Comparison of ion energy distributions from ns- and ps-laser produced tin plasmas from solid and droplet targets

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(1)MSc Physics and Astronomy Track: AMEP. Master Thesis. Comparison of ion energy distributions from ns- and ps-laser produced tin plasmas from solid and droplet targets by. Sjoerd van der Heijden 10336001. November 4, 2017 60 EC 01-09-2016 through 31-08-2017. Supervisor: Prof. dr. Wim Ubachs. Assessor: Prof. dr. Paul Planken. Advanced Research Center for Nanolithography.

(2) Abstract Ion energy distributions from laser produced tin plasmas have been experimentally investigated using time-of-flight data obtained from charge collecting Faraday cups, and compared for nanosecond- and picosecond-long laser pulses both on solid tin targets and liquid tin droplet targets. We find large differences between the four experimental cases, foremost that solid targets produce more total ioncharge than droplets, and that ps-pulses produce far less ion-charge, but with far greater kinetic energy than in the case of ns-pulses. Further, we compare two theoretical models describing hydrodynamic expansion of plasma into the vacuum to the experimentally obtained distributions. The first theoretical model, that considers laser-plasma interactions, fits reasonably well to the ion-charge energy distribution for nanosecond-long pulses on solid. However, this model fails to describe much of the distribution for nanosecond pulses on droplets, for which it was in fact developed. The second model, which does not consider such laser-plasma interactions, fits well to ion-charge energy distributions of the picosecond experiments with reasonable accuracy. For more detailed interpretations of the physical mechanisms driving the fast ions, charge-state-resolved measurements are a necessity.. Contents 1 Introduction 2 Experimental setup 2.1 Laser . . . . . . . 2.2 Target . . . . . . 2.3 Ion detection . . 2.4 Other setups . .. 2. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 3 3 4 5 9. 3 Experimental results 3.1 Ion energy distributions . . . 3.1.1 Nanosecond-on-solid . 3.1.2 Nanosecond-on-droplet 3.1.3 Picosecond-on-solid . . 3.1.4 Picosecond-on-droplet 3.2 Comparing cases . . . . . . . 3.2.1 Charge yield . . . . . 3.2.2 Fast ion peak . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 10 10 10 11 12 12 14 14 15. vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 17 17 18 18 19 20 20 21. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 4 Experiment vs theory 4.1 Theory: hydrodynamic expansion into the 4.2 Comparing the four cases to two models . 4.2.1 Nanosecond-on-solid . . . . . . . . 4.2.2 Nanosecond-on-droplet . . . . . . . 4.2.3 Picosecond-on-solid . . . . . . . . . 4.2.4 Picosecond-on-droplet . . . . . . . 4.3 Concluding remarks . . . . . . . . . . . . 5 Conclusion. 22. Appendix I. 25. Derivation of data correction equation. Appendix II System parameters for the different experimental setups. 26. Appendix III Effect of errors during data analysis. 27. Appendix IV Necessity of data correction. 28 1.

(3) 1. Introduction. Laser-produced tin plasmas currently serve as sources for extreme ultraviolet (EUV) light for nanolithography. These plasmas are generated from microscopic tin droplets illuminated by short laser pulses, with pulse lengths ranging from femtoseconds to hundreds of nanoseconds. Aside from EUV light, however, the plasma also generates energetic debris, which limits the lifetime of the nanolithography machines. This debris consists of neutral tin and tin ions ranging Sn1−14+ . Aside from being a practical problem, plasma expansion into the vacuum is also a challenge from a theoretical point of view [1, 2, 3, 4]. As such, obtaining the energy distribution of ions from laser produced tin plasmas is of particular interest from both practical and fundamental considerations. In this thesis, we discuss and compare experimentally obtained ion energy distributions for nanosecond and picosecond-long pulses on both solid tin targets and tin droplet targets. These four cases will be referred to as nanosecond-on-solid, nanosecond-on-droplet, picosecond-on-solid and picosecond-on-droplet (nanosecond and picosecond may be abbreviated to ns and ps respectively). The energy distributions are obtained through time-of-flight (TOF) measurements, taken with Faraday cup charge collectors. To facilitate the comparison between these cases, new experimental data is obtained for the nanosecond-on-solid case, while previously obtained data from the ARCNL groups is (re)analyzed for the other three cases. We show that there are significant differences between ion energy distributions of the four cases, as well in the total charge emitted by the plasmas. An important conclusion from this is that experiments performed with solid targets have little predictive value for experiments using droplets, even when using similar pulse lengths and fluences. This could possibly discredit present literature. A specific difference we found between solid-target experiments and droplet experiments is that solid targets tend to produce more total charge. Further, we show that picosecond-long pulses on both kinds of target result in fewer ions produced, but at higher kinetic energies than nanosecond-long pulses, for the same kind of target and laser fluence, consistent with existing literature [5, 6]. Aside from analyses and comparisons of experimental data, two relatively recent theoretical models for plasma expansion into the vacuum are discussed and fitted to our data. We find that the model proposed by Mora [2], which neglects laser-plasma interactions, fits excellently to the data from the picosecond-on-droplet case, as can be expected. The model also describes some of the energy distributions for the picosecond-onsolid case, but this does not hold for all laser energies used. The model derived by Murakami et al. [4], which does consider laser-plasma interactions, fits well to the data from the nanosecond-on-solid case, again as expected. For higher laser energy, however, a high energy feature emerges that is poorly described by either model. The energy distributions for the nanosecond-on-droplet case appear to be dominated by this feature, which neither model can describe, hinting at more complicated underlying physics that is missed by both approaches. While the high energy parts of the energy distributions of the latter two cases may appear similar, towards lower ion energy the plasma from the solid target appears to be of a much more thermal nature. This might originate from the difference between solid and droplet-targets. We begin by describing the experimental setup for the investigation of the nanosecond-on-solid case, as well as the data analysis method that was developed as part of this thesis. The setups for the other three cases are briefly discussed here as well. Then, the results from the four cases are presented, and the total charge yield and high ion-energy features are discussed and compared between the cases. Finally, the theoretical models are compared and fitted to the ion energy distributions.. 2.

(4) 2. Experimental setup. In this chapter the experimental setup used for the nanosecond-on-solid case is discussed in detail after which we briefly discuss the crucial difference with the experimental setups used for the other three cases. Further, the data analysis procedure, going from time-of-flight to the final ion energy distribution, is discussed. Fig. 1 shows the experimental setup. To study the ions emitted by laser-produced tin plasma, we illuminate a solid tin plate with a pulsed laser. Ion detection is done using Faraday cups (FCs).. Figure 1: Schematic drawing of the experimental setup for the study of tin plasma and the ions emitted. A Nd:YAG laser, operating at its fundamental wavelength of 1064 nm, is used to create laser-produced plasma (LPP) from a solid tin plate target. First, the unpolarized light also emitted by the laser system is partially removed from the beam using a thin-film polarizer (TFP). Afterwards the remaining light can be attenuated by a rotatable λ/2 waveplate and another TFP. A telescope triples the beam diameter after which a lens with a focal distance of 1 m focuses the beam down to a width of 90 µm at full-width at half-maximum (FWHM) on the tin target. Three Faraday cups (FCs) are installed to detect the ions emitted from the tin plasma that is then generated (only the two in the horizontal plane are shown here). Biasing circuits, with common voltage sources, are connected to all FCs. A 500 MHz-bandwidth oscilloscope records their output.. 2.1. Laser. The light is generated by a 10 Hz repetition rate Nd:YAG laser, operating at its fundamental wavelength of 1064 nm. The maximum available pulse energy is 410 mJ; beam diameter is 6 mm and the temporal profile of the pulses is approximately Gaussian with a FWHM of about 6 ns. The laser light is 98 % linearly polarized, the other 2 % is partially filtered out of the beam by means of a thin-film polarizer (TFP). After that the laser can be attenuated using a λ/2 wave plate and a second TFP; the laser energy can be reduced to ∼0.8 mJ by rotating the wave plate. A telescope is used to triple the beam diameter because a wide beam can be focused more tightly. A lens with a focal distance of 1 m then focuses the laser on the tin target, at the target surface. 3.

(5) the full-width at half-maximum (FWHM) of the beam is roughly 90 µm. We estimate that about 70 % of the attenuated beam energy reaches the target (comparing energy measurement right after the attenuation stage with energy measured inside the chamber), the rest is lost along the laser path probably due to poorly reflecting or transmitting optical elements. The spatial beam profile in focus is well approximated with a Gaussian function with calculated peak fluences 6.1 J cm=2 for 0.8 mJ pulses and 3.1 × 103 J cm=2 for 410 mJ pulses.. 2.2. Target. The target is a tin plate of 99.999 % purity (Goodfellow) with dimensions 32 mm × 32 mm × 1 mm. The target is mounted on computer controlled x- and y-stages so that the target can be moved in the plane perpendicular to the beam axis. The motion stages with the tin target are placed in a vacuum chamber. Measurements are performed at a pressure of ∼10=7 mbar such that the mean-free-path of the ions is longer than the distance between the target and the ion detectors, the mean-free-path is estimated to be ∼100 m. Measurements were done at 10 Hz repetition rate, matching the laser repetition rate. Measurements are repeated 50 times to allow for sufficient averaging. Before the measurements start, several pulses are shot at the tin to clean the target of contaminations. This is necessary as the target is covered by a layer of tin oxide. Ablation of this layer causes a clear oxygen-related fast-ion feature in the measurements, which is not the subject of this study. During continuous operation the tin target is moved by a fraction of the beam spot size between every measurement to obtain stable results; if the target is stationary between shots the ion traces fluctuate and craters with protruding walls appear quite quickly [5]. Keeping the target stationary between shots results in significant signal variations (up to ±30 % in voltage) in the measured data for different shots. Moving the target between shots yields more reproducible results but influences the signal in other ways. The effects on the data and underlying processes are discussed here. The fluctuations in the traces for a stationary target can be explained by the melting of the tin. Pulses of several ns are long enough for heat to spread into the target, causing melting to occur alongside ablation [7]. The molten tin re-solidifies after the pulse, leaving an irregular surface for the next pulse. Part of the molten material also forms into crater walls due to thermal expansion and recoil vapor pressure. These walls may then influence the geometry of the plasma expansion. This process also makes it very difficult to find the total ablated volume. This contrasts with ps-on-solid case, where heat is efficiently transferred from the laser heated electrons to the lattice rather than diffused into the material. This results in little melting and relatively high ablation volumes compared to ablation by ns pulses, leaving a clean crater [7, 8, 9]. We find experimentally that for nanosecond laser pulses more stable results are obtained by moving the target by a fraction of the beam diameter (∼60 µm) after every shot. In this way the tin hit by the laser is always relatively flat and deep craters cannot form. Higher energy pulses deform the target more strongly, causing stronger voltage fluctuations in the data. Increasing the distance the target moves between shots reduces this, again due to fewer shots hitting the same area on the target. The tin oxide layer is not an issue when moving the target between shots, as most of the oxide has been ablated by the wings of the pulse previous shots and no fast-ion feature, related to such low-Z elements, is apparent. However, an unwanted effect of moving the target between shots is that every shot hits the side of the crater created by the previous shots. The angle of this slope becomes greater for smaller step size and larger pulse energies, as this gives deeper craters with steeper edges. The result of this is that the peak emission related to the (naturally anisotropic [10]) angular ion distributions shift away from the ion detector that is near the original normal of the target, and away or towards a detector that looks at the target under a 30°-angle, depending on the direction of motion (detectors and their locations are discussed below). Measurements of this effect are shown in Fig. 2, where positive step sizes tilt the surface normal towards the 30° detector. We employ the model of Anisimov et al. [10] to find the angle of the modified surface normal through the. 4.

(6) expression Y (θ) = Y (0). . 1 + tan2 (θ) 1 + k 2 tan2 (θ). 3/2 (1). with polar angle with respect to the surface normal θ, Y (θ) the charge yield along θ, and k a parameter signifying anisotropy [5, 10]. For k = 1 the ions are distributed isotropically, for higher values the ions are emitted more and more along the surface normal. The k parameter decreases for increasing laser energy. After comparing different experiments performed with different step sizes and directions of motion, the angle between the new surface normal and the laser path is determined to be ∼4° for step sizes of both 40 µm and 60 µm, which are generally used when obtaining the main results; further, we find k ≈ 2 from the same analysis for laser-pulse energies of 300 mJ. This angle is small enough to not significantly influence the coordinate system: we can still compare data from the 30° detectors in the ns-on-solid case with those from 30° detectors in the other cases, since the anisotropy factor k is of order unity. 300mJ 6ns 30° 10 µm per step 40 60 -60 -40 -10. Ion current (A/sr). 15. 10. 5. 0 0. 20. 40. 60. 80 100 120 140 160 180 200 220 240. Time per meter (µs). Figure 2: Dependence of the FC(=30,0) time-of-flight ion signal on the direction and step size of the motion stages for the ns-on-solid experiment.. 2.3. Ion detection. To detect the ions from the tin plasma, three Faraday cups (FCs) are used (Fig. 3). These cups consist of a charge collector, a suppressor and a shield electrode. The collector is a copper cone connected to a 500 MHzbandwidth oscilloscope (Agilent) via readout electronics. The oscilloscope records the time-of-flight (TOF) spectra of the ions. The oscilloscope input impedance is set at 10 kW by a variable resistance (Thorlabs VT1), weighing the benefits of improved voltage-signal size against the associated RC-time increase (see below). The (analog) system bandwidth of the oscilloscope limits its time resolution to ∼2 ns, but to measure at 10 Hz repetition rate the scope is set to record 10,000 points for each trace, lowering the resolution to ∼200 ns. The vertical (voltage) scale has an 8-bit resolution. The suppressor is a copper plate with an aperture of 8 mm diameter, placed in front of the collector. A negative voltage is applied to this plate to prevent electrons in the plasma from entering and secondary electrons (electrons liberated from a surface by the impact of an energetic particle) from leaving the cup, while letting ions pass. The shield electrode is a grounded aluminum cover for the cup and suppressor to prevent the electric field from extending into the vacuum system and to shield the FC from particles not coming straight from the illuminated spot on the target (mostly electrons). The shield has an aperture of 6 mm diameter, this diameter determines the opening angle of the cup. Two FCs are installed in the horizontal plane, a third is situated in the vertical plane. In the horizontal plane, one FC is at a 2°-angle with respect to the laser beam, at a distance of 73 cm from the target, the other has a 30°-angle and is at 64.5 cm. The FC in the vertical plane has a 30°-angle and is 77 cm away from 5.

(7) Figure 3: Cross section schematic of the Faraday cups as used in the experiments. The funnel shaped part is the charge collector, the metal piece above it is the suppressor and covering both is the shield. The charge collector is connected to an oscilloscope via readout electronics, the suppressor is connected to a power supply and the shield is grounded.. the target. The FCs will be referred to as FC(=2,0), FC(=30,0), and FC(0,30) respectively. We find that the suppressor is by itself insufficient to fully prevent plasma electrons from reaching the collector. For this reason a negative bias voltage is applied to the collector cup itself as well. To prevent this bias voltage from also being applied to the oscilloscope, the oscilloscope input is isolated from the bias by a high-pass filter, providing “AC-coupling” to the oscilloscope (Fig. 4). A further low-pass filter is installed between bias voltage supply and the cup. These filters are built together into a box that is connected directly to an FC. These boxes will from here on be referred to as “bias box”. The bias boxes are connected to the voltage supply and the oscilloscope with 50-Ω coaxial cables. The bias voltage on the charge collector must be smaller than the voltage on the suppressor to prevent secondary electrons from escaping the collector and influencing the measurements. For this experiment we used a suppressor voltage VS = =100 V and the voltage on the collector VFC = =60 V, pending a systematic study of the influences thereof on the measurements. For now we operate under the assumption that these voltages provide full separation of positive (ion) and negative (electron) charges. Thus, the time-of-flight charge recordings are assumed to give a direct measure of the total ion charge arriving per unit time. These assumptions need to be checked in more detail in future work as, for instance, systematic errors may be expected especially for the lower velocity ranges [11].. Figure 4: The circuit used to read out a Faraday cup (FC) and apply a bias voltage to it at the same time. The electronic filters are built into one box together, a “bias box”, as denoted in the figure. The bias box is connected directly to an FC, and to the voltage supply and oscilloscope with coax cables.. Data analysis In this section we discuss the corrections applied to the measurements to retrieve their physical meaning and to compare the data from the different experiments. The conversion from TOF graphs to energy distribution 6.

(8) Figure 5: Simplified version of the circuit from Fig. 4. This simplified circuit was used to derive an algebraic expression for the ion flux at the FCs.. plots is also discussed. In Fig. 6 the relevant steps are detailed for the four experimental cases for comparable laser fluence. All circuitry changes signal going through it because components with capacitive properties charge up and have to discharge through resistors, which takes time. Raw TOF traces that are affected by this process are shown in Fig. 6(a). This process effectively convolves the signal with an exponential decay τ1 e−t/τ , where RC-time τ = RC. However, the circuitry used in this experiment is complex and a more detailed treatment is required to be accurate (see below). In addition to deforming signal due to RC-time, the decoupling capacitor has a distinct effect on the signal. The capacitor will charge as signal passes through and subsequently it discharges through the oscilloscope, reducing the recorded voltage. To retrieve ion current at the FCs from oscilloscope data, an algebraic approach is employed. Using Ohm’s ˙ and Kirchhoff’s laws the ion current at the cup can be law U = IR, the equation for capacitance U = IC derived as a function of measured voltage and system constants. For practical reasons a simplified model of the circuit is considered (Fig. 5), combining the resistors on the left side and considering an ideal volt source supplying 0 V (which is simply a connection to ground). The derivation then yields: 1 CFC 1 1 1 CC RC Iions (t) = + + + + Uout (t)+ Rosc CB Rosc Rbias Rbias CB Rbias Rosc RC CFC RC CC ˙ CC CFC Uout (t)+ + + + CFC + CC + CB Rosc Rbias Z t 1 öut (t) + Uout (τ ) dτ, (2) + CFC CC RC U Rbias CB Rosc 0 with ion current entering the FC Iions and voltage recorded by the oscilloscope Uout (t). All other terms are system constants, denoted in RFig. 5. The TOF traces are offset corrected before the correction is applied to obtain the correct values for U dτ . By plugging in the voltages from Fig. 6(a) into Eq. (2) the original ion current at the cup can be calculated back, shown in Fig. 6(b). To better compare the results from the different experiments, TOF traces are also corrected for the distance between the tin targets and the FCs by dividing the ion-current amplitude by the opening angle of the FCs and time is divided by the distance the ions have to travel to the FCs. Additionally, the data is smoothed ¨ terms in Eq. (2) greatly amplify noise. Smoothed and corrected TOF traces are shown as the U˙ and U in Fig. 6(c). Finding the energy distribution of individual ions, dN/dE, is not possible when using FCs, as only total charge can be measured. Instead we look at the energy distribution of the collected charge, 7.

(9) 0.14. (a). 0.12. -2. -2. Ion signal ( A). -2. 0.10. -2. µ. . (b). 20. 3mJ 10ns droplet 30° (18.4 J cm peak fluence) 3mJ 6ns solid FC(-2,0) (21.2 J cm peak fluence) 2.5mJ 4.5ps solid FC(0,30) (22.1 J cm peak fluence) 4mJ 15ps droplet 30° (21.1 J cm peak fluence). 0.08 0.06 0.04. 15. 10. 5. 0.02 0.00. 0. 20. 40. 60. 0. 80 100 120 140 160 180 200 220 240. 0. 20. 40. 60. 80 100 120 140 160 180 200 220 240. µ. µ. 0.12. 100. (c). (d). 10. -1. µ

(10) . 0.08. -1. Ion current (A/sr). 0.10. 0.06 0.04. 1 0.1 0.01. 0.02. 0.001 0.00. 0. 20. 40. 60. 10. 80 100 120 140 160 180 200 220 240. µ

(11). 100. 1000. . 10000. Figure 6: Graphs from different steps in the process of going from oscilloscope measurements to ion energy distribution. From each experiment the average of 50 traces is shown, the colormap is the same for (a) through (d). Measurements were done with peak laser fluence around 20 J cm=2 . Figure (a) shows raw ion traces as recorded by the oscilloscope. Figure (b) shows the ion current as calculated with Eq. (2). Note that the gray curve is now more prominent than in the raw data. This is due to the oscilloscope input impedance being 100 W during the ns-on-droplet measurements, instead of 10 kW as in the other experiments. In Fig. (c) the traces are smoothed, the ion current is corrected for the FC opening angle and the time axis is converted to time per meter. Having time on the x-axis makes it difficult to compare the traces, as the distances to the FCs differ between the experiments. Finally, Fig. (d) shows the energy distribution of the ions, dQ/dE plotted against E. These values are calculated from ion current and flight time using Eqs. (3) and (4).. dQ/dE, plotted against energy. Assuming the thermal energy of the ions is negligible when the ions reach the detector, ion energy is simply their kinetic energy, E(t) =. 1 x2 1 mv 2 = m FC , 2 2 t2. (3). with tin ion mass m, distance from the tin target to the FCs xFC and time-of-flight t. The energy distribution. 8.

(12) is derived as follows: −1 dQ dQ dt dE t3 = = Iions (t) = Iions (t) . dE dt dE dt mx2FC. (4). TOF traces plotted with this new scaling are shown in Fig. 6(d). An artifact introduced by this transformation is that noise on the signal at long time-of-flight (low ion energies) appears as the dominant feature in the energy distributions. This happens because the transformation turns a horizontal line in TOF graphs into a curve ∝ t−3/2 ; the same happens to the noise floors in the TOF graphs. The prefactor to the curve increases with the height of the vertical line, which means that stronger noise in the measurements also gives a more dominant noise curve in the energy distribution plots. This is clearly illustrated in the gray curves in Fig. 6. Going from TOF spectra to ion energy distribution also makes it very important to correct the data for any offset (illustrated in Appendix III).. 2.4. Other setups. The picosecond-on-droplet and nanosecond-on-droplet cases were performed using the droplet generator experimental setup described by Kurilovich et al. [12]. For the ns-on-droplet experiment the used laser pulse length was 10 ns; laser wavelength was 1064 nm; the laser pulse energy was varied between 0.5 mJ and 371 mJ and the diameter of the droplet targets was roughly 45 µm. For the ps-on-droplet experiments, the pulse length was varied between 15 and 115 ps, and the pulse energy was varied between 0.1 mJ and 10 mJ; the droplet diameter was roughly 30 µm (data for the same droplet size were corrupted due to incomplete laser-pulse contrast related to a problem with the electro-optic modulator). For both droplet experiments only ion data from an FC at 30° with respect to the laser beam and 36.6 cm away from the droplet will be considered. An additional FC was installed at ∼60°, but is not considered in this thesis. The picosecond-on-solid cases is described by Deuzeman et al. [5], performed using the same experimental setup described earlier in this chapter. However, this particular experiment used an 800 nm wavelength laser while the other experiments used 1064 nm light, but data from Freeman et al. [13] suggests that this difference only slightly influences the angular- and energy-distributions of the ions. Pulse durations for this experiment ranged 0.5–4.5 ps and pulse energies ranged 0.1–2.5 mJ. Ion measurements were done with three FCs under the same angles as for the ns on solid experiment, hence they will be referred to as FC(=2,0), FC(=30,0) and FC(0,30) as well. The distances between the target and the cups was different between the experiments however, with FC(=30,0) and FC(0,30) being 26 cm away for the ps experiment. FC(=2,0) was at 73.5 cm as it is in the ns on solid experiment. To better compare the two ps experiments, only 4.5 ps-long pulses on-solid and 15 ps-long pulses on droplets are considered. For the ns-on-droplet experiment an oscilloscope input impedance of 100 W was used, for the other experiments this impedance was 10 kW. The experiments were performed independently, resulting in different pulse energies used. An elaborate table detailing the chosen system parameters can be found in Appendix II.. 9.

(13) 3. Experimental results. In this chapter we present the experimental results for the four studied cases: nanosecond-on-droplet, nanosecond-on-solid, picosecond-on-solid, and picosecond-on-droplet. Each case is discussed separately first, starting with a generalized time-of-flight traces, expressed as time-per-meter to correct for the various flight path lengths (see previous chapter) for a wide range of laser pulse energies. Next to each such trace, the corresponding charge energy distribution is plotted. After these detailed discussions, we phenomenologically study the behavior of the total charge and fast ion energy with changing laser pulse energy for the four cases.. 3.1. Ion energy distributions. In this section, we discuss the ion energy distributions for the four studied cases starting with the data specifically obtained as part of this thesis. 3.1.1. Nanosecond-on-solid. Figs. 7(a) and (b) respectively show the ion TOF (per meter) graphs and energy distributions for the ns on solid experiment. Measurements are shown for 10, 100, 200, 300, and 389 mJ, using 6 ns long pulses focused down to a ∼90 µm diameter (FWHM) spot. Results are shown for both FC(=2,0) and FC(0,30)..

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(18) . . 10. -1. Ion current (A/sr). 1000. (a). . 1 0.1 0.01 0.001. . 1E-4 . . . . . . . . . . . . 10. µ

(19). 100. 1000. . 10000. 100000. Figure 7: Ion traces (a) and energy distribution of ions (b) from solid tin ablated by 6 ns pulses for FC(=2,0) (black) and FC(0,30) (red) for various pulse energies. Data is corrected for distance between tin target and FC, opening angle of FC and electronic response, as discussed in Sec. (2.3). Pulse energies are 10, 100, 200, 300, and 389 mJ; increase in pulse energy is denoted by the arrows. Please note the different scalings of the x-axes for Subig. (a) and Fig. 8(a).. Firstly, we observe an increase in ion current (at all times) when increasing laser pulse energy for both FC(=2,0) and FC(0,30). In all cases, there is more charge collected on FC(=2,0) than on FC(0,30). Simultaneously, there is a noticeable trend towards relatively larger signals at short TOF, indicating a strong increase in the relative and absolute contribution in high-energy ions. Ions generally have longer TOF moving towards FC(0,30) than FC(=2,0). It is interesting to note that the total increase in signal at 100 eV is of about two orders of magnitude, whereas at 3000 eV this is closer to three. Very noticeable Fig. 7(a) is the doubly-peaked structure for FC(=2,0) at the larger pulse energies. The nature of the transformation of TOF to energy distribution makes that these features are much less distinctive in the latter, although clearly a. 10.

(20) “shoulder” seems to appear in the high-energy tail of the distribution. The FC(0,30) traces also give a hint that such a high-energy feature is present, though much less noticeable. The high energy tails seem to follow an exponential decay and cross the noise floor at ∼10 keV. The wavy features beyond that are smoothed noise. The clear trend of increasing total charge with laser energy is well known from literature, although the specific quantitative scaling with this energy was not a priori known. This is discussed separately below. The anisotropy which is clearly proven by comparison of the FC(0,30) and FC(=2,0) traces is also well-known (e.g., see [10] and associated Eq. (1)), with an anisotropy factor k ranging k ≈ 4 − 2 for laser energies ranging 10 − 389 mJ respectively, consistent with available literature [5, 6]. Doubly peaked structures are well known from experimental work on picosecond pulsed ablation [5, 14, 15], where the fast ion feature is attributed to acceleration in a time-dependent ambipolar field. This field is created because energetic electrons escape at the plasma edge and set up a space charge layer which accelerates the ions. Less is known of such fast ion features in nanosecond-pulsed ablation, although a similar fast ion feature was indeed observed by Farid et al. [16], who also show that the peak is not caused by the target surface being contaminated by light elements. At this point, it is unclear what is the underlying reason for having two features, however, as we will show below, it is this fast ion feature that is reproduced at least partially in the nanosecond-on-droplet case. The second, slower ion feature is thus specific for the solid-target case(s). 3.1.2. Nanosecond-on-droplet. Figs. 8(a) and (b) respectively show the ion TOF graphs and energy distributions for the ns on droplet experiment. Measurements were repeated for pulse energies between 0.5 mJ and 371 mJ (specific values are given in the figure), using 10 ns long pulses focused down to a ∼100 µm diameter (FWHM) spot. . . (b). 1. -1 -1. . µ

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(23) . . Ion current (A/sr). 10. (a). . 0.1 0.01 0.001. . 1E-4 . . . . . . . . . . 10. µ

(24). 100. 1000. . 10000. 100000. Figure 8: Ion traces (a) and energy distribution of ions (b) from liquid tin droplets ablated by 10 ns pulses from an FC at 30°. Data is corrected for distance between tin target and FC, opening angle of FC and electronic response, as discussed in Sec. (2.3). The periodic noise in (a) before the ion peak, corresponding to the high energy signal in (b), is electrical pickup from the laser system. The distinct cutoffs in the graphs for laser energies >120 mJ are due to the oscilloscope being set to a shorter time range for those measurements. The increased amplitude of the noise is because the noise floor imposed by the oscilloscope scales with the voltage range it is set to.. Apparent in Fig. 8(a) are sharp peaks that shift to shorter times-of-flight for higher pulse energies. As before, the total charge collected increases with increasing laser energy. It is compared with the other three experimental cases in Sec. (3.2.1). The ion signal fades much faster with time than for the other three 11.

(25) experimental cases studied, which is why here the x-axis only ranges 0–50 µs m=1 , contrasting the previous and following ranges, which span 0–250 µs m=1 . The sharp peaks seen in the TOF traces are also clearly visible in Fig. 8(b), where they still clearly shift in peak energy with increasing laser power. We clearly observe a sharp, distinct peak ion-energy for most of the recorded traces, which is unique when comparing the four cases studied. The energies of these peaks are compared to the energies of fast ions from the other experiments in Sec. (3.2.2). Beyond this most-likely, peak energy, we find that the distribution decays very rapidly, with a much sharper drop than for the previous nanosecond-on-solid case. Towards lower “ion” energy the distributions remain almost constant, again contrasting with the previous case. While ns-long laser pulses on tin droplets have been studied both experimentally [12, 17] and theoretically [1, 3, 4], but these works have not focused on ion energy distributions. To the best of our knowledge only the works of Chen et al. [18] and Chen et al. [19] investigate this topic. In those works qualitatively similar ion TOF traces are shown as in Fig. 8(a). It is interesting to point out that in these works the tin droplets had a diameter of 150 µm, contrasting 45 µm for the ns-on-droplet case in this document. We note that these works lack detail on the precise treatment of the FC data, such as on the data correction for electronic responses as discussed in this thesis. Unfortunately, neither study shows ion spectra as a function of laser energy. In the following, we briefly point out several analysis features that are illustrated in Fig. 8(b). As discussed in Sec. (2.3), the noise floor follows a power-law curve ∝ t−3/2 (see 0.5 mJ gold-colored curve in Fig. 8(b)). The height of this curve increases with noise amplitude, which in turn increases with the voltage-scaling of the oscilloscope. This means that traces with higher amplitude have stronger noise as we rescaled the oscilloscope voltage scale accordingly during the data taking. For laser energies ≥120 mJ there is an apparent cutoff in the data left of energy ∼60 eV due to the oscilloscope being set to measure over a shorter time-range (thus limiting the total recorded TOF). Further, in Fig. 8(a) there appears to be signal before 5 µs m=1 , which was established to be pickup from the laser system. This noise is also visible as ringing in Fig. 8(b) at high ion energy. 3.1.3. Picosecond-on-solid. Figs. 9(a) and (b) respectively show ion TOF graphs and energy distributions for the ps-on-solid case, using 4.5 ps-long pulses focused down to a ∼100 µm diameter (FWHM) spot. Measurements are shown for laser energies from 0.5 mJ to 2.5 mJ with 0.4 mJ increment, for both FC(=2,0) and FC(=30,0). Data from FC(0,30) is not shown due its similarity to that of the FC(=30,0), being set up symmetrically. As before, we see that the total charge collected increases with increasing laser energy (see Sec. (3.2.1)), with the difference between the two FCs decreasing with increasing laser energy due to a decreasing anisotropy parameter, as pointed out previously by Deuzeman et al. [5]. The FC(=2,0) TOF data show a simple distribution with a single peak, while for the FC(=30,0) there are two distinct peaks. In the energy distribution (Fig. 9(b)), the faster of the two peaks is visible as a “shoulder” at high energy. This particular feature shifts quickly to higher ion energy for higher laser energy, with the low-energy peak much more slowly following a similar trend. The energy distributions for FC(=2,0) are smooth, monotonically decreasing functions. On the low-energy end the energy distribution for the FC(=2,0) becomes indistinguishable from the noise faster than for FC(=30,0), because the former was further away from the target and as such had a lower signal-to-noise ratio. In the TOF graphs, the measurements from both FCs appear to converge to a similar decay curve. For the 2.5 mJ measurement this happens around t = 70 µs m=1 , but convergence is reached after increasing times for decreasing laser energy. At more detailed discussion of the data presented here on the ps-on-solid case, and interpretations thereof, can be found in the published work of Deuzeman et al. [5]. 3.1.4. Picosecond-on-droplet. Figs. 10(a) and (b) respectively show ion TOF graphs and energy distributions for the ps-on-droplet case, using 15 ps-long pulses focused down to a ∼95 µm diameter (FWHM) spot. Measurements are shown for. 12.

(26) . 10. (a) . (b).

(27) . . -1. Ion current (A/sr). -1. µ

(28) . 1. . 0.1 0.01 0.001 1E-4. . 1E-5 . . . . . . . . . . . . 10. . 100. µ

(29). 1000. . 10000. 100000. Figure 9: Ion traces (a) and energy distribution of ions (b) from solid tin ablated by 4.5 ps pulses for FC(=2,0) (black) and FC(=30,0) (red). Data is corrected for distance between tin target and FC, opening angle of FC and electronic response, as discussed in Sec. (2.3). Pulse energies go from 0.5 mJ to 2.5 mJ with 0.4 mJ increments. Signal increases with increasing laser energy, denoted by the arrows. Please note that in (a) the black curves have more noise than the red because FC(=2,0) was further away from the target than FC(=30,0), reducing the available solid angle.. laser energies between 1 and 10 mJ; specific values are mentioned in the figure. . 1. (a). 0.1 -1. µ

(30) .

(31) . . . -1. Ion current (A/sr). (b). 0.01 0.001 1E-4 1E-5. 1E-6 . . . . . . . . . . . . 10. . µ

(32). 100. 1000. . 10000. 100000. Figure 10: Ion traces (a) and energy distribution of ions (b) from liquid tin droplets ablated by 15 ps pulses for an FC at 30°. Data is corrected for distance between tin target and FC, opening angle of FC and electronic response, as discussed in Sec. (2.3).. Again, the total charge collected increases with increasing laser energy (see Sec. (3.2.1)) but is much lower, close to an order of magnitude so, than for the intuitively similar case of the ps-on-solid. The TOF graphs feature a single, very fast peak, with an extremely short rise time (much shorter than for the ps-on-solid case in Fig. 9(a)). This is naturally reflected in the energy distributions shown in Fig. 10(b), where there is still significant signal for ion energies >100 keV for the highest pulse energies. The peaks in the TOF traces do not show up as distinct peaks in the energy distribution. The energy of the fast peak in the TOF traces 13.

(33) will be compared to the fast ions in the other experiments in Sec. (3.2.2). To the best of our knowledge, no experimental information is available in the open literature about the ions emitted from such systems.. 3.2. Comparing cases. In the following, we compare the four experimental cases on two “emergent” properties obtained from the recorded TOF traces: total charge and fast-peak ion energies as a function of laser pulse energy. 3.2.1. Charge yield. Figs. 11(a) and (b) show the total charge yields as function of laser energy for the ns experiments and ps experiments respectively. Nanosecond pulses generally, for similar laser fluence, generate more charge than picosecond pulses. Also, generally, more charge is produced from solid targets than on droplets, again taken at similar laser fluence. Apparent in (a) is that most data points follow a power law dependence on the laser pulse energy, most beautifully apparent for the 10 ns-on-droplet case. This particular case shows a power law behavior that is very similar, especially regarding the apparent power-law energy “offset”, to the behavior of the momentum kick delivered to the droplets by laser pulses as described by Kurilovich et al. [12]. Such a dependence also appears to describe the behavior of the ps experiments, but due to limited laser energy it cannot yet be confirmed whether a power law dependence will be reached. The power law being offset by a certain constant makes sense physically, as a certain threshold laser pulse energy is required to start efficient ablation. As total charge yield depends on many variables (e.g. ablation volume, ionization processes, plasma processes, laser-plasma interactions) it is difficult to formulate a model and there is no clear expectation for the dependence of charge yield on laser energy yet. This is reflected in the available public literature, where we were unable to find any clear predictions. As we did not record ablated volumes, we also cannot make any statements regarding the ionization fraction. There is experimental data on the charge yield but varying conclusions are drawn from them, with Amoruso et al. [15] describing both a logarithmic and a power law scaling, Toftmann et al. [6] describing a linear scaling and Deuzeman et al. [5] describing an either logarithmic or linear dependence.. (a). 100. (b). µ . µ . 1 10. 0.1. 1 6ns solid 2° 6ns solid 30° 10ns droplet 30°. 0.1. 1. 10. 4.5ps solid 2° 4.5ps solid 30° 15ps droplet 30°. 0.01. 100. 0.1.

(34) . 1. 10.

(35) . Figure 11: Charge yields from the different FCs for the ns experiments (a) and the ps experiments (b) as a function of laser-pulse energy. The red solid line shown in (a) is a fit of an offset power law as given in [12]. The power found from the fitting procedure is 0.68(1).. 14.

(36) 3.2.2. Fast ion peak. In Sec. (3.1), we observe that picosecond pulses produce faster ions than nanosecond pulses, when compared at similar laser fluence, both on solid and droplet targets. The fast peaks in the TOF graphs on the nanosecond-on-solid case seem to arrive at a time similar to the single feature found in the nanosecond-ondroplet case. Therefore, we hypothesize that they have similar physical origins. It is instructive to compare such features as a function of laser pulse energy. However, these features are not equally apparent, or even quantifiable, in the energy distributions. Therefore, we infer a “peak energy” corresponding to the maximum of the fast features in the TOF traces (see Fig. 12). From Subfig. (a) we conclude that these apparently similar features have very different absolute energies and scalings thereof with laser pulse energy. Correcting for the different pulse lengths used by transforming from laser-pulse energy to laser-pulse intensity (see Fig. 12(b)) only makes this difference larger. A more detailed analysis is required to make more definite statements..

(37) . 10. 6ns-on-solid 10ns-on-droplet 4.5ps-on-solid 15ps-on-droplet. (a) 10.

(38) . 6ns-on-solid 10ns-on-droplet 4.5ps-on-solid 15ps-on-droplet. 1. (b). 1. 1. 10. 100. 1E10. 1E11. 1E12.

(39) .

(40) . -2. Figure 12: “Peak energy” of charge as inferred from TOF peaks for the fastest peaks visible for the four different experimental cases, as a function of laser pulse energy (a) and laser peak intensity (b). Peak intensities are calculated by dividing peak fluence by FWHM pulse duration. The powers found from the fitting procedure are 1.1(1) for ns-onsolid, 0.51(1) for ns-on-droplet, 1.10(3) for ps-on-solid and 0.73(8) for ps-on-droplet. Similarly, we expect that the physics behind the acceleration of the fastest of ions is similar for the ps-ondroplet and ps-on-solid cases. A quick inspection of the TOF graphs Figs. 9 and 10, however, shows that the rise time is much shorter for the ps-on-droplet case. We compare the peaks of the fastest TOF features (as before expressed in corresponding kinetic energies) in Fig. 12(a). From the ps-on-solid experiment only the FC(=30,0) data is considered, as the fast peaks (presumed to be there) are always indistinguishable from the thermal signal for FC(=2,0). For these ps-cases, correcting for the difference in laser pulse length (see Fig. 12(b)) does in fact move the two data sets closer together although the scaling with laser pulse intensity (and energy) is still quite different. Here too, more detailed analysis is required to make more definite statements but this difference in scaling could well be due to plasma “vapor” absorption [5] that is more relevant for the longer (ps-)pulse length and energies. This should become apparent with future analysis of the longer pulse-length ps-on-droplet data, already available. Apparent in all plots in Fig. 12 is that the data sets appear to follow power laws. To demonstrate this, we fit power laws to the data. We observe larger powers for the solid target experiments compared the droplet experiments for both the ps and ns cases. Comparison with literature shows that both Hora et al. [20] and F¨ ahler and Krebs [21] find a linear dependence of peak ion energy of fast ions for ns pulses, which is close to what we find for the ns-on-solid experiment. For ps pulses Hora et al. [20] claim that the ion peak energy is not dependent on laser energy, but this is not in line with observations from either ps experiment. In literature 15.

(41) both ambipolar-field diffusion and Coulomb explosion are mentioned as possible causes for the generation of high energy ions [5, 15, 22, 23, 24]. A Coulomb explosion occurs when electrons are removed from their respective ions in a tight region, after which the ions repel each other due to Coulomb interactions. When the particles initially are closer together, the acceleration of the ions becomes greater. As such, Coulomb explosions can only occur at surfaces (also at liquid surfaces), while ambipolar-field diffusion can also be maintained/enhanced when the plasma plume is illuminated by the persisting laser pulse. On the origins of fast ions in the ns cases, Farid et al. [16] claim that the front of the plasma plume is efficiently heated through inverse bremsstrahlung, further accelerating the already energetic ions located there. While Chang and Warner [25] also see that laser-plasma interaction further accelerates the plasma expansion, the rate of expansion they observe still is around one order of magnitude slower than the ion velocity found in e.g. Fig. 8 for comparable laser fluence. In the next chapter, we will compare typical ion energy distribution functions for the four experimental cases studied here to theoretical models describing plasma expansion into the vacuum. We show that, overall, there is reasonable agreement between theory and experiment and that “simple” hydrodynamic principles provide reasonable explanation of the spectra over a broad ion-energy range.. 16.

(42) 4. Experiment vs theory. In this chapter, we briefly introduce two theoretical models that describe the expansion of a plasma into the vacuum and compare these models to the four experimental cases studied in this thesis. Those works consider ion energy distribution functions in a purely hydrodynamical framework (where the ion velocity is simply identified with the macroscopic hydro-velocity of a single-fluid quasi-neutral plasma, given some average charge state only) for a particular and well-known type of hydrodynamic self-similar solutions. Because the hydrodynamic solution is not augmented by the coupled solution of the equations of ionization/recombination kinetics, the resulting hydrodynamic energy distribution function must be compared with the measured distribution function, where all the charge states, including the neutrals, are summed up together. Here, we again stress that our FC measurement only serve to give an approximation of the plasma flow as the separation of electrons from the ions in the quasi-neutral expansion of the plasma cannot be assumed to be complete and is dependent on e.g. the set bias voltages and local magnetic fields [26]. The correspondence between the calculated hydrodynamic energy distribution function and the measured energy distribution of either a single charge state, or of the total charge of all ions captured by a Faraday cup as is done in the following, may be very questionable. Also, the applicability of the hydrodynamic approach may be very questionable itself. Generally, ion energy spectra could be adequately inferred from the hydrodynamic density and velocity fields whenever the condition regarding typical length scales L λD is fulfilled, where L is a typical plasma length scale, and λD is the local value of the Debye radius [2]. The “hard”, high-energy tails of the ion energy spectra, particularly visible in the nanosecond-on-droplet case (see below), may originate from those parts of the hydrodynamic flow where the above condition fails [2, 27].. 4.1. Theory: hydrodynamic expansion into the vacuum. The first model used for our comparisons was proposed by Mora [2] to describe plasma expansion after the interaction of a solid target with a short laser pulse (∼ ps), yielding the distribution function (“spectrum”) √ 1 dN ∝ √ e− E/E0 dE E. (5). with N the particle count and E0 a measure for the energy in the system ZkB Te with Z the charge state and Te the electron temperature. An example of such a spectrum is plotted in Fig. 13 (gray curve). Assumed is that the plasma is a quasi-infinite plane of infinite thickness, in which cold ions are pulled by hot electrons. This model does not consider laser-plasma interaction and is thus only relevant for short pulse interactions such as our ps-cases. The second model, introduced by Murakami et al. [4], considers the hydrodynamic, isothermal expansion of mass-limited plasmas with different geometries, ranging from infinite-planar to spherical cases. This model considers laser-plasma heating and can thus be applied to cases involving longer pulse lengths such as our ns-cases. Also, this model may be relevant for the 15 ps-long pulses as significant plasma “vapor” absorption is to be expected for these pulse lengths [5]. The model predict a ion energy distribution given by √ α−2 −E/E dN 0 ∝ E e , dE. (6). with α the geometry of the plasma expansion (α = 1 for planar, 2 for cylindrical, 3 for spherical) and E0 ˙ For α = 3, Eq. (6) similarly defined as above as 21 mR˙ 2 with ion mass m and plasma expansion rate R. reduces to the well-known Maxwellian energy distribution and is the only model function here that can explain a peaked ion energy distribution, with a maximum at E = 12 E0 . Examples of this spectrum are shown in Fig. 13 (colored curves); one curve is plotted for each of the mentioned values of α. When looking 17.

(43) at the curves, we see that towards low energy Mora’s model converges to Murakami’s model with α = 1, since their similar prefactor dominates the distribution. Towards high energy the three implementations of Murakami’s model converge, because their similar exponential factor dominates there. 100. Mora Murakami α=1 Murakami α=2 Murakami α=3. 10. -1. -1. μ

(44) . 1 0.1 0.01 0.001 1E-4 1E-5 1E-6 1E-7. 0.01. 0.1. 1. . 10. 100. 1000. Figure 13: Comparisons of the theoretical models, each with E0 = 1 and unity value for the normalization factors. Murkami’s model is plotted for three different values of α.. We can compare the typical energies E0 to those obtained from laser-plasma absorption theory giving an electron temperature Te in units of eV, 2/3. Te = 27(A/Z)1/3 λ4/3 IL ,. (7). assuming unity absorption, where A is the ion mass number at charge state Z, λ is the laser wavelength normalized by 1 µm, and IL is the laser intensity normalized by 10 × 1011 W/cm2 . This yield typical temperatures of ∼13 eV at 10 × 1010 W/cm2 and (taking A = 120; Z = 10; λ = 1) and ∼62 eV at 10 × 1011 W/cm2 corresponding to typical E0 ≈ ZTe = 130 and 620 eV, respectively. In the work of Murakami et al. [4] the authors arrive at even higher E0 = 2−3 keV for plasma temperatures of 30−50 eV. This serves to give us a feeling of the order of magnitude of ion energies that we can expect in the following.. 4.2. Comparing the four cases to two models. We proceed by comparing typical experimental spectra per experimental case with the two models presented above. 4.2.1. Nanosecond-on-solid. Fig. 14(a) shows four experimental data sets: two laser pulse energies (3 and 380 mJ) each for two FCs. As described above, we expect that the model of Murakami et al. [4] would be best suited to describe these data and we use Eq. (6) to fit the four functions. The traces for low laser pulse energy have no clear maximum and seem to be best described by choosing α = 2. The traces from both FCs are well fit by this model, yielding E0 = 164(1) and 129(2) eV for the 2° and 30°-FC, respectively. These values could be interpreted using Eq. (5) In the work of Murakami et al. [4] very similar experimental traces are presented which are very well fit by Eq. (6). Those experimental traces deal with Xe-LPPs but references to the original experimental works are incomplete and/or refer to unpublished work. 18.

(45) . 1000. (a). (b) µ

(46) -1. 10. -1. -1. -1. µ

(47) . 100. 1 0.1. .

(48) .

(49) . . . 0.01 0.001 1. 10. 100. . . . . 1000. . 10000.

(50) . . . . Figure 14: Different data sets (points) for the ns-on-solid case (a) and the ns-on-droplet case with fits of Murakami’s theoretical model (dashed), Eq. (6). Fitting parameters are E0 , α and the normalization constant. For each fit α is set to be either 1, 2 or 3. In (a), E0 for the fits to the higher laser energy data is 116(2) eV for FC(=2,0) and 114(3) eV for FC(=30,0), for both FCs α = 3. For the lower laser energy fits E0 = 164(1) eV for FC(=2,0) and E0 = 129(2) eV for FC(=30,0), here α = 2 for both FCs. In (b), E0 = 3550(29) eV and 358(4) eV for the higher and lower laser energy cases, respectively. For both, α = 2.. The high energy traces in Fig. 14(a) have a clear maximum and can thus only be expected to be fit well by using α = 3. This is a reasonable assumption as the relevant length scale of the plasma at these energies is larger than the laser spot size. However, the fit of Eq. (6) to the data does not yield a satisfactory agreement. At first glance, it is especially the high energy tail that is not captured at all by the model, although this is strongly dependent on the details of the chosen fit procedure (here, and in the following, we choose weighting proportional to the variance which is chosen to equal the y-axis value). Also the fitted values E0 =116(2) and 114(3) eV do not match our expectation to find larger energies at larger laser-pulse energies. Here, it should be noted that these comparisons only make sense for the same geometry. The high-energy shoulder visible for the larger laser pulse energy case are very similar to the main feature in the ion energy distribution seen for the nanosecond-on-droplet case (see below) and we thus expect similar physical origins. 4.2.2. Nanosecond-on-droplet. Fig. 14(b) shows two experimental data sets: two laser pulse energies (3 and 371 mJ) each for single FC placed at 30°. Visible inspection of the curves immediately makes us discard Mora’s model and the “plateau” visible for both cases at the low ion energies, is only described by any measure of accuracy by choosing α = 2 in Murakami’s model. Nevertheless, they are very poorly described by the model which is not at all able to explain the sharp peak structure at high kinetic energy visible for all but lowest of laser pulse energies. This peak structure is followed by a very rapid drop-off toward higher energies that cannot be described by any of the models. This apparent energy “cutoff” occurs could be compared (in future work) with theory works describing such cutoff energies, e.g., given by Mora [2], Murakami et al. [4], and Murakami and Basko [3]. We conclude that more work is required on the theory front and hypothesize that the charge state distribution, which is not taken into account in any way in the aforementioned models, plays an important role. Currently, we are developing experimental facilities to measure the charge-state-resolved ion energy distribution which would shed more light on the origins of the ion energy distribution and influence thereof.. 19.

(51) 4.2.3. Picosecond-on-solid. Fig. 15(a) shows four experimental data sets: two laser pulse energies (0.5 and 2.5 mJ) each for two FCs placed at 2° and 30° (see setup). The short nature of the laser-solid interaction would indicate that Mora’s model, which does not take laser-plasma heating into account, is best suited here. We fit Eq. (5) to the four data sets and find reasonable agreement, especially in the case of the FC(=2,0) trace at the higher laser pulse energy where we obtain E0 = 400 eV. As we have no further information regarding the average charge state, we cannot further interpret this value, but the order of magnitude certainly makes sense invoking Eq. (7). 100. 1. (a). 10.

(52) . -1. -1. µ

(53) . -1. -1. µ

(54) . 0.1 1 0.1 0.01. . . 0.001 1E-4 1E-5. 1. 10. (b). 0.01 0.001 1E-4 1E-5 1E-6 1E-7. 100. 1000. . 10. 10000. 100. 1000. . 10000. 100000. Figure 15: Different data sets (points) for the ps-on-solid case (a) and the ps-on-droplet case with fits of Mora’s theoretical model (dashed), Eq. (5). Murakami’s model is also fitted (red dashed), but only to the lowest data set in (a). The two fitting parameters in Mora’s model are E0 and the normalization constant. In (a), E0 for the fits to the higher laser energy data is 400(11) eV for FC(=2,0) and 114(5) eV for FC(=30,0). For the lower laser energy fits E0 = 68(2) eV for FC(=2,0) and E0 = 9.6(6) eV for FC(=30,0), here α = 2 for both FCs. In (b), E0 = 921(41) eV and 254(8) eV for the higher and lower laser energy cases, respectively.. At lower laser pulse energies, especially at larger observation angles, Mora’s model does not appear to hold, which is why we choose to fit Murakami’s model here, too. Some improvement in agreement is reached, but not sufficiently so to make any hard statements. We hypothesize that large observation angle (which thus favor small anisotropy factors k) could be more sensitive to the more “thermal” part of the plasma expansion and not the fast ion front which typically flows along the surface normal [1, 2, 10]. We note that the FC(=30,0) traces dip on the low-energy side of their apparent respective maxima near several 10 eV. This could possibly be related to incomplete separation of negative and positive plasma charges entering the FC. Again, most hydrodynamical models (as are discussed in this chapter) would predict an ion spectrum that is monotonically decreasing with ion energy. 4.2.4. Picosecond-on-droplet. Fig. 15(b) shows two experimental data sets: two laser pulse energies (1 and 10 mJ) each for two FCs placed at 2° and 30° (see setup). We chose here slightly different laser pulse fluence to facilitate comparisons with the above picosecond-on-solid case which, at a shorter 4.5 ps pulse length, would otherwise have a higher intensity. As before, the short nature of the laser-solid interaction implies that Mora’s model, which does not take laser-plasma heating into account, is best suited here. It indeed fits very well to the two shown data sets. The low energy 1 mJ case yields a fit parameter E0 = 254(8) eV, the high energy 10 mJ case gives. 20.

(55) E0 = 921(41) eV. A further high-energy tail >10 keV seems to be present for the larger laser pulse energies, although the available experimental data is limited here and is prone to (albeit small) systematic errors in the deconvolution procedure (see Sec. (2.3) and Appendix III). From simultaneous EUV-spectroscopic investigations, we understand that there is a significant ionization degree at least up to Z = 13. At the same time, Eq. (7) implies a plasma electron temperatures scaling ∼100–1000 eV over the here studied range of laser intensities IL = 6 × 1011 –7 × 1012 W/cm2 , in line with the found values for E0 but only when assuming unity value for Z. This is not in agreement with the EUV-spectral observations. A straightforward analysis is hampered not only by the missing information about the charge state distribution, but also by the non-flat spatial laser beam profile and curved droplet surface which means that we are actually dealing with a large spread in intensities. Nevertheless, we conclude that the shape of the ion energy distributions are very well described by Mora’s model.. 4.3. Concluding remarks. Murakami’s model fits well to the low-laser-energy ns-on-solid data. The model also describes parts of the data for higher laser energy, but fails to do so for the higher ion-energy part. The model poorly describes the ion energy distributions for the ns-on-droplet case. Mora’s model excellently describes the ps-on-droplet data, and works in varying degrees to describe data from the ps-on-solid case. In the latter case, especially towards lower laser energy and greater angle from the target normal, Mora’s model fails to describe the experimental results. This shows that the foundations of the theoretical approach to plasma expansion into the vacuum is present but that there is still work to be done to reach a complete understanding. What might help in getting a deeper understanding would be to consider ionic charge states, both in the experiment and in the theory. In the experimental case, work is currently underway to use an electrostatic analyzer with the ns-on-solid setup to gather charge state resolved ion data. Concerning theory, we note that Kelly and Dreyfus [28] and Tallents [29] postulate, rather than derive, Maxwellian functions, one per charge state, to describe plasma expansion. Burdt et al. [30] have done charge-resolved measurements and fit such functions to their results, claiming good agreement between them. However, the underlying physics is not yet understood fully.. 21.

(56) 5. Conclusion. This thesis contains a study of ion energy distributions from laser-produced-plasmas, inferred from experimentally obtained time-of-flight Faraday cup traces, for four different experimental cases. These cases were nanosecond-on-solid, nanosecond-on-droplet, picosecond-on-solid, and picosecond-on-droplet. The data collection for the nanosecond-on-solid case was also part of this thesis, as well as development of a data analysis method and application thereof to the data of all four cases. On comparison of the four cases, it is clear that the ion energy distributions pertaining to them are not at all similar. Of these differences especially the difference between the solid-target cases and the droplet cases is an important finding, as often the simpler solid-target setups are (mis)used to gain knowledge about processes surrounding droplet-targets. There are also large differences between the nanosecond-pulsed experiments on the one hand, and picosecondpulsed experiment on the other. The foremost differences are that, at similar fluence, ps-pulses produce far fewer ions but at far higher kinetic energy than ns-pulses. Scaling for intensity does not bridge the divide. It is to be expected that the two are so inherently different, as ps-pulses lack the laser-plasma interaction, and therefore laser-plasma heating, that is present for ns-pulses. To investigate the effects of the aforementioned laser-plasma heating, we compare the four cases to two theoretical models, one in which this laser-plasma heating is considered, one for which it is not. As expected, the former model fits well to the ns-cases and the latter better fits to the ps-cases. The models do not always describe the data, however, indicating that the theory is not complete as of yet. Especially the apparent cutoff at large kinetic energies in the ns-on-droplet case is of particular practical importance and theoretical interest and should be further studied. Further comparisons between experimental and theoretical work can be made when charge state resolved energy distributions are obtained, as additional material exists in which charge states are considered individually.. Acknowledgements I would like to thank the EUV Plasma Dynamics group at ARCNL for hosting the work done for this thesis and for many insightful discussions. I also want to thank the ARCNL group EUV Generation and Imaging for the use of their laser for studying the ps-on-droplet case. Furthermore I want to thank the AMOLF workshop and ARCNL technicians for aid and technical support during the experiments and Mikhail Basko for enlightening communications.. 22.

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(59) Appendix I. Derivation of data correction equation. Basic laws we can use are U = IR and I = C U˙ . After applying Kirchhoffs laws on our system we know the following: URbias = −UFC UFC = UB + URC + Uout Uout = UCC. (8). IRbias + Iions = IB + IFC. (11). (9). IB = Iout + ICC. (12). IB = IRC. (13). (10). Using these equations, we want to find Iions as a function of system constants and the measured voltage, Uout. Iions = IB + IFC − IRbias. Iout =. (14). Uout Rosc. (15). (10) ICC = CC U˙ CC = CC U˙ out Uout (12) IB = CC U˙ out + Rosc. (16). (9) IFC = CFC U˙ FC = CFC (U˙ B + U˙ RC + U˙ out ) IB + RC I˙B + U˙ out = CFC CB Uout RC U˙ out CC ˙ (16) ¨ ˙ Uout + = CFC + RC CC Uout + + Uout CB CB Rosc Rosc. UFC URbias =− Rbias Rbias Z t 1 CC 1 =− Uout + Uout (τ ) dτ + Rbias CB Rosc CB 0 RC + RC CC U˙ out + Uout + Uout Rosc. (17). IRbias = using eq. 17. 1 CFC 1 1 1 CC RC + + + + Uout (t)+ Rosc CB Rosc Rbias Rbias CB Rbias Rosc CC CFC RC CFC RC C C ˙ + (CFC + CC + + + )Uout (t)+ CB Rosc Rbias Z t 1 öut + + CFC CC RC U Uout (τ ) dτ Rbias CB Rosc 0. (18). . =⇒ Iions (t) =. 25. (19).

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