80 mm focal length telecentric f-theta lens. The beam waist diameter of 10.8 ${\rm \mu}$m was measured by the Liu's method on a chrome-coated glass sample. Machining was performed on a 100 × 100 × 8.2 mm3 fused silica block. Initially, the laser beam was focused below the rear side of the sample ((figure 3a). The laser beam was scanned in spiral, contour or hatch trajectories (figure 3b). After scanning one layer, the sample was lowered by a specific value of dz (figure 3a). In each subsequent layer, the scan path of the beam was rotated relative to it by an angle of 33°. This was done to achieve a more homogeneous machining regime. Beam scanning and shifting of a sample in the Z direction were repeated until the object was cut entirely. It should be noted that the channels were milled by blowing an air current towards them. Air was blown through a 3 mm outlet diameter nozzle, 8.4 mm below the rear side of the sample. After each layer, the air was blown for 100 ms with a pressure of 0.1 MPa. The processing products were removed from the ablation zone with the help of an extraction system (AD Oracle iQ, Bofa International Ltd).

Figure 3. (a) The scheme of milling and laser scanning; (b) contour, spiral and hatch laser scanning algorithms. The darkest colour represents the beginning of the scan, and the brightest colour represents the end. (c) The steps of nozzle milling. The grey colour represents the areas to be removed in the corresponding stage, blue – the material not affected by the laser, and red – the already processed part.

Nozzle milling was divided into 5 stages (figure 3c). In the first step, the conical part of the nozzle (step 1) was formed by the spiral scanning algorithm. The spiral algorithm is the fastest because jumps between scan lines could be avoided. The inclined channel for injection was then formed by the spiral scanning algorithm (step 2). Next, the plane part of the nozzle was milled by the hatching algorithm (step 3). The hatching is advantageous for ablating uniform surfaces throughout the processing area. For other algorithms, the thermal accumulation effects are higher for the centre area than for the outer parts. The larger channel for acceleration was milled with a contour algorithm (step 4). It was desired to mill the channels with a scanning algorithm, with which the scanning would be started from the centre to the outside. Since both channels had some parts where the following layers had larger milling areas than the layer before, outward scanning ensured that the ablated particles had a chance to be removed from the milled area. Scanning to the outside can be done with both a spiral and a contour algorithm. The spiral algorithm is optimal for parts with circular shapes. However, the contour scan algorithm was chosen to produce the acceleration channel due to the more complex shape. Finally, the nozzle body was cut entirely out using a spiral algorithm (step 5).

The quality of the milled nozzles was evaluated using an optical microscope (Eclipse LV100NDA, Nikon) and an optical profilometer (S Neox, Sensofar). The roughness of the linear profiles was measured according to ISO 4288:1996.

4. Nozzle fabrication

The nozzle was milled using 65 ${\rm \mu}$J pulse energy and a 20 ${\rm \mu}$m distance between the centres of successive irradiated spots along the scanning trajectory. The distance between the sequent scanning lines was 25 ${\rm \mu}$m for the nozzle body and 20 ${\rm \mu}$m for the channels. The nozzle body was milled with a pulse repetition rate of 200 kHz (step 1). This allowed us to reach a high milling rate of 0.75 mm3 s−1 and a milling efficiency of 0.06 mm3 J−1. Other parts were milled with a lower pulse repetition rate to minimise the thermal accumulation effects. The inclined injection channel (step 2) was milled with a relatively low 1 kHz pulse repetition rate. The plane part (step 3) and acceleration channel (step 4) were milled with repetition rates of 10 kHz and 2 kHz, respectively. The processing time of grey parts in each step is indicated in images above in figure 3c. The total milling time was 1 hour and 13 minutes.

A photo of the milled nozzle is shown in figure 4(a), and the optical microscope image of the top surface of the nozzle is shown in figure 4(b). The top and bottom sides of the injection channel are shown in figures 4(c) and 4(d), and the top, waist and bottom of the acceleration channel are seen in figures 4(e)–4(g). The edge chipping at the bottom and top side of the injection nozzle was up to 130 ${\rm \mu}$m and 60 ${\rm \mu}$m, respectively. The chipping of the acceleration nozzle was smaller – up to 120 ${\rm \mu}$m on the bottom side and up to 30 ${\rm \mu}$m on the top side. Both channels had larger chipping on the bottom surface. The reason for this may have been that high air pressure was blown into the channels, and at the end of the milling, a thin layer of glass at the bottom surface was fractured.

Figure 4. Photo of the milled nozzle (a). Optical microscope photographs (b)–(g). Photograph of the top surface of the nozzle (b). The entire image was made from a programmatically stitched sequence of photographs taken at different Z-heights. The injection channel at the top (c) and bottom (d). The acceleration nozzle at different positions is shown in (e)–(g). Section of nozzle drawing (h) and topography of cut nozzle channels (i).

A drawing of the entire section of the nozzle channels is shown in figure 4(h). The nozzle was cut through using the same bottom-up milling technique to view the milled channels. The topography of the cut channels can be seen in figure 4(i). After cutting, the surface roughness of the channels in the vertical direction (along the flow direction) was measured. The acceleration channel had an Ra of 1.6 ${\rm \mu}$m and Rz of 11.3 ${\rm \mu}$m. The surface quality of the inclined injection channel was poorer, with Ra of 4.1 ${\rm \mu}$m and Rz of 21.4 ${\rm \mu}$m. This was comparable to the roughness of the plane part of the nozzle with Ra of 4.1 ${\rm \mu}$m and Rz of 31.7 ${\rm \mu}$m.

To connect the nozzle to two different gas supply systems, using two Parker type 9 valves, two adapters and a conical mounting ring were made of stainless steel by three-dimensional laser printing (figure 5a,b). An image of the assembled nozzle with adapters is shown in figure 5(c).

Figure 5. Two-valve adapter for supplying different gases to the nozzle (a,b) and image of the assembled nozzle with two different gas supply systems (c).

5. Concluding remarks

In this work, a two-stage supersonic nozzle was proposed to optimise the injection and acceleration of electrons. The first stage of the nozzle is used for ionisation injection of electrons and adjustment of the position of the injected electrons into the rear part of the plasma bubble. The second stage of the nozzle is dedicated to the LWFA acceleration of electrons and ensures the maximal acceleration distance and low energy spread of the electrons. The gas density was simulated using the OpenFOAM CFD software and the designed nozzle was manufactured from a single fused silica block using a three-dimensional nanosecond rear-side processing technique.

Acknowledgements

Editor Luís O. Silva thanks the referees for their advice in evaluating this article.

Funding

The research leading to these results was funded by the Research Council of Lithuania under grant agreement No. S-MIP-21-3.

Declaration of interest

The authors report no conflict of interest.

Data availability

The authors confirm that all of the data and codes used in this study are available from the corresponding author upon reasonable request.

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Figure 0

Figure 1. Schematic of a two-stage nozzle (a,b), consisting of a first electron injection stage (1), a second acceleration stage (2) and a one-half straight wall section (3) that reduces the gas concentration drop between the two nozzles. Longitudinal profile of gas concentration (c) to form an increasing gas density gradient.

Figure 1

Figure 2. Helium gas density diagram of the two-stage supersonic nozzle (a) and longitudinal gas concentration profile at the backing pressure of 27 bar along the laser propagation path at a 0.5 mm distance above the outlet of the injection stage of the nozzle (b) simulated using the OpenFOAM CFD software.

Figure 2

Figure 3. (a) The scheme of milling and laser scanning; (b) contour, spiral and hatch laser scanning algorithms. The darkest colour represents the beginning of the scan, and the brightest colour represents the end. (c) The steps of nozzle milling. The grey colour represents the areas to be removed in the corresponding stage, blue – the material not affected by the laser, and red – the already processed part.

Figure 3

Figure 4. Photo of the milled nozzle (a). Optical microscope photographs (b)–(g). Photograph of the top surface of the nozzle (b). The entire image was made from a programmatically stitched sequence of photographs taken at different Z-heights. The injection channel at the top (c) and bottom (d). The acceleration nozzle at different positions is shown in (e)–(g). Section of nozzle drawing (h) and topography of cut nozzle channels (i).

Figure 4

Figure 5. Two-valve adapter for supplying different gases to the nozzle (a,b) and image of the assembled nozzle with two different gas supply systems (c).