Liquid Sheet Jet Target

• Liquid sheet jet targets are free-flowing, self-refreshing planar films formed by colliding liquid jets that provide renewable targets for high-intensity laser experiments.
• They exhibit inverse thickness scaling laws governed by Reynolds and Weber numbers, with interferometry confirming sub-micron and nanometer scale dimensions.
• Integrated diagnostics and closed-loop Bayesian optimization enhance laser coupling and proton acceleration efficiency through precise control of hydrodynamic stability.

A liquid sheet jet target is a free-flowing planar liquid film used as a self-refreshing interaction medium for high-field lasers, spectroscopy, scattering, and beam-driven targetry. In the laser-ion context, a converging liquid-sheet platform has been demonstrated as a multi-Hz target for laser-driven proton acceleration, combining a sub-$\mu$m water sheet, online diagnostics, and closed-loop wavefront optimization (Glenn et al., 8 Aug 2025). More broadly, the term encompasses colliding-jet flatjets, gas-compressed sheets, multilayer liquid heterostructures, 3D-printed sheet injectors for XFELs, and high-power liquid-metal films, all of which exploit the capacity of liquid flow to generate continuously renewed targets with controlled thickness, large lateral extent, and vacuum compatibility over experimentally relevant timescales (Chang et al., 2021, Hoffman et al., 2022, Konold et al., 2023, Halfon et al., 2013).

1. Physical basis and hydrodynamic formation

The canonical liquid sheet jet is produced when two laminar cylindrical jets collide obliquely and redirect axial momentum into a laterally expanding film. In the fluid-dynamics literature, collision of two identical jets of diameter $D$ and velocity $U$ generates a thin, radially spreading sheet whose morphology is governed primarily by the Reynolds number $Re = \rho U D / \mu$ and Weber number $We = \rho U^2 D / \sigma$ (Chen et al., 2012, Chen et al., 2011). Depending on $(Re, We)$ and the impingement angle, the sheet may exhibit a closed rim, open rim, unstable rim, flapping sheet, or atomizing ligament regime (Chen et al., 2012).

For idealized impinging-jet sheets, leading-order thickness laws follow from mass conservation. One representative expression is

$h(r) = \frac{D^2}{4\,r\,\sin(\theta/2)},$

with $r$ the downstream distance and $\theta$ the half-angle of the spreading geometry (Chen et al., 2012). In laser-target studies using orthogonal micro-jet collision, the same inverse-distance behavior appears as

$t(x) \simeq \frac{\pi d^2}{4\,x\,\tan\theta},$

and was verified by white-light interferometry for minimum thicknesses as low as $D$0 near the collision point (Cao et al., 2023). In vacuum flatjet measurements, a colliding-jet thickness profile of the form $D$1 was reported, with measured minimum thickness $D$2 for suitable nozzle sizes and flow conditions (Chang et al., 2021).

A persistent theoretical issue is that several classical inviscid thickness solutions satisfy the continuity and momentum constraints simultaneously. A minimum-energy analysis argues that the physically admissible thickness law is the Hasson and Peck solution, because it minimizes the lateral surface area among the Ranz, Miller, and Hasson–Peck candidates (Kebriaee et al., 2021). This is significant because it frames sheet thickness not only as a kinematic consequence of impingement, but also as a constrained variational selection problem.

A related misconception is that sheet formation is necessarily a surface-tension-dominated phenomenon. In an idealized two-dimensional impact problem, coherent ejecta-sheet formation was shown to follow from incompressibility, momentum conservation, and the free-surface condition $D$3, rather than from surface tension or viscous stresses (Ellowitz et al., 2012). This does not eliminate the role of $D$4 and $D$5 in real liquid-sheet stability, but it does separate sheet generation from later rim stabilization and breakup dynamics.

2. Converging liquid-sheet generation and target geometry

In the proton-acceleration platform of Korte et al., the target is generated by a tungsten microfluidic converging nozzle with a $D$6 aperture that produces two colliding jets and a flat sheet (Glenn et al., 8 Aug 2025). Water is fed at $D$7, corresponding to $D$8, and the jets expand at $D$9 after striking at a half-angle set by the nozzle geometry (Glenn et al., 8 Aug 2025). The local sheet width grows approximately linearly,

$U$0

and a first-order thickness estimate follows from mass conservation,

$U$1

At the laser interaction point, $U$2 below the nozzle, white-light interferometry measured $U$3 (Glenn et al., 8 Aug 2025).

This geometry places the target in the sub-$U$4m overdense regime relevant to TNSA while preserving continuous refresh. The platform also incorporates an in-vacuum catcher and heated trap to reduce vapor load, and sheet-edge jitter in the plane of the sheet is reported as $U$5, within the $U$6 Rayleigh range of the $U$7 focusing optic (Glenn et al., 8 Aug 2025). This relation between hydrodynamic stability and optical depth of focus is central to liquid-sheet deployment in relativistic laser experiments.

Other sheet-producing geometries elaborate the same hydrodynamic principle. In liquid heterostructures, three lithographically etched borosilicate channels meet near the chip exit: two outer channels of diameter $U$8 impinge at $U$9 around a central $Re = \rho U D / \mu$0 jet, producing a multilayer laminar sheet in which the inner liquid is completely enveloped by the outer sheet (Hoffman et al., 2022). In gas-compressed flatjets, a $Re = \rho U D / \mu$1 liquid orifice is compressed by He at $Re = \rho U D / \mu$2–$Re = \rho U D / \mu$3 bar into a $Re = \rho U D / \mu$4–$Re = \rho U D / \mu$5 sheet, demonstrating that collisional impingement is not the only route to stable planar targets (Chang et al., 2021).

3. Laser coupling and proton-acceleration regime

The converging-sheet ion source is driven by a Ti:Sa CPA laser delivering up to $Re = \rho U D / \mu$6 on target in $Re = \rho U D / \mu$7 FWHM at $Re = \rho U D / \mu$8, with repetition rate up to $Re = \rho U D / \mu$9, although most reported data were taken at $We = \rho U^2 D / \sigma$0 (Glenn et al., 8 Aug 2025). An $We = \rho U^2 D / \sigma$1 off-axis parabola produces a focal spot radius $We = \rho U^2 D / \sigma$2, giving peak intensity up to $We = \rho U^2 D / \sigma$3 and $We = \rho U^2 D / \sigma$4. For a Gaussian pulse,

$We = \rho U^2 D / \sigma$5

Spatial phase is measured by a HASO wavefront sensor and controlled by a 52-actuator deformable mirror, while temporal shaping of GDD and TOD is performed with a Dazzler (Glenn et al., 8 Aug 2025).

Under these conditions, with a sub-$We = \rho U^2 D / \sigma$6m overdense target and $We = \rho U^2 D / \sigma$7, proton production is described as operating in the TNSA regime (Glenn et al., 8 Aug 2025). Laser–plasma coupling generates a hot-electron population that traverses the sheet, escapes at the rear surface, and establishes a sheath field up to $We = \rho U^2 D / \sigma$8, accelerating protons normal to the rear surface from water’s hydrogen or from an adsorbed layer (Glenn et al., 8 Aug 2025). The usual scaling logic is retained: an empirical form $We = \rho U^2 D / \sigma$9 is quoted, with $(Re, We)$0–$(Re, We)$1 and $(Re, We)$2–$(Re, We)$3, alongside the Mora-type estimate

$(Re, We)$4

The measured beam characteristics place the source in the regime of compact, high-flux laser-driven proton beams. Reported beam metrics include typical divergence $(Re, We)$5, on-shot peak dose up to $(Re, We)$6, and low shot-to-shot flux variation with $(Re, We)$7 (Glenn et al., 8 Aug 2025). These figures are relevant because they connect sheet-target hydrodynamics directly to accelerator observables rather than only to target morphology.

4. Diagnostics and closed-loop optimization

A defining feature of the converging liquid-sheet platform is the integration of an extensive suite of online diagnostics. Plasma expansion is probed with an independent $(Re, We)$8, $(Re, We)$9, $h(r) = \frac{D^2}{4\,r\,\sin(\theta/2)},$0 pulse delayed up to $h(r) = \frac{D^2}{4\,r\,\sin(\theta/2)},$1, enabling optical shadowgraphy and interferometry of plasma expansion and shock formation (Glenn et al., 8 Aug 2025). Time-resolved images averaged over $h(r) = \frac{D^2}{4\,r\,\sin(\theta/2)},$2 shots track the overdense plasma radius $h(r) = \frac{D^2}{4\,r\,\sin(\theta/2)},$3, yielding early expansion speeds of $h(r) = \frac{D^2}{4\,r\,\sin(\theta/2)},$4, approximately $h(r) = \frac{D^2}{4\,r\,\sin(\theta/2)},$5 (Glenn et al., 8 Aug 2025).

Fast-electron diagnostics employ a permanent-magnet dipole spectrometer with $h(r) = \frac{D^2}{4\,r\,\sin(\theta/2)},$6, Lanex screen, and CCD, with $h(r) = \frac{D^2}{4\,r\,\sin(\theta/2)},$7 acceptance along the laser-forward direction and sensitivity above $h(r) = \frac{D^2}{4\,r\,\sin(\theta/2)},$8 (Glenn et al., 8 Aug 2025). Exponential tails are fit to $h(r) = \frac{D^2}{4\,r\,\sin(\theta/2)},$9 to extract $r$0. Proton-beam profiling is performed with a ZnS(Ag) scintillator and CCD at $r$1 from the target rear, shielded by $r$2 Al-Mylar and calibrated with a slotted RCF stack (Glenn et al., 8 Aug 2025). The principal optimization observable is provided by a time-of-flight spectrometer using a $r$3 diamond detector at $r$4 and $r$5 off rear normal, with proton energy reconstructed as

$r$6

where $r$7 (Glenn et al., 8 Aug 2025).

Closed-loop control is implemented by Bayesian optimization of six Zernike-mode wavefront terms on the deformable mirror: $r$8, $r$9, $\theta$0, and $\theta$1 (Glenn et al., 8 Aug 2025). The surrogate model is a Gaussian process with an RBF-plus-white-noise kernel, and the acquisition function is Expected Improvement modified to separate measurement noise $\theta$2 (Glenn et al., 8 Aug 2025). The workflow uses bursts of 10 shots at $\theta$3, computes burst-averaged $\theta$4 from the ToF as the fitness, updates the posterior, maximizes EI, and applies the next wavefront setting.

The optimization raised the average maximum proton energy from $\theta$5 for the manually optimized focus to $\theta$6, an $\theta$7 gain (Glenn et al., 8 Aug 2025). During the same run, integrated ToF flux increased by $\theta$8, although flux was not part of the reward function, and the radius containing $\theta$9 of focal energy shrank by $t(x) \simeq \frac{\pi d^2}{4\,x\,\tan\theta},$0 (Glenn et al., 8 Aug 2025). The direct interpretation given is that the gain arose from enhanced energy concentration within the focal spot. A plausible implication is that the liquid-sheet format is not only debris-lean and self-refreshing, but also algorithmically compatible with autonomous accelerator tuning.

5. Stability, vacuum operation, and thermodynamic constraints

For high-NA laser interaction, stability must be evaluated relative to the Rayleigh range and not only by macroscopic appearance. In a dedicated study of free-flowing thin liquid sheets for laser-ion acceleration, motion amplitudes in the surface-normal direction were stabilized below $t(x) \simeq \frac{\pi d^2}{4\,x\,\tan\theta},$1 in the stable region, and even below $t(x) \simeq \frac{\pi d^2}{4\,x\,\tan\theta},$2 after parameter optimization (Cao et al., 2023). The relevant decomposition separates high-frequency vibration, associated with pump vibrations and mechanical resonance, from low-frequency jitter caused by residual pump pulsation and hydrodynamic instability growth (Cao et al., 2023). The practical prescriptions reported were to reduce flow rate, optimize ethylene-glycol concentration near $t(x) \simeq \frac{\pi d^2}{4\,x\,\tan\theta},$3, keep jet lengths below $t(x) \simeq \frac{\pi d^2}{4\,x\,\tan\theta},$4, include a pulsation dampener, and increase the collision angle from $t(x) \simeq \frac{\pi d^2}{4\,x\,\tan\theta},$5 to $t(x) \simeq \frac{\pi d^2}{4\,x\,\tan\theta},$6 in the low-$t(x) \simeq \frac{\pi d^2}{4\,x\,\tan\theta},$7, high-$t(x) \simeq \frac{\pi d^2}{4\,x\,\tan\theta},$8 regime (Cao et al., 2023).

Operation in vacuum introduces evaporative cooling and vapor-load constraints. Raman thermometry of sub-$t(x) \simeq \frac{\pi d^2}{4\,x\,\tan\theta},$9m flatjets in vacuum showed that the temperature of the water sample reaches around $D$00 and the ethanol around $D$01, with faster cooling for higher-vapor-pressure liquids (Chang et al., 2021). In colliding-jet flat sheets, smaller nozzles and lower flow rates gave thinner, shorter sheets and faster cooling per millimeter, while gas-dynamic jets reached cooling rates up to $D$02 (Chang et al., 2021). These results matter because they establish that target thickness, vacuum compatibility, and thermodynamic state are coupled design variables rather than separable subsystems.

Vacuum operation is nevertheless well established across several liquid-sheet implementations. Liquid heterostructure sheets are described as self-refreshing and vacuum stable, with low-vapor-pressure liquids such as water/toluene operable in $D$03–$D$04 Torr and more volatile solvents requiring differential pumping or cryo-catchers (Hoffman et al., 2022). At the European XFEL, a 3D-printed sheet jet operated with a shroud and catcher at $D$05, and the sheet fully regenerated between pulses at $D$06–$D$07, whereas $D$08 produced overlapping explosions and rim fraying (Konold et al., 2023). For the proton-acceleration platform, sub-$D$09m thickness, minimal debris, and stable jet-cell refreshing at $D$10 were identified as a path to kHz operation with PW-class lasers (Glenn et al., 8 Aug 2025). This suggests that repetition-rate scaling is limited not only by target replenishment but also by micro-hydrodynamics, vapor handling, and pulse-to-pulse interaction with the disturbed downstream sheet.

6. Variants, applications, and system-level scope

Liquid sheet jet targets now span a wide range of geometries and use cases.

Implementation	Key target characteristic	Reported use
Converging water sheet (Glenn et al., 8 Aug 2025)	$D$11 at interaction point	Laser-driven proton acceleration
Liquid heterostructure (Hoffman et al., 2022)	Buried inner layer thinner than $D$12	Interface-specific spectroscopy
3D-printed XFEL sheet (Konold et al., 2023)	$D$13–$D$14 wedge thickness	Megahertz liquid sample delivery
LiLiT lithium film (Halfon et al., 2013)	$D$15-thick, $D$16-wide film	Neutron production and beam dump

In multilayer liquid heterostructures, the two high-momentum outer jets form a thin leaf-shaped sheet while an immiscible inner jet is flattened into a buried central layer, yielding three discrete laminar layers (Hoffman et al., 2022). The inner-layer thickness follows an empirical master curve with $D$17 and $D$18, and the buried layer can be tuned from microns to below $D$19 (Hoffman et al., 2022). The significance is not merely structural: such sheets transmit from the IR through the visible/UV into soft- and hard-X-rays and support interface-specific spectroscopy using standard transmission or reflection geometries (Hoffman et al., 2022).

At XFELs, the 3D-printed sheet jet addresses interaction-volume fluctuations that affect cylindrical jets. Its wedge profile rises from $D$20–$D$21 at the upper rim to $D$22–$D$23 near the lower rim over $D$24, with horizontal variation below $D$25 across the central $D$26 (Konold et al., 2023). Shot-to-shot intensity fluctuations of the normalized AGIPD response were $D$27 over 5 min at $D$28, compared with $D$29 for a GDVN under comparable conditions (Konold et al., 2023). The sheet geometry therefore functions as a stability-control strategy as much as a thickness-control strategy.

The LiLiT system represents a different branch of the liquid-sheet target class: a high-power, windowless liquid-lithium film for neutron production (Halfon et al., 2013). Here the target is not sub-$D$30m but a $D$31-thick film flowing at up to $D$32 and designed to remove $D$33 beam power by forced convection (Halfon et al., 2013). Electron-beam tests sustained areal power densities $D$34 and volume power density $D$35, while the $D$36 thick-target yield at $D$37 was $D$38 (Halfon et al., 2013). The broader point is that “liquid sheet jet target” denotes a target class unified by continuous renewal and controlled free-surface geometry, but diversified across radically different thickness, material, and beam-loading regimes.

Across these variants, several future directions were explicitly identified. For the converging-sheet accelerator platform, on-the-fly thickness tuning via $D$39 or nozzle temperature was proposed as a route to traverse from TNSA to radiation-pressure regimes, while the same Bayesian framework was suggested for cryogenic jets, tapes, GDD/TOD control, pulse-shape tuning, and additional diagnostics such as beam profile or emittance (Glenn et al., 8 Aug 2025). Outstanding challenges include early filamentation and shock formation, kHz-level recovery times, vapor-load management, and nonlinear coupling among many laser and target parameters, motivating PIC or hybrid simulations and more advanced acquisition strategies such as “Bayesian exploration” and multi-fidelity models (Glenn et al., 8 Aug 2025).