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

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.


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 2020).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. 2009;Mangles et al. 2012;Wan et al. 2022), ionisation injection (McGuffey et al. 2010;Pak et al. 2010) and density down-ramp injection (Schmid et al. 2010;Buck et al. 2013;Foerster et al. 2022).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. 2010;Buck et al. 2013;Wang et al. 2021).A lot of research has been done on density down-ramp injection in beam-driven plasma wakefield acceleration (Suk et al. 2001;De La Ossa et al. 2017;Xu et al. 2017).This technique has been employed recently for free-electron lasing using a high-quality LWFA electron beam of 10-50 pc (Schmid et al. 2010;Buck et al. 2013;Wang et al. 2021).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. (2016).Nevertheless, due to the nonlinear laser propagation in LWFA plasma, it is challenging to analyse the injection mechanisms (Vieira et al. 2010).
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 deph ∝ n −3/2 ) and depletion lengths (L depl ∝ 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. 2011;Liu et al. 2011;Pollock et al. 2011;Kim et al. 2013;Wang et al. 2013;Vargas et al. 2014;Golovin et al. 2016;Steinke et al. 2016) 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. (2020).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. 2019;Oubrerie et al. 2022).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 2014;Ekerfelt et al. 2017;Massimo et al. 2017).
Nowadays, emerging kilohertz laser systems, generating few-cycle laser pulses with tens of mJ of energy, provide peak intensities in the range of 10 18 -10 19 W cm −2 and can be efficiently used for particle acceleration (He 2014;Zhang et al. 2022;Tóth et al. 2023).To drive the charged particles in the LWFA self-guiding bubble regime, relatively high plasma concentrations of n = 3-5 × 10 19 cm −3 and tighter focusing of the laser beam to the diameter of 3-5 µm 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 µm (Rovige et al. 2020).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. 2001;Pukhov & Kostyukov 2008;Guillaume et al. 2015;Oubrerie et al. 2022).
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. 2015(Cai et al. , 2016;;Cai, Liu, & Shi 2017).However, the converging and diverging parts are manufactured in separate steps (Cai et al. 2015(Cai et al. , 2017)).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. 2016;Andrianaki et al. 2023).Femtosecond laser-induced chemical etching (FLICE), laser trepanning and electrical discharge machining (EDM) can be used to produce diameters smaller than 100 µm (Takahashi et al. 2013;Tomkus et al. 2019;Chiomento et al. 2021;Zuffi et al. 2022).However, de Laval-shaped channel formation is typically a two-step process for laser trepanning and EDM techniques (Li et al. 2018;Chiomento et al. 2021;Zuffi et al. 2022).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. 2021).However, FLICE, laser trepanning or EDM of micrometre-scale diameters are limited to low millimetre-scale depths (Takahashi et al. 2013;Tomkus et al. 2019;Chiomento et al. 2021).For larger nozzles, nanosecond rear-side milling is an appealing approach, as centimetre-sized nozzles can be formed with channel diameters down to 100 µm (Tomkus et al. 2018;Chaulagain et al. 2021).Additionally, this technique can form converging-diverging channels up to some angle in a single step (Chaulagain et al. 2021).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.

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. 2014;OpenFOAM, 2022).The first stage of the nozzle with a diameter of 200-300 µm is used for the formation of a 1 % N 2 + 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. 2020) 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 µm maximum from the nozzle outlet was correspondingly between 1.0 × 10 19 and 5.2 × 10 19 cm −3 .Also, the gas concentration in figures 1 and 2 corresponds to the backing pressure of 27 bar.

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. 2021).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 2015;Tomkus et al. 2018Tomkus et al. , 2020)).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 µm was measured by the Liu's method on a chrome-coated glass sample.Machining was performed on a 100 × 100 × 8.2 mm 3 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).
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.

Nozzle fabrication
The nozzle was milled using 65 µJ pulse energy and a 20 µm distance between the centres of successive irradiated spots along the scanning trajectory.The distance between the sequent scanning lines was 25 µm for the nozzle body and 20 µm 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 mm 3 s −1 and a milling efficiency of 0.06 mm 3 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 µm and 60 µm, respectively.The chipping of the acceleration nozzle was smaller -up to 120 µm on the bottom side and up to 30 µm 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.
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 R a of 1.6 µm and R z of 11.3 µm.The surface This was comparable to the roughness of the plane part of the nozzle with R a of 4.1 µm and R z of 31.7 µm.
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).

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.

FIGURE 1 .
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 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 redthe already processed part.

FIGURE 4 .
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 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).