Mass Partitioning in Fragmenting Tin Sheets

We experimentally study the mass partitioning of a fragmenting liquid sheet formed after the impact of a ns-laser pulse on a tin microdroplet, and its dependence on laser pulse energy and droplet size. We present the temporal evolution of individual liquid fractions: the sheet and its bounding rim, ligaments protruding from the rim, and droplets shed by the ligaments, applying machine learning to analyze sub-resolution fragments. Our results show that the temporal evolution of the mass partitioning between the sheet, rim, ligaments, and fragments is independent of the deformation Weber number—following Wang and Bourouiba [J. Fluid Mech. 935 , A29 (2022)] for the analogous droplet-pillar impact case, extending the work to larger Weber numbers and to a system where the timescale of deformation is fully decoupled from impact. The full mass partitioning is accounted for by quantifying the further contributions unique to the laser-droplet impact case: that of a centrally located mass remnant, and the mass ablated by the laser pulse. These ﬁndings can be employed to optimize the mass utilization of the liquid tin that is used as target material in the production of extreme ultraviolet light for nanolithography.


I. INTRODUCTION
State-of-the-art extreme ultraviolet (EUV) sources for nanolithography generate EUV light from laser-produced plasma fed with liquid tin microdroplets irradiated by highenergy laser pulses [1][2][3][4][5]. EUV lithography enables the production of the most advanced semiconductor devices and is critically dependent on the production of EUV light. EUV light is generated in a two-step process, in which a first laser prepulse is employed to shape the initial spherical droplet into a pizza shape: a thin axisymmetric sheet with a thickness of just several tens of nanometers and a thick bounding rim with a diameter of several micrometers [6,7]. In a single lithography machine, this process is repeated at an approximately 50-kHz rate-thus firing on a trillion droplets over a year of operation. At the bounding rim, ligaments protrude radially outward; the ligaments, in turn, shed droplets [8,9]. This complex liquid "target" further interacts with a second laser pulse. This main pulse turns the liquid tin into a plasma that emits EUV radiation with a wavelength centered around 13.5 nm [10][11][12][13]. Undoubtedly, the local liquid-mass distribution of the tin target heavily influences its response to a subsequent laser pulse. Furthermore, the shed droplets may carry a considerable amount of tin from the liquid system away from the laser interaction zone, contributing to the risk of contaminating nearby light-collecting optics with tin "debris." A deeper understanding of the temporal and spatial evolution of the liquid mass of the prepulse-shaped targets is valuable for further optimization of EUV light sources to increase EUV output power and to minimize the tin-mass consumption and generation of tin debris.
The fluid dynamic response of a tin microdroplet to a laser-pulse impact resembles the ubiquitous cases of water droplets impinging on solids, see, e.g., Refs. [6,8,9,[14][15][16][17][18][19][20][21][22][23][24][25][26]. Wang and Bourouiba [27] elucidated in detail the mass partitioning in the sheet, bounding rim, ligaments, and fragments produced from droplet-pillar impact. It was found that the mass fractions contained in these channels are independent of the impact conditions (captured by the Weber number) and can be described as a sole function of time. Despite the similarity to its mechanical counterpart of droplet impact onto an obstacle, laser-pulse impact onto a droplet does provide some key insights into the droplet dynamics. First, during laser-pulse impact, the timescales of impact and droplet deformation are completely decoupled, which allows investigation of the late-time fluid dynamic response to perturbations caused by the early time laser pulse [14,28]. Second, laser impact provides access to much larger Weber numbers, which enables us to significantly extend the range of investigation and test the robustness of scaling relations. Inspired by Ref. [27], we propose that the initial volume of a laser-impacted droplet V 0 = π D 3 0 /6 (with a diameter of D 0 ) is distributed over a sheet, a bounding rim, ligaments that radially protrude from this rim, and fragments generated by the breakup of ligaments. One further feature, unique to the laser-droplet impact case, is the presence of a center mass-a disk located in the center of the sheet [6,7]. Apart from these channels, a certain ablated mass fraction of tin will be heated up by the laser pulse and converted to a plasma or vapor phase, propelling the droplet and setting the impact conditions [28][29][30]. The mass distribution of the tin from the initial droplet is thus given by where the volume-or, equivalently, mass-in the sheet, rim, ligaments, fragments, center mass, and the mass ablated by the laser pulse are represented by V s , V r , V l , V f , V c , and V a , respectively. Several studies have been carried out to better understand the dynamics of the various contributions to Eq. (1). However, so far only the contribution of V s was explicitly experimentally quantified [6,7]. Ligaments were investigated by Klein et al. [8] in terms of their number and the scaling thereof with the impact Weber number. The dynamics of tin-fragment shedding was experimentally investigated in Ref. [9], focusing on the speed of the fragments but leaving out information on the fragment-size distribution. Although the main channels for tin partitioning have thus been identified and studied individually, an explicit quantification of the volume fraction in the channels of V r , V f , V l , V c , and V a has not yet been undertaken, and the validity, or rather the completeness of Eq. (1) remains unattested.
In this work, we present the global volume partitioning among the channels listed in Eq. (1) by combining our experimental results on droplet-laser impact with existing analytical models developed for the similar case of a droplet impacting a pillar extending the range to much larger Weber numbers (from the approximately 300-900 range of Ref. [27] to approximately 1000-5500 in the current work), and using machine learning techniques. We present our results on the individual volume fractions taken up by the various channels and thus provide the full volume distribution of tin in a target, and discuss application perspectives.

II. EXPERIMENTS AND METHODS
Our experimental setup is described in detail in Ref. [9]. Key aspects and related updates are summarized below [see Fig. 1(a)]. A vertically aligned train of liquid tin microdroplets (at 260 • C, with density ρ = 6960 kg/m 3 , surface tension σ = 0.54 N/m, and dynamic viscosity μ = 1.8 × 10 −3 Pa s [31,32]) is dispensed by a droplet generator mounted on top of a vacuum vessel (10 −7 mbar). In experiments, we systematically vary the diameter of droplets D 0 from 34 to 67 µm. The droplet train, traveling at a speed of approximately 10 m/s and a repetition rate ranging from 4 to 80 kHz, passes through a horizontal light sheet, which is generated by a helium-neon (He-Ne) laser and is positioned about 4 mm above the center of the vacuum vessel, i.e., the laser-droplet interaction point. The scattered He-Ne light is captured by a photomultiplier tube, from which the kilohertz-frequency signal is downconverted to a 10-Hz signal that serves as the trigger to the data acquisition and laser systems.
A circularly polarized laser beam generated from a Nd:YAG laser-pulse system operating at a wavelength of 1064 nm is aligned onto the droplets at the center of the vacuum vessel. The laser pulse has a Gaussian temporal shape and a duration of 10 ns at FWHM. The laser beam is focused down at the droplet location to a focal spot with a size of 110 µm at FWHM. The energy of the laser pulse is varied using a half-wave plate in series with a thin-film polarizer (cf. Fig. 1), which, together with the given range of droplet sizes, allows access to a Weber number range spanning approximately 1000-5500 in the current work.
The response of droplets is inspected using a stroboscopic shadowgraph imaging system. We record the expansion and fragmentation behavior of droplets by using two CCD cameras attached to long-distance microscopes (K2 DistaMax, Infinity Photo-Optical) at 90 • and 30 • with respect to the laser propagation direction, thus obtaining side-and front-view images, respectively. Both microscopes are equipped with a CF-3 objective (Infinity Photo-Optical) and are positioned at a distance of 110 mm to the center of the vessel. For the front view, we employ a double-frame camera (pco.4000), which is capable of capturing two consecutive frames of the same tin sheet with a minimum interframe time of approximately 280 ns. We use a single-frame camera (Manta G145-B, AVT) for the side view. The target is back lit by two consecutive and identical, incoherent light pulses with a wavelength of 560 ± 10 nm and pulse duration of 5 ns (FWHM). The first pulse is positioned within the exposure windows of both the Manta camera (side view) and the first frame of the pco camera (front view). The second illumination pulse, which 014048-2 Top-view schematic of the experimental setup comprising a vacuum chamber, synchronization and trigger systems, Nd:YAG laser, and shadowgraphic imaging systems. The energy of the laser pulse is varied using a half-wave plate (λ/2) in series with a thin-film polarizer (TFP) and a beam dump (BD). A quarter-wave plate (λ/4) is used before the final focusing lens to set a circular polarization. The imaging systems provide front-(pco) and side-view (Manta) images of the expanding tin sheet using longdistance microscopes (see the main text). Two consecutive shadowgraphy probe pulses SP 1 and SP 2 are used as illumination sources for the front-view microscope, whereas a single pulse SP 1 is used for the side-view system. Figure adapted from Liu (2022) [9]. (b) A front-view shadowgraphy image showing a typical expanding sheet following the impact of a 20-mJ laser pulse on a droplet with an initial diameter of 67 µm. Insets provide an enlarged view of fragments and ligaments. (c) Side-and front-view shadowgraphy images of a liquid tin sheet after being irradiated with a 50-ns-length vaporization pulse (for details see Ref. [7]). The vaporization pulse removes the liquid from the thin sheet, enabling visualization of the relatively thick features, including the bounding rim and the center mass. The individual volumes contributing to Eq. (1) are indicated in (b),(c).
follows the first pulse with a tunable time delay (typically set from 1 to 5 µs in the experiment), is collected by the second-frame exposure window of the pco camera only. Our dual-camera imaging system has been upgraded to provide improved resolution not only for the front-view [9], but also for the side-view camera. The images are used to determine the diameter of (subresolution) spherical fragments d f through machine-learning techniques. Furthermore, by employing a double-frame camera, we obtain the detachment time t d for each individual fragment according to the method described in Ref. [9]. Relating the detachment time of each fragment to its diameter provides the instantaneous size of the shed fragments d f (t d ). These results, in combination with the shedding rate (the rate of increase of the fragment number), yield the volume of liquid accumulated in the fragments V f over time. Next, our postprocessing of the front-view images allows us to accurately detect the formation of ligaments at the edge of the sheet [see Fig. 1(b)]. By finding the position of the bounding rim, the length of each ligament L l growing from the rim is determined. To obtain the instantaneous thickness of the ligaments w l , we invoke analytical models from Refs. [16,33]. The data of w l , L l , and the ligament number N l allow for a determination of the volume of ligaments V l .
We obtain the volume of the center mass V c [visualized in Fig. 1(c), see caption for details] by the associated remnant of liquid situated in the center of the sheet, as assessed from side-view images. By reviewing the experimentally validated numerical simulations performed in Ref. [30], originally designed to study the early time droplet deformation, we are able to estimate the volume of ablated material V a . Next, we briefly recap the experimental study on the volume of the sheet V s previously presented in Refs. [6,7]. We infer the volume of the rim V r obtained by using our data of the radial location of the rim R r in tandem with a robust analytical model from Ref. [33].

A. Assumptions
We note that our findings regarding the global mass distribution are based on the assumption that the density of liquid tin remains constant in our observation time regime (on the order of µs). To support this assumption, an extreme example case is investigated where a droplet with D 0 = 30 µm is irradiated with a particularly energetic laser pulse with E p = 80 mJ, a beam diameter of 60 µm (flattop), and a pulse duration of 20 ns (square) in a radiationhydrodynamic simulation using RALEF-2D (see below).
At t = 21 ns (i.e., 1 ns after the laser is turned off), most of the liquid has a temperature < 1021 K (plasma shielding prevents laser light from directly heating the surface), at which the density of liquid tin is 6643 kg/m 3 , 4.6 % less than the density at the initial liquid-tin temperature in our experiments (6960 kg/m 3 at 533 K, cf. Sec. II) [31]; meanwhile the surface tension decreases by a similar 5.8 % [32] and the ratio ρ/σ , which appears in the pertinent equations in the following, by just 1 %. We also assume that the response of the droplet to a 10-ns-length (FHWM) laser pulse is dominated by incompressible flow. The validity of this assumption is supported by Ref. [30] where our results of kinetic energy partitioning were shown to agree with an analytical fluid-dynamic model developed by Gelderblom et al. [14] for an incompressible liquid droplet responding to an instantaneous pressure impulse. The assumption is further supported by the studies by Reijers et al. [34] and Meijer et al. [35], which conclude that an incompressible flow will dominate the droplet response when τ p /τ a 1 for a Gaussian-temporally-shaped pulse; τ a = R 0 /c s is the acoustic time, which characterizes the acoustic wave that traverses the droplet at the speed of sound c s ≈ 2500 m/s [36]. In our case, the pulse duration satisfies τ p τ a . Still, we do find some compressibility effects as we hypothesize that the center-mass feature is a result of the collapse of a cavity generated by a rarefaction wave (cf. Sec. III B).

A. Volume of the fragments and ligaments
To obtain the volume of fragments and ligaments, we first need to assess their diameters. In addition, other parameters such as fragment number, ligament number, and ligament length are required to fully determine the volume of fragments and ligaments. In this section, we first discuss the temporal evolution of the volume accumulated in fragments V f . The volume accumulated in the fragments over time can be expressed as where t onset is the onset time of fragmentation,Ṅ f the shedding rate related to the fragment number N f , and d f the diameter of the newly formed fragments. Equation (2) can be approximated by its discretized form where the subscript i represents the index for the sampling point in time, and d f (t d ) is the diameter of the fragments shed at t d . Equation (3) requires as input the fragment number N f at each time delay, and the instantaneous diameter of the fragments d f detached at that time. The experimental results on these two parameters will now be discussed. Subsequently, we discuss the length and number of ligaments, and infer their diameters. Relating these quantities allows us to present the instantaneous volume contained in the ligaments V l . Figure 2(a) presents the fragment number N f as a function t/τ c for different combinations of droplet diameter D 0 and pulse energy E p ; τ c = [ρD 3 0 /(6σ )] 1/2 is the capillary time. Each data point represents the mean value of the fragment number obtained from several images (typically 20 images) recorded at the same time delay. For each image, the fragment number is obtained by our fragment-detection algorithm with an auxiliary correction from manual inspections. The performance of this postprocessing is illustrated in the inset of Fig. 2(a). It is shown that our algorithm could fail to correctly identify a fragment when it is (i) too close to its neighbor fragment, (ii) still in a ligament breakup process, and (iii) too far out of focus. These errors are corrected by our manual check. Our data in Fig. 2(a) shows that for a fixed D 0 , a higher pulse energy results in a higher fragmentation rateṄ f , and also an earlier onset of fragmentation t onset , leading to a larger N f at a given t/τ c . Figure 2(b) presents t onset /τ c as a function of the deformation Weber number (referred to as the Weber number hereafter) We d = ρṘ 2 0 D 0 /σ whereṘ 0 is the initial expansion speed of the droplet;Ṙ 0 is determined by a linear fit of the R r data (following Klein et al. [28]) using the first three time frames after the onset of the impact, typically up to 300 ns to 1 µs. In Ref. [8] the deformation Weber number We d was first introduced to capture the dynamics of the sheet expansion upon laser impact to a better accuracy than the Weber number based on dropletpropulsion speed [8,9]. Indeed, as motivated in Ref. [8], it is the radial expansion speed of the droplet and not its impact speed that sets the deformation dynamics, and the ratio of these two speeds strongly depends on the impact conditions [14,30]. A similar result was found, in fact, in droplet-pillar impact experiments [37]. Our data indicate an earlier onset of fragmentation with an increasing Weber number. This observation has also been reported for the case of droplet impact on a pillar in Ref. [37], in which a scaling law t onset /τ c ∼ We −1 d was found. A fit of this power law to our data is plotted in Fig. 2(b) in a solid line. The prefactor from the fit is 263(23), with the number in the bracket representing the standard error of the fit. This number is in reasonable agreement with the prefactor approximately equal to 320 analytically obtained in Ref. [37]. to the data as proposed in Ref. [16]. The fit yields a prefactor of 1.0(1). A linear-linear representation of the power-law fits and data is provided in the Appendix.

Fragment number
We observe that for most of the time during the shedding process, the increase in fragment number follows a linear trend, hence there is a constant shedding rateṄ f that depends only on the Weber number. Figure 2(c) presentṡ N f τ c as a function of We d , whereṄ f is obtained by a linear fit to the time domain of N f where a constant shedding rate is evident. Our graph shows an increase inṄ f with increasing We d . In Ref. [16], the fragments are generated with a continuously decreasing rateṄ f , in contrast with our results in Fig. 2 It is yet unclear what is the origin of this difference between our results and those of Ref. [16]. We do note that we also found a disagreement with the same work on the topic of fragment speed in our previous study in Ref. [9]. Therefore, we speculate that both discrepancies have the same root cause and that a detailed study of the ligament growth and fragment pinch-off dynamics is called for. In Ref. [16] a scaling law ofṄ f τ c ∼ We 3/4 d is analytically identified. This scaling is also plotted in Fig. 2(c) and is shown to be in good agreement with our data, where a prefactor of 1.0(1) is used as obtained from a fit to our data.

Fragment diameter
Typical fragment diameters studied in this work are at or below the resolution of our imaging system (approximately 3 µm). To still quantify these diameters under such challenging conditions, we apply machine learning based on a convolutional neural network (CNN). The CNN training data set is obtained from a custom calibration target. This calibration target consists of a series of optically opaque, 200-nanometer-thick gold disks fabricated on a silica substrate with stepped diameters ranging from 1.4 to 10 µm. The calibration target is imaged using the exact same imaging setup and configuration as used to record the images of the tin target fragments of interest. By recording an image of the calibration target at several focal positions, the training data set is assembled. By including such out-offocus images, the network is able to correctly identify the size of fragments, even when they are positioned slightly out of focus. More information on the machine-learning method, including the CNN structure, the preprocessing steps applied to each image, and the validation studies performed can be found in the Appendix.  Figure 3(a) presents images of the gold disks on the calibration target captured by our imaging system at the focal plane (top row), and selected fragments detected in our experiments (bottom row). The images of the tin fragments are, visually, indistinguishable from those of gold disks, indicating that the machine-learning method can accurately determine the size of the fragments within the size range of the calibration targets. The results obtained with the machine-learning method are further supported by Fig. 3(b), which presents a horizontal lineout for each individual calibration object and fragment shown in Fig. 3(a). To obtain the final fragment-size data, we constrain the eccentricity of the identified fragments. This filter enables us to separate the individual fragments-located within a finite distance from the focal plane (approximately ±30 µm)-from aggregated fragments or any ligaments that were mistakenly identified as fragments. Figure 3(c) presents the fragment diameter d f , as determined with our machine-learning analysis, as a function of the radial distance to the center of the sheet r. The data is obtained for the case where D 0 = 67 µm and E p = 20 mJ. The radial distribution of d f at two different time delays ( t = 5 µs and t = 10 µs) are plotted. Our results show that more outward-located fragments have (i) smaller sizes and (ii) travel faster (as indicated by a longer traveling distance in the r direction). These observations are consistent with the conclusions presented in Ref. [9] on fragment speeds and, as we discuss below, with previous work on droplet-pillar impact [16,33,38]. Since detached fragments travel at constant speeds, the radial distribution of fragment diameters at t = 10 µs can be obtained by propagating the distribution d f (r) at an earlier moment t = 5 µs following This radial distributiond f (r) | t=10µs is also plotted in Fig. 3(c) in which the fragment speed u f at t = 5 µs is obtained by our double-frame camera, in line with the steps established in Ref. [9]. It is shown thatd f (r) | t=10 µs as predicted by Eq. (4) agrees well with the data from direct measurements at t = 10 µs. This agreement shows our ability to connect each individual fragment with its diameter d f and speed u f that define the trajectory of the fragments. Next, by finding the intersection of the ballistic trajectories of fragments with the radial expansion of the ligament, we acquire the detachment time t d of each corresponding fragment. Figure 4 presents d f as a function of t d /τ c . Our observations are consistent with the studies by Wang et al. [16,33,38] on a water droplet impacting a pillar. In these studies, the relation between the diameter of the bounding rim, ligaments, and fragments has been studied in detail. First, by invoking momentum conservation on an extrusion of a ligament growing from a continuously decelerating rim, Wang et al. [33] found a universal criterion that governs the instantaneous thickness of the bounding rim b r where Bo is the local Bond number of the rim based on its radial decelerationR r . During the expansion of the sheet, the deceleration scales as −R r ∼ R max /τ 2 c ∼  5) and (6). The error bars represent the standard deviation of the binned data.
(σ/ρR 2 0 )We , consistent with the previous study by Villermaux et al. [15]. Next, the relation between the diameters of ligaments w l and fragments d f was experimentally investigated in Ref. [16,38] to yield where α(t) = 0.32(t/τ c ) 2 + 0.24(t/τ c ) + 0.92 is a function of t/τ c solely and varies from 1 to 1.2 during the expansion of the sheet [16]. Values of d f (t d ) predicted by Eqs. (5) and (6) are depicted in Fig. 4 by a solid line. The deceleration of the sheetR r as input for the relation Bo = 1 is obtained by fitting a third-order polynomial to the expansion trajectory of the rim R r for the corresponding data set. Figure  4 presents good agreement between the analytical prediction and our results. This agreement may be seen to either verify our determination of fragment diameter or, alternatively, to demonstrate the robustness of Eq. (6). Given the agreement shown in Fig. 4, we decide to employ Eq. (6) for the discussion below on the global volume fractions.

Ligament length
Next, we determine the instantaneous volume contained in the ligaments V l . A single ligament can be described as a cylinder with a length L l and a diameter w l , hence a volume π w 2 l L l /4. We first discuss the experimental results for the ligament length. The length of each ligament is determined by the radial distance from the tip of the ligament to the rim. The detection of the rim and the tip of the ligament follows the method detailed in Ref. [9]. The values of ligament diameter w l are extracted from Eq. (6).  after laser-pulse impact. The results of two pulse energies are presented (E p = 7 and 20 mJ). Our results show that for both pulse energies the ligament length increases with time, with an increasing most-probable-length L l,PDF . The spreading of L l (i.e., the PDF width) increases with time as old ligaments grow, while new ones are generated. For low energies (low We numbers) this spreading width is smaller, in particular at early times, given that fewer ligaments grow and fragment at a lower rate (see Secs. III A 1 and III A 4). The increase in L l,PDF over time is shown in Fig. 5(b) in which L l,PDF for three different pulse energies are presented, that is, E p = 7, 15, and 20 mJ, with the corresponding deformation Weber number We d = 1200, 2300, and 3300, respectively. Figure 5(b) also shows that the growth of L l,PDF is rather independent of the pulse energy (also see Sec. III A 5). It should be noted that an accurate estimate of L l by our postprocessing requires that the ligaments protrude radially from the rim. Any deviation from an outward-oriented cylindrical shape will lead to an underestimate of the actual length of the ligament. Any such angular deviations are typically well below 10 • leading to a systematic error of just 2 % (in few, extreme and late-time cases, angles of up to 30 • are observed, leading to a still modest underestimation of 13 %).

Ligament number
In addition to ligament length, the postprocessing illustrated in Fig. 5(c) yields ligament number N l for each processed image. Figure 6(a) presents N l as a function of t/τ c for different combinations of droplet diameter D 0 and pulse energy E p . It is shown that for a given droplet size, a larger number of ligaments is induced by a higher pulse energy. Furthermore, once the first ligaments are detected by our postprocessing, we observe a slight increase in their number over time, reaching a maximum at t ≈ 0.3 − 0.5 τ c . Later, N l decreases, mainly due to the merging of ligaments [8,9,38]. Figure 6(b) presents N l,max as a function of We d . The value of N l,max for each data set is obtained by averaging the ligament number ranging from two frames in time preceding the maximum of N l to two frames after that moment. A power-law fit of N l ∼ We 3/8 d is performed to the data and yields a prefactor of 2.4(1). This power law has been analytically proposed in Ref. [8] where it has been further confirmed experimentally for the case of laser impact on a droplet with D 0 = 50 µm. These data are reproduced in Fig. 6(b), showing their agreement with the results of our measurements.
The scaling N l ∼ We 3/8 d has also been reported by Wang et al. [16] for the case of pillar impact. We note, following Ref. [16], that measurements of the number of ligaments should be distinguished from those of initial corrugations-the early time manifestation of capillary instabilities arising on the rim. Only a fraction of the corrugations develop into ligaments. This phenomenon is also observed in our case (cf. Ref. [9]) where both the temporal and spatial dimensions are drastically smaller than those in Ref. [16]. The inset in Fig. 6(b) shows the number of corrugations N cor and ligaments N l for the case with D 0 = 67 µm and E p = 7 mJ. We note that N cor is determined through a direct visual inspection. It is clearly shown that the corrugation number decreases, with only part of them becoming ligaments at later moments.

Results of fragment and ligament volume
With the measurements of the key parameters at hand, we are now in the position to determine the volume taken up by fragments and ligaments. Figure 7(a) shows the volume ratio V f /V 0 as a function of t/τ c . Each V f data point is determined following Eq. (3) with N f retrieved from the experimental data shown in Fig. 2, and d f (t d ) obtained from Eq. (6). The application of Eq. (6) is supported by the agreement between the model and our machine-learning results on d f (t d ) presented Fig. 4. Given the temporal range of the comparison in Fig. 4 (i.e., t 0.45τ c ) and also the validity range t 0.5τ c for Bo = 1 for which the rim requires a sufficiently large deceleration [6,33], we apply Eq. (6) to the determination of d f and V f up to t = 0.5τ c . Figure 7(a) presents a continuous increase in V f over time: a progressive detachment of liquid into fragments. Furthermore, the results for the E p = 15 and 20 mJ cases are almost indistinguishable although the E p = 20 mJ case presents nearly twice the amount of fragments compared to the E p = 15-mJ case at any given time cf. Fig. 2, with the higher-impact energy case producing smaller fragments. For the case of E p = 7 mJ, less liquid is channeled into the fragments at early times. This peculiarity is mainly due to the strongly delayed fragmentation onset moment for the small energy case: t onset /τ c ≈ 0.3 for E p = 7 mJ, as compared to t onset /τ c ≈ 0.1 and 0.06 for E p = 15 and 20 mJ, respectively [cf. Fig. 2(b)]. Figure 7 (2), yielding V f /V 0 (t/τ c ) that is determined by t/τ c solely, consistent with Wang et al. [27]. The volume of ligaments scales as V l ∼ w 2 l N l L l ∼ D 3 0 We  Fig. 7(b), the Weber number ranges from 1200 to 3300, thus giving a very limited, and indeed unobservable predicted change of approximately 10 % on L l [cf. Fig. 5(b)] given the postulated weak dependence on the Weber number We More data, taken over a wider range of Weber numbers, would be required to experimentally verify the scaling of the ligament length.
The temporal evolution of V f and V l has been studied previously by Wang et al. [27] for water droplet impact on a pillar. In that work, the sum of ligament and fragment volume represents the liquid shed from the rim V f +l = V l + V f . The ratio V f +l /V 0 , and the ratio V f /V 0 contributing to it were separately shown to be functions of t/τ c only and are independent of the droplet's Weber number. As a corollary, V l /V 0 should also be a function of t/τ c only, independent of the Weber number. Figures 7(a) and 7(b) show reasonable agreement between our experimental results and the predictions from Ref. [27]. The slightly lower volume channeled into the fragments, comparing to . The enlarged inset of (d.1) shows some fine droplets, which detach from the jet at early times. (e) A side view of a MEK liquid droplet impact by a laser pulse, taken and adapted from Ref. [8]. Note the change in length scale. the model predictions, is shown to be offset by a modestly larger volume channeled into the ligaments-to yield a near-perfect agreement between model and experiment for V f +l /V 0 in the applicable time domain, as shown in Fig. 7(c).

B. Volume of the center mass
During the expansion of the droplet, a disk-shaped feature may appear in the center of the sheet [6,7]. The formation of such a center mass is hypothesized to originate from early-time laser-induced cavitation and cavity collapse inside the droplet [8,35]. In some cases, laser impact may lead to jetting. Such jetting has also been observed by Klein et al. [8] for laser impact upon a liquid methyl-ethyl-ketone (MEK) droplet [see Fig. 8(e)].
In our previous work in Ref. [6], we identified this center mass as a separate channel in the liquid distribution in the target and excluded its volume from that of the sheet V s . The thickness of the center-mass disk could not be directly assessed due to experimental limitations. However, our initial observations did indicate that the center disk is significantly thicker than the surrounding sheet, and its contribution to the overall mass balance should thus be taken into account. During the sheet expansion, the center mass is bounded by a much thinner halo [6]. Meanwhile, spontaneously formed holes will appear on the sheet when the amplitude of capillary instabilities exceeds the thickness of the sheet [8]. These holes could open in the bounding halo of the center mass. Once the holes appear, the edges of these holes expand with the Taylor-Culick speed, which increases with decreasing local thickness. Therefore, the relatively thin halo around the center mass retreats promptly, separating the center mass from the rest of the sheet. This phenomenon ceases the exchange of liquid between the center mass and the sheet, effectively maintaining the amount of liquid of the center mass during the rest of the expansion. Figure 8(c.1) presents a front view of a sheet to exemplify the separation of the center mass from the rest of the liquid. The corresponding sideview image is shown in Fig. 8(c.2). As time passes, the detached center mass collapses as a result of the surface tension, forming either a spherical droplet or a center jet that can be described by the shape of a cylinder [see panels (c.2) and (d. 1)]. The jet could further break into a series of small droplets due to a Rayleigh-Plateau instability, as shown in Fig. 8(d.2) with its inset showing an enlargement on these droplets. We determine V c from the volume of these associated geometries (i.e., the cylindrical liquid jet or spherical droplets) from our 90 • side-view images with their improved resolution compared to our previous studies, and using the largest, 67-µm droplet size for optimum visibility. Figures 8(a) and 8(b) show the volume ratio of the center mass to the initial droplet V c /V 0 as a function of t/τ c , and the Weber number We d , respectively. Our results in Fig. 8(a) indicate that a 5-20 % fraction of V 0 is channeled to the center mass, with the precise number depending on the experimental parameters. There is no evident dependence of V c /V 0 on time. Figure 8(b) shows that a larger pulse energy appears to reduce the fraction of tin in V c . Further studies are required to fully quantify and to clarify the origin of the center-mass feature. In particular, the influence of the initial droplet size needs to be addressed in future work: the droplet size dictates the acoustic timescale [34] and the deposited laser energy per droplet volume (cf. Ref [39]), which in turn characterize the degree of compressibility of the flow and could influence the formation of the center mass.
Averaging the available data in Fig. 8(b) yields a fraction of 9(3) % for the center mass, which we use in the following discussion for the global volume distribution. We note that the obtained volume of the center mass may be slightly smaller than the actual volume as some of the liquid shed from the center jet may appear in the form of fine particles, see the enlarged inset of Fig. 8(d.1). These particles detach at an early time from the center of the sheet and travel forward at high speeds. Therefore, they quickly move out of the field of view of our side-view images and thus cannot be captured by our measurements.

C. Volume of ablated material
When a tin microdroplet is irradiated with a laser pulse, a fraction of liquid material at the surface of the droplet is transformed into plasma that expands away from the droplet. This expansion provides a momentum kick to the droplet, which results in a radial expansion and a propulsion velocity of the droplet along the laser direction [8,14,29,30,40]. The volume of ablated material V a can be obtained from the data underlying our numerical study in Ref. [30], where detailed radiation-hydrodynamic simulations by RALEF-2D were performed to examine the partitioning of kinetic energy between propulsion and expansion. Based on simulations for two different droplet sizes D 0 = 27 and 45 µm, impacted by laser pulses with energy E p ranging in 0.5-30 mJ and 20-104 µm (FWHM) beam size, the ablated material accounts for 5-23 % of the initial amount of tin. With these numerical settings, the droplet is conditioned with a propulsion and expansion speed ranging from 30 to 250 m/s, yielding a wide range of Weber number from 600 to 15 000. The scaling of V a /V 0 with We d and D 0 is highly nontrivial given the complex interplay of the underlying physical processes. The simulations show a monotonic increase of V a /V 0 with increasing We d and a decrease with increasing D 0 . A fit to the simulation data results in an approximate, heuristic scaling relation of the ablated fraction ∼D −β 0 We γ d , with 0.6 ≤ β ≤ 0.8 and 0.3 ≤ γ ≤ 0.5. This scaling relation is only applicable for a limited parameter range and does not capture the full set of dependencies. For one of the cases studied, our simulations indicate that approximately 16 % of the total material will be ablated when a droplet with a diameter of 27 µm is impacted by a laser pulse with energy of 4 mJ and beam diameter of 50 µm (FWHM). As a result, the droplet is accelerated to a speed of 144 m/s along the laser direction. Propelling the droplet with the same size instead to 219 m/s would require a laser pulse energy of 8 mJ, ablating 23 % of the initial mass, consistent with the approximate scaling relation.

D. Volume of the sheet
In this section, we recapitulate our previous experimental measurements on the volume of the sheet [6,7]. Given that the droplet expands into an axisymmetric sheet, the volume of the sheet can be determined by spatial integration of the thickness profile. These thickness profiles-for a wide range of pulse energies and three different droplet sizes-were experimentally measured in Refs. [6,7] by (i) determining the opening velocity (Taylor-Culick speed) of spontaneously formed holes on the sheet, and (ii) the partial transmissivity of the backlight through the sheet, and (iii) by using a probe pulse that gradually removes the liquid from the sheet. Our results revealed that a sheet with a thickness of several tens of nanometers is formed a few microseconds after the laser impact. The thickness profile h(r) presents a slender spatial gradient, where h decreases with radial distance to the center of the sheet r. With further spatial integration, we provided the volume of the sheet V s in a time range of t/τ c ∈ (0.15, 0.7). The data of V s indicates a continuous loss of tin from the sheet. Furthermore, we showed that the volume ratio V s /V 0 depends on the nondimensional time t/τ c only and is independent of the Weber number.
This self-similar curve for the thickness in combination with the sheet expansion trajectory produces a description of the time evolution of the sheet volume V s . We note that at early times, the slender-slope approximation [15,[23][24][25], underpinning the self-similarity, no longer applies to the sheet. As a result, we cannot expect that our self-similar curve for the thickness profile and the corresponding description for the sheet volume are valid for early-time dynamics.

E. Volume of the rim
The bounding rim can be described by the shape of a curved cylinder with a length of 2π R r and a diameter of b r . Therefore, its volume reads V r = (π 2 /2)R r b 2 r . The length of the rim 2π R r has been well captured by our experiments with the postprocessing that determines the radial position of the rim [see Fig. 5(c)]. However, direct measurements of the thickness of the rim b r are still lacking. Our experiments elaborated in Ref. [7] confirmed the presence of the rim, where it is directly visualized with a vaporization pulse-a pulse designed to remove the fluid from the thin sheet [see Fig. 1(c)]. Regardless, the yet unknown mass fraction removed from the rim by this vaporization pulse prevents an accurate determination of b r . Moreover, the subresolution size of b r would make an estimation of its volume through direct optical inspection particularly challenging.
To still determine the volume of the rim, we employ the robust relation for the local Bond number Bo = ρb 2 r (−R r )/σ = 1 [i.e., Eq. (5)] that universally governs the instantaneous thickness of the rim [33]. The relation Bo = 1 is valid for inviscid flow, which is applicable in our case where the Reynolds number Re = D 0Ṙ0 /ν ∼ 10 4 given the viscosity ν = 0.26 × 10 −6 m 2 /s of liquid tin [41]. As a result, the volume of the rim can be determined as Equation (7) indicates that V r /V 0 is a function of t/τ c only. This is expected as b 2 r ∼ D 2 0 We −1/2 d and R r ∼ D 0 We 1/2 , rendering b 2 r R r /D 3 0 independent of the Weber number.

F. Global mass partitioning
So far, we discuss the volume fraction of liquid taken up by various channels. In our experiments, we systematically alter the Weber number We d = ρṘ 2 0 D 0 /σ by varying the  9. Volume fraction of different channels with respect to the initial droplet V 0 as a function of t/τ c . The volume of the sheet V s is reproduced from the experimental data presented in Refs. [6,7], along with a heuristic linear fit. The summed volume of fragments and ligaments V f +l is from Fig. 7(c), along with the prediction on V f +l from Ref. [27] ( †). The red line represents Eq. (7) for the volume of the rim V r . The volume of the center mass V c is reproduced from Fig. 8(a), along with a horizontal line at 9 % representing the mean values of the data shown in Fig. 8(b). The sum of all these channels V sum = V s + V f +l + V r + V c is plotted by a green line. The shaded area indicates the temporal range of t > 0.5τ c where the assumptions underpinning the models are invalid (see the main text). droplet diameter D 0 and the pulse energy E p , which jointly determine the different initial expansion speeds of dropletṡ R 0 .
Our observations allow us to summarize the complete liquid distribution in Fig. 9, which presents the volume ratio of all channels (except V a ) listed in Eq. (1) as sole functions of t/τ c . We exclude the volume of ablated material V a from the graph given that the material ablation occurs instantly (within 10 ns) after the start of the laser pulse and hence V a is not a variable changing with t/τ c throughout the droplet expansion process. The data for V s from Refs. [6,7] are reproduced in the graph. Furthermore, we sum up the volume of these channels, that is, To plot V sum /V 0 , a heuristic linear fit is performed to the data of V s /V 0 in order to produce a continuous description of V s based on experimental data. For a continuous description of V f +l , the model from Ref. [27] is used. This model has been shown to agree with our experimental data [cf. Fig. 7(c)]. For the volume of the center mass, a constant fraction of 9 % is added to V sum /V 0 given the fact that there is no evident dependence For all three targets, the volume fraction taken up the center mass is set as 9 % (cf. Fig. 8). The segment curves representing the sheet volume are determined by integrating the thickness profile h(r) reproduced from Ref. [7]. The volume fractions taken by the rim and ligaments are determined by Eq. (7), and by V l shown in Fig. 7(b), respectively. Curves that represent fragment volume are obtained by combining the descriptions of fragment speeds provided by Ref. [9] and the data of V f presented in Fig. 7(a).
of V c /V 0 on t/τ c for the fixed droplet size D 0 = 67 µm presented in Fig. 8. The obtained V sum /V 0 is represented by a green curve in Fig. 9. Our results indicate that V sum /V 0 captures approximately 90 % of the original droplet volume in the applicable domain. The "missing" volume fraction of approximately 10 % to V sum /V 0 is consistent with the expected volume of the ablated material V a /V 0 , which accounts for 5-23 % (cf. Sec. III C). Therefore, we validate Eq. (1) with reasonable accuracy. This observation is remarkable given the complexity of the interplay between the underlying physical processes, and the experimental and theoretical approaches as represented in Fig. 9. We note that V sum becomes larger with increasing time. This observation could be an intrinsic consequence of the systematically higher pulse energies and smaller droplets that correspond to the data of V s for the early times taken from Refs. [6,7]. Such conditions lead to a larger fraction of ablated material, leading to smaller V s and, hence, a smaller V sum .

IV. APPLICATION PERSPECTIVE
The volume ratio of all loss channels with respect to the original droplet volume can be characterized by the nondimensional time t/τ c = t/[ρD 3 0 /(6σ )] 1/2 solely. In other words, given a droplet diameter D 0 , time is the key parameter determining the tin volume distribution in a target. As a corollary, to minimize the tin lost from the sheet to the fragments, it is best that the successive laser pulse (e.g., the main pulse) following the prepulse with a minimum time delay (i.e., "shoot early"). To allow the tin target to reach a sufficient size in this short time to provide an optimal geometrical overlap with the large main pulse, an energetic prepulse is needed to induce a higher expansion rate of the droplet (i.e., "shoot hard"). Furthermore, as a direct result of a larger V s , a larger sheet thickness will be available (i.e., more tin is available for interacting with the main pulse), and scaling to larger main-pulse energies-and more output EUV light-may be feasible in this case. Figure 10 presents the spatial distribution of liquidtin volume on three targets, which correspond to targets dubbed A, B I , and B II as presented in Ref. [7]. Briefly, these targets are obtained by using laser pulses with E p = 20 mJ [target A (1.2 µs)] and 12 mJ [target B I (2.1 µs), B II (3.1 µs)], respectively, focused down to a spot size of 55 µm at the location of the droplet with a diameter D 0 = 29 µm. The targets share a common sheet diameter relevant for application-but present very different morphologies. For example, our results show a steep gradient of the mass distribution at the rim and ligaments: for target B II almost 40 % of the total mass is located within a narrow ring with a width of approximately 10 µm. This "localized mass" could be a source of inefficient usage of the tin by the main pulse or the second prepulse. Furthermore, the shoot early, shoot hard solution (target A) will lead to a smaller area over which the liquid debris is dispersed: at equal sheet "target" size, the outer fragments are closer to the rim [7] and cf. Fig. 10.
Another application may be found in the ability to define a predefined spatial shape of the main pulse or other secondary pulses [7], implemented in such a way that it relates to the spatial characteristics of the sheet (see Fig. 10). This approach may lead to a more efficient utilization of the tin material and reduce the required laser energy in the EUV light-generation process. For example, the local fluence of such pulses may be matched to the local thickness of the sheet.

V. CONCLUSION
We present results on the mass partitioning on a liquidtin target produced by laser-pulse impact on a tin microdroplet. Experimental studies are combined with a preexisting model from Refs. [16,33] that governs the thickness of the sheet's rim, significantly extending the range of Weber number over which the robustness of the model is tested. The temporal evolution of the mass fraction in the sheet, the rim, and the summed channel of the fragments and ligaments is demonstrated to be independent of the droplet's Weber number, and can be described as a sole function of time t/τ c . The full mass distribution is obtained when also accounting for the mass situated in a center-mass feature and for the mass lost through laser ablation.
Our results of the mass partitioning of fragmenting tin sheets provide a guide toward optimum utilization of the tin mass in EUV sources for nanolithography using a dual-laser recipe (i.e., prepulse-main pulse). We provide recommendations concerning both the delay between the prepulse and successive pulses, and the spatial fluence profile of such pulses. These recommendations aim to minimize any contaminating debris, and to maximize the utilization of tin mass for EUV generation. In addition to the industrial applications, the fundamental insight gained by the comparisons between the cases of laser impact and pillar impact is helpful to expand the scope of applications of the related universal laws.

ACKNOWLEDGMENTS
This work has been carried out at the Advanced Research Center for Nanolithography (ARCNL). ARCNL is a public-private partnership with founding partners UvA, VU, NWO-I and ASML, and associate partner RUG. This project has received funding from European Research Council (ERC) Starting Grant No. 802648 and is part of the VIDI research program with Project No. 15697, which is financed by the Dutch Research Council (NWO).

Fragment size determination with machine learning
The typical diameter of the tin fragments of interest, those expelled by the expanding sheet as presented in Sec. III A 2, are close to or below the resolution of our imaging system. This, combined with the partial coherence of the back-lighting source, leads to a nontrivial size determination of the fragments. However, every combination of a fragment size and its position in the focal plane will result in a unique image. Therefore, a rigorous physical model, taking into account the specifics of the experiment, would thus still be able to determine the size of such fragments. Such a model would have to include, among others, the point spread function and the precise temporal and spatial coherence properties of our imaging system and illumination, respectively. Instead of developing such a highly complex model we turn to machine learning, and more specifically a CNN, to obtain the fragment diameters d f . To train the CNN we make use of a custom-manufactured calibration target. This target consists of 200-nm-thick, optically opaque Au disks of various diameters (characterized with an atomic force microscope), fabricated on a silica substrate. All CNN training is performed using the images obtained from these Au disks. We further perform validation studies on two independent data sets. One of Ni particles deposited on a silica substrate and one with freefalling Ag-coated silica spheres. In the remaining part of this section, all objects of various types, Au disks, Ni and Ag-coated-sphere solid particles, and liquid Sn fragments will all be referred to as particles unless specified.

a. Image preprocessing
The image data is preprocessed before passing it to the CNN for training and evaluation. This is done to simplify the data and to remove several properties of the image that do not contain any information about the size of the particle. Firstly, the full-frame images are processed following P ij = (P ij − P 0 )/(P b,ij − P 0 ) where P ij is the raw image of the object and P b,ij is a reference image, containing only the bare illumination. The nonzero parameter P 0 represents the dark noise of the camera. Secondly, each particle is detected, cropped, and centered in the image by a welltuned detection and centering procedure, combining noise filtering and thresholding steps. Then, as our illumination intensity fluctuates significantly shot to shot, each cropped image is normalized by its own background level, follow-ingP ij = P ij /P ij , withP ij the center value of a Gaussian fit to the image histogram. After normalization we apply a final crop with a square size of 30 by 30 pixels. Lastly, ( for the tin-fragment data, where we have no control over neighboring, close-by objects, a masking step is applied. In this step, all pixels of any object but the most central particle of interest are masked with a value ofP ij . This allows us to include many more images that would otherwise not give any valid result.

b. Architecture and training
The architecture of the employed network is presented in Fig. 11. The network consists of a single convolutional layer followed by a max-pooling step and a single dense layer after flattening, all with ReLU activation. Minor tuning of the network structure was done using the Ni-particle images as a testing data set. This investigation showed a sensitivity to overfitting for more complex network structures and motivated the use of this, more basic structure. The final employed network is trained on the Au-disktarget images for 2000 epochs with a batch size of 32 and  included a total of 2772 training images. Figure 12 illustrates a selection of these Au-disk training images after preprocessing. The focal-depth range z included in the training process spans 0.1 mm in steps of 10 µm. This range is intentionally chosen to be larger than the accepted focal-depth range of the final evaluated data set (see Sec. III A 2). Figure 13(a) shows the relative error of the particle diameter of the training data set after training. To test the robustness, i.e., the dependence of the network on specific training input, we perform two test cases where a random selection of sizes or focal positions is left out of training. As can be seen in Figs. 13(b) and 13(c), for both cases comparable evaluation results are obtained for the data excluded from training when compared to Fig. 13(a).

c. Validation
The physical differences between the Au-disk training targets and spherical tin fragments concern geometry, material, and the presence of a supporting substrate. First, given our plane-wave illumination, we expect no influence of any glass substrate on particle imaging. Furthermore, the limited numerical aperture of our imaging system (NA = 0.24) should restrict any possible dependence on the materials' refractive index (for λ = 560 nm, Sn: n + ik = 1.61 + i5.57 [42], and Au: 0.38 + i2.59 [43]) and geometry on the diffraction pattern. Refractive-index and shape-related differences in light (Mie) scattering by particles are most prominent at large angles [44,45].
The Ni particles are deposited on a silica substrate and recorded with our imaging system. As the Ni particles are deposited on a substrate, a systematic focal-depth scan is also performed. The validation data set for these particles includes six focal positions over a 50-µm range. We find that the Ni particles range significantly in size and shape. We therefore investigate them using a scanning electron microscope from which we obtain an accurate, exact size. Particles whose shapes clearly deviate from spherical are discarded.
The Ag-coated silica microspheres, supplied in monodisperse powder form, are sprayed in air and therefore are not supported by any substrate. The recorded images contain many particles at many positions within the focal plane. From these images several particles are picked by manual selection, choosing only those that are close to focus and spatially well isolated. Two particle diameters are used, 4.5 µm (38 particles), and 9 µm (19 particles) with coefficients of variation of 5 % each as specified by the manufacturer. As shown in Fig. 13(d) all validation results show a mean size determination error within 10 %, which validates the use of Au-disk targets as training objects for evaluation of spherical tin fragments.

Linear-linear representation of power-law fits
For clarity the power-law fits and underlying data from Figs. 2 and 6 are here repeated in linear-linear representation, see Fig. 14.