MSc Physics and AstronomyTrack: AMEP

Master Thesis

Comparison of ion energy distributions fromns- and ps-laser produced tin plasmas

from solid and droplet targets

by

Sjoerd van der Heijden10336001

November 4, 2017

60 EC01-09-2016 through 31-08-2017

Supervisor:Prof. dr. Wim Ubachs

Assessor:Prof. dr. Paul Planken

Advanced Research Center for Nanolithography

Abstract

Ion energy distributions from laser produced tin plasmas have been experimentally investigated usingtime-of-flight data obtained from charge collecting Faraday cups, and compared for nanosecond- andpicosecond-long laser pulses both on solid tin targets and liquid tin droplet targets. We find largedifferences between the four experimental cases, foremost that solid targets produce more total ion-charge than droplets, and that ps-pulses produce far less ion-charge, but with far greater kinetic energythan in the case of ns-pulses. Further, we compare two theoretical models describing hydrodynamicexpansion of plasma into the vacuum to the experimentally obtained distributions. The first theoreticalmodel, that considers laser-plasma interactions, fits reasonably well to the ion-charge energy distributionfor nanosecond-long pulses on solid. However, this model fails to describe much of the distribution fornanosecond pulses on droplets, for which it was in fact developed. The second model, which does notconsider such laser-plasma interactions, fits well to ion-charge energy distributions of the picosecondexperiments with reasonable accuracy. For more detailed interpretations of the physical mechanismsdriving the fast ions, charge-state-resolved measurements are a necessity.

Contents

1 Introduction 2

2 Experimental setup 32.1 Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Ion detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Other setups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Experimental results 103.1 Ion energy distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1.1 Nanosecond-on-solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.2 Nanosecond-on-droplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1.3 Picosecond-on-solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.1.4 Picosecond-on-droplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Comparing cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.1 Charge yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.2 Fast ion peak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Experiment vs theory 174.1 Theory: hydrodynamic expansion into the vacuum . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Comparing the four cases to two models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.2.1 Nanosecond-on-solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2.2 Nanosecond-on-droplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2.3 Picosecond-on-solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2.4 Picosecond-on-droplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.3 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Conclusion 22

Appendix I Derivation of data correction equation 25

Appendix II System parameters for the different experimental setups 26

Appendix IIIEffect of errors during data analysis 27

Appendix IV Necessity of data correction 28

1

1 Introduction

Laser-produced tin plasmas currently serve as sources for extreme ultraviolet (EUV) light for nanolithog-raphy. These plasmas are generated from microscopic tin droplets illuminated by short laser pulses, withpulse lengths ranging from femtoseconds to hundreds of nanoseconds. Aside from EUV light, however, theplasma also generates energetic debris, which limits the lifetime of the nanolithography machines. Thisdebris consists of neutral tin and tin ions ranging Sn1−14+. Aside from being a practical problem, plasmaexpansion into the vacuum is also a challenge from a theoretical point of view [1, 2, 3, 4]. As such, obtainingthe energy distribution of ions from laser produced tin plasmas is of particular interest from both practicaland fundamental considerations.

In this thesis, we discuss and compare experimentally obtained ion energy distributions for nanosecond andpicosecond-long pulses on both solid tin targets and tin droplet targets. These four cases will be referred toas nanosecond-on-solid, nanosecond-on-droplet, picosecond-on-solid and picosecond-on-droplet (nanosecondand picosecond may be abbreviated to ns and ps respectively). The energy distributions are obtained throughtime-of-flight (TOF) measurements, taken with Faraday cup charge collectors. To facilitate the comparisonbetween these cases, new experimental data is obtained for the nanosecond-on-solid case, while previouslyobtained 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 inthe total charge emitted by the plasmas. An important conclusion from this is that experiments performedwith solid targets have little predictive value for experiments using droplets, even when using similar pulselengths and fluences. This could possibly discredit present literature. A specific difference we found betweensolid-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 athigher kinetic energies than nanosecond-long pulses, for the same kind of target and laser fluence, consistentwith existing literature [5, 6].

Aside from analyses and comparisons of experimental data, two relatively recent theoretical models forplasma expansion into the vacuum are discussed and fitted to our data. We find that the model proposed byMora [2], which neglects laser-plasma interactions, fits excellently to the data from the picosecond-on-dropletcase, as can be expected. The model also describes some of the energy distributions for the picosecond-on-solid 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, againas expected. For higher laser energy, however, a high energy feature emerges that is poorly described byeither model. The energy distributions for the nanosecond-on-droplet case appear to be dominated by thisfeature, which neither model can describe, hinting at more complicated underlying physics that is missed byboth approaches. While the high energy parts of the energy distributions of the latter two cases may appearsimilar, towards lower ion energy the plasma from the solid target appears to be of a much more thermalnature. 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 wellas the data analysis method that was developed as part of this thesis. The setups for the other three casesare briefly discussed here as well. Then, the results from the four cases are presented, and the total chargeyield and high ion-energy features are discussed and compared between the cases. Finally, the theoreticalmodels are compared and fitted to the ion energy distributions.

2

2 Experimental setup

In this chapter the experimental setup used for the nanosecond-on-solid case is discussed in detail after whichwe 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. 1shows the experimental setup. To study the ions emitted by laser-produced tin plasma, we illuminate a solidtin 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:YAGlaser, operating at its fundamental wavelength of 1064 nm, is used to create laser-produced plasma (LPP) from a solidtin plate target. First, the unpolarized light also emitted by the laser system is partially removed from the beam using athin-film polarizer (TFP). Afterwards the remaining light can be attenuated by a rotatable λ/2 waveplate and anotherTFP. A telescope triples the beam diameter after which a lens with a focal distance of 1 m focuses the beam down toa width of 90µm at full-width at half-maximum (FWHM) on the tin target. Three Faraday cups (FCs) are installedto detect the ions emitted from the tin plasma that is then generated (only the two in the horizontal plane are shownhere). Biasing circuits, with common voltage sources, are connected to all FCs. A 500 MHz-bandwidth oscilloscoperecords their output.

2.1 Laser

The light is generated by a 10 Hz repetition rate Nd:YAG laser, operating at its fundamental wavelength of1064 nm. The maximum available pulse energy is 410 mJ; beam diameter is 6 mm and the temporal profile ofthe 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 lasercan be attenuated using a λ/2 wave plate and a second TFP; the laser energy can be reduced to ∼0.8 mJ byrotating the wave plate. A telescope is used to triple the beam diameter because a wide beam can be focusedmore tightly. A lens with a focal distance of 1 m then focuses the laser on the tin target, at the target surface

3

the full-width at half-maximum (FWHM) of the beam is roughly 90µm. We estimate that about 70 % ofthe attenuated beam energy reaches the target (comparing energy measurement right after the attenuationstage with energy measured inside the chamber), the rest is lost along the laser path probably due to poorlyreflecting or transmitting optical elements. The spatial beam profile in focus is well approximated with aGaussian function with calculated peak fluences 6.1 J cm=2 for 0.8 mJ pulses and 3.1× 103 J cm=2 for 410 mJpulses.

2.2 Target

The target is a tin plate of 99.999 % purity (Goodfellow) with dimensions 32 mm× 32 mm× 1 mm. Thetarget is mounted on computer controlled x- and y-stages so that the target can be moved in the planeperpendicular 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 longerthan 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 arerepeated 50 times to allow for sufficient averaging. Before the measurements start, several pulses are shotat the tin to clean the target of contaminations. This is necessary as the target is covered by a layer of tinoxide. Ablation of this layer causes a clear oxygen-related fast-ion feature in the measurements, which is notthe subject of this study. During continuous operation the tin target is moved by a fraction of the beamspot size between every measurement to obtain stable results; if the target is stationary between shots theion 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 resultsbut 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 ofseveral 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 moltenmaterial also forms into crater walls due to thermal expansion and recoil vapor pressure. These walls maythen influence the geometry of the plasma expansion. This process also makes it very difficult to find thetotal ablated volume. This contrasts with ps-on-solid case, where heat is efficiently transferred from thelaser heated electrons to the lattice rather than diffused into the material. This results in little melting andrelatively high ablation volumes compared to ablation by ns pulses, leaving a clean crater [7, 8, 9]. We findexperimentally that for nanosecond laser pulses more stable results are obtained by moving the target bya fraction of the beam diameter (∼60µm) after every shot. In this way the tin hit by the laser is alwaysrelatively flat and deep craters cannot form. Higher energy pulses deform the target more strongly, causingstronger voltage fluctuations in the data. Increasing the distance the target moves between shots reducesthis, again due to fewer shots hitting the same area on the target. The tin oxide layer is not an issue whenmoving the target between shots, as most of the oxide has been ablated by the wings of the pulse previousshots 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 cratercreated by the previous shots. The angle of this slope becomes greater for smaller step size and larger pulseenergies, as this gives deeper craters with steeper edges. The result of this is that the peak emission relatedto the (naturally anisotropic [10]) angular ion distributions shift away from the ion detector that is near theoriginal 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 ofthis 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

expression

Y (θ)

Y (0)=

(1 + tan2(θ)

1 + k2 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 parametersignifying anisotropy [5, 10]. For k = 1 the ions are distributed isotropically, for higher values the ions areemitted 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 anglebetween the new surface normal and the laser path is determined to be ∼4° for step sizes of both 40µmand 60µm, which are generally used when obtaining the main results; further, we find k ≈ 2 from thesame analysis for laser-pulse energies of 300 mJ. This angle is small enough to not significantly influence thecoordinate system: we can still compare data from the 30° detectors in the ns-on-solid case with those from30° detectors in the other cases, since the anisotropy factor k is of order unity.

0 20 40 60 80 100 120 140 160 180 200 220 2400

5

10

15

Ion

curre

nt(A

/sr)

Time per meter (µs)

300mJ 6ns 30°10 µm per step4060-60-40-10

Figure 2: Dependence of the FC(=30,0) time-of-flight ion signal on the direction and step size of the motion stagesfor 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 acharge collector, a suppressor and a shield electrode. The collector is a copper cone connected to a 500 MHz-bandwidth 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 Hzrepetition rate the scope is set to record 10,000 points for each trace, lowering the resolution to ∼200 ns. Thevertical (voltage) scale has an 8-bit resolution. The suppressor is a copper plate with an aperture of 8 mmdiameter, placed in front of the collector. A negative voltage is applied to this plate to prevent electrons inthe plasma from entering and secondary electrons (electrons liberated from a surface by the impact of anenergetic particle) from leaving the cup, while letting ions pass. The shield electrode is a grounded aluminumcover for the cup and suppressor to prevent the electric field from extending into the vacuum system and toshield 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. TwoFCs 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 otherhas 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

Figure 3: Cross section schematic of the Faraday cups as used in the experiments. The funnel shaped part is thecharge collector, the metal piece above it is the suppressor and covering both is the shield. The charge collector isconnected to an oscilloscope via readout electronics, the suppressor is connected to a power supply and the shield isgrounded.

the target. The FCs will be referred to as FC(=2,0), FC(=30,0), and FC(0,30) respectively. We find thatthe suppressor is by itself insufficient to fully prevent plasma electrons from reaching the collector. For thisreason a negative bias voltage is applied to the collector cup itself as well. To prevent this bias voltagefrom also being applied to the oscilloscope, the oscilloscope input is isolated from the bias by a high-passfilter, providing “AC-coupling” to the oscilloscope (Fig. 4). A further low-pass filter is installed between biasvoltage 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 voltagesupply and the oscilloscope with 50-Ω coaxial cables. The bias voltage on the charge collector must be smallerthan the voltage on the suppressor to prevent secondary electrons from escaping the collector and influencingthe measurements. For this experiment we used a suppressor voltage VS ==100 V and the voltage on thecollector VFC ==60 V, pending a systematic study of the influences thereof on the measurements. For nowwe 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 thetotal ion charge arriving per unit time. These assumptions need to be checked in more detail in future workas, 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. Theelectronic filters are built into one box together, a “bias box”, as denoted in the figure. The bias box is connecteddirectly 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 andto compare the data from the different experiments. The conversion from TOF graphs to energy distribution

6

Figure 5: Simplified version of the circuit from Fig. 4. This simplified circuit was used to derive an algebraicexpression 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 comparablelaser fluence.

All circuitry changes signal going through it because components with capacitive properties charge up andhave to discharge through resistors, which takes time. Raw TOF traces that are affected by this process areshown in Fig. 6(a). This process effectively convolves the signal with an exponential decay 1

τ e−t/τ , where

RC-time τ = RC. However, the circuitry used in this experiment is complex and a more detailed treatment isrequired to be accurate (see below). In addition to deforming signal due to RC-time, the decoupling capacitorhas a distinct effect on the signal. The capacitor will charge as signal passes through and subsequently itdischarges through the oscilloscope, reducing the recorded voltage.

To retrieve ion current at the FCs from oscilloscope data, an algebraic approach is employed. Using Ohm’slaw U = IR, the equation for capacitance U = IC and Kirchhoff’s laws the ion current at the cup can bederived as a function of measured voltage and system constants. For practical reasons a simplified model ofthe circuit is considered (Fig. 5), combining the resistors on the left side and considering an ideal volt sourcesupplying 0 V (which is simply a connection to ground). The derivation then yields:

Iions(t) =

(1

Rosc+CFC

CB

1

Rosc+

1

Rbias+

1

Rbias

CCCB

+RC

RbiasRosc

)Uout(t)+

+

(CFC + CC +

CCCFC

CB+RCCFC

Rosc+RCCCRbias

)Uout(t)+

+ CFCCCRCUout(t) +1

RbiasCBRosc

∫ t

0

Uout(τ) dτ, (2)

with ion current entering the FC Iions and voltage recorded by the oscilloscope Uout(t). All other terms aresystem constants, denoted in Fig. 5. The TOF traces are offset corrected before the correction is applied toobtain 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 distancebetween the tin targets and the FCs by dividing the ion-current amplitude by the opening angle of the FCsand time is divided by the distance the ions have to travel to the FCs. Additionally, the data is smoothedas the U and U terms in Eq. (2) greatly amplify noise. Smoothed and corrected TOF traces are shownin 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

µ

!

" " !

"

(a)

µ

µ

(b)

µ

(c)

µ

(d)

Figure 6: Graphs from different steps in the process of going from oscilloscope measurements to ion energy dis-tribution. 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 bythe oscilloscope. Figure (b) shows the ion current as calculated with Eq. (2). Note that the gray curve is now moreprominent than in the raw data. This is due to the oscilloscope input impedance being 100W during the ns-on-dropletmeasurements, instead of 10 kW as in the other experiments. In Fig. (c) the traces are smoothed, the ion current iscorrected for the FC opening angle and the time axis is converted to time per meter. Having time on the x-axis makesit difficult to compare the traces, as the distances to the FCs differ between the experiments. Finally, Fig. (d) showsthe energy distribution of the ions, dQ/dE plotted against E. These values are calculated from ion current and flighttime using Eqs. (3) and (4).

dQ/dE, plotted against energy. Assuming the thermal energy of the ions is negligible when the ions reachthe detector, ion energy is simply their kinetic energy,

E(t) =1

2mv2 =

1

2mx2FCt2

, (3)

with tin ion mass m, distance from the tin target to the FCs xFC and time-of-flight t. The energy distribution

8

is derived as follows:

dQ

dE=dQ

dt

dt

dE= Iions(t)

(dE

dt

)−1

= Iions(t)t3

mx2FC. (4)

TOF traces plotted with this new scaling are shown in Fig. 6(d). An artifact introduced by this transforma-tion is that noise on the signal at long time-of-flight (low ion energies) appears as the dominant feature inthe energy distributions. This happens because the transformation turns a horizontal line in TOF graphsinto a curve ∝ t−3/2; the same happens to the noise floors in the TOF graphs. The prefactor to the curveincreases with the height of the vertical line, which means that stronger noise in the measurements also givesa more dominant noise curve in the energy distribution plots. This is clearly illustrated in the gray curvesin Fig. 6. Going from TOF spectra to ion energy distribution also makes it very important to correct thedata 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 generatorexperimental setup described by Kurilovich et al. [12]. For the ns-on-droplet experiment the used laserpulse length was 10 ns; laser wavelength was 1064 nm; the laser pulse energy was varied between 0.5 mJ and371 mJ and the diameter of the droplet targets was roughly 45µm. For the ps-on-droplet experiments, thepulse 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 30µm (data for the same droplet size were corrupted due to incompletelaser-pulse contrast related to a problem with the electro-optic modulator). For both droplet experimentsonly ion data from an FC at 30° with respect to the laser beam and 36.6 cm away from the droplet will beconsidered. 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 experimentalsetup described earlier in this chapter. However, this particular experiment used an 800 nm wavelengthlaser while the other experiments used 1064 nm light, but data from Freeman et al. [13] suggests that thisdifference only slightly influences the angular- and energy-distributions of the ions. Pulse durations for thisexperiment ranged 0.5–4.5 ps and pulse energies ranged 0.1–2.5 mJ. Ion measurements were done with threeFCs 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 theexperiments however, with FC(=30,0) and FC(0,30) being 26 cm away for the ps experiment. FC(=2,0) wasat 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 dropletsare considered. For the ns-on-droplet experiment an oscilloscope input impedance of 100W was used, for theother experiments this impedance was 10 kW. The experiments were performed independently, resulting indifferent pulse energies used. An elaborate table detailing the chosen system parameters can be found inAppendix II.

9

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 flightpath lengths (see previous chapter) for a wide range of laser pulse energies. Next to each such trace, thecorresponding charge energy distribution is plotted. After these detailed discussions, we phenomenologicallystudy 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 dataspecifically 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 onsolid experiment. Measurements are shown for 10, 100, 200, 300, and 389 mJ, using 6 ns long pulses focuseddown to a ∼90µm diameter (FWHM) spot. Results are shown for both FC(=2,0) and FC(0,30).

µ

(a)

µ

(b)

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 angleof FC and electronic response, as discussed in Sec. (2.3). Pulse energies are 10, 100, 200, 300, and 389 mJ; increasein 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 bothFC(=2,0) and FC(0,30). In all cases, there is more charge collected on FC(=2,0) than on FC(0,30). Si-multaneously, there is a noticeable trend towards relatively larger signals at short TOF, indicating a strongincrease in the relative and absolute contribution in high-energy ions. Ions generally have longer TOF mov-ing towards FC(0,30) than FC(=2,0). It is interesting to note that the total increase in signal at 100 eV isof about two orders of magnitude, whereas at 3000 eV this is closer to three. Very noticeable Fig. 7(a) is thedoubly-peaked structure for FC(=2,0) at the larger pulse energies. The nature of the transformation of TOFto energy distribution makes that these features are much less distinctive in the latter, although clearly a

10

“shoulder” seems to appear in the high-energy tail of the distribution. The FC(0,30) traces also give a hintthat such a high-energy feature is present, though much less noticeable. The high energy tails seem to followan exponential decay and cross the noise floor at ∼10 keV. The wavy features beyond that are smoothednoise.

The clear trend of increasing total charge with laser energy is well known from literature, although thespecific 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 ranging10−389 mJ respectively, consistent with available literature [5, 6]. Doubly peaked structures are well knownfrom experimental work on picosecond pulsed ablation [5, 14, 15], where the fast ion feature is attributedto acceleration in a time-dependent ambipolar field. This field is created because energetic electrons escapeat the plasma edge and set up a space charge layer which accelerates the ions. Less is known of such fastion features in nanosecond-pulsed ablation, although a similar fast ion feature was indeed observed by Faridet al. [16], who also show that the peak is not caused by the target surface being contaminated by lightelements. At this point, it is unclear what is the underlying reason for having two features, however, as wewill show below, it is this fast ion feature that is reproduced at least partially in the nanosecond-on-dropletcase. 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 dropletexperiment. Measurements were repeated for pulse energies between 0.5 mJ and 371 mJ (specific values aregiven in the figure), using 10 ns long pulses focused down to a ∼100µm diameter (FWHM) spot.

µ

(a)

µ

(b)

Figure 8: Ion traces (a) and energy distribution of ions (b) from liquid tin droplets ablated by 10 ns pulses from anFC 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 dueto the oscilloscope being set to a shorter time range for those measurements. The increased amplitude of the noise isbecause 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. Asbefore, the total charge collected increases with increasing laser energy. It is compared with the other threeexperimental cases in Sec. (3.2.1). The ion signal fades much faster with time than for the other three

11

experimental cases studied, which is why here the x-axis only ranges 0–50µs m=1, contrasting the previousand following ranges, which span 0–250µs m=1. The sharp peaks seen in the TOF traces are also clearlyvisible in Fig. 8(b), where they still clearly shift in peak energy with increasing laser power. We clearlyobserve a sharp, distinct peak ion-energy for most of the recorded traces, which is unique when comparingthe four cases studied. The energies of these peaks are compared to the energies of fast ions from the otherexperiments in Sec. (3.2.2). Beyond this most-likely, peak energy, we find that the distribution decays veryrapidly, 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-longlaser pulses on tin droplets have been studied both experimentally [12, 17] and theoretically [1, 3, 4], butthese works have not focused on ion energy distributions. To the best of our knowledge only the works ofChen et al. [18] and Chen et al. [19] investigate this topic. In those works qualitatively similar ion TOFtraces are shown as in Fig. 8(a). It is interesting to point out that in these works the tin droplets had adiameter of 150µm, contrasting 45µm for the ns-on-droplet case in this document. We note that these workslack detail on the precise treatment of the FC data, such as on the data correction for electronic responsesas 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 inSec. (2.3), the noise floor follows a power-law curve ∝ t−3/2 (see 0.5 mJ gold-colored curve in Fig. 8(b)). Theheight of this curve increases with noise amplitude, which in turn increases with the voltage-scaling of theoscilloscope. This means that traces with higher amplitude have stronger noise as we rescaled the oscilloscopevoltage scale accordingly during the data taking. For laser energies ≥120 mJ there is an apparent cutoff inthe data left of energy ∼60 eV due to the oscilloscope being set to measure over a shorter time-range (thuslimiting the total recorded TOF). Further, in Fig. 8(a) there appears to be signal before 5µs m=1, which wasestablished to be pickup from the laser system. This noise is also visible as ringing in Fig. 8(b) at high ionenergy.

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, using4.5 ps-long pulses focused down to a ∼100µm diameter (FWHM) spot. Measurements are shown for laserenergies 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)), withthe difference between the two FCs decreasing with increasing laser energy due to a decreasing anisotropyparameter, as pointed out previously by Deuzeman et al. [5]. The FC(=2,0) TOF data show a simple distri-bution 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 shiftsquickly to higher ion energy for higher laser energy, with the low-energy peak much more slowly followinga 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 noisefaster than for FC(=30,0), because the former was further away from the target and as such had a lowersignal-to-noise ratio. In the TOF graphs, the measurements from both FCs appear to converge to a similardecay curve. For the 2.5 mJ measurement this happens around t = 70µs m=1, but convergence is reachedafter increasing times for decreasing laser energy. At more detailed discussion of the data presented here onthe 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 ∼95µm diameter (FWHM) spot. Measurements are shown for

12

µ

(a)

µ

(b)

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 andelectronic 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 havemore noise than the red because FC(=2,0) was further away from the target than FC(=30,0), reducing the availablesolid angle.

laser energies between 1 and 10 mJ; specific values are mentioned in the figure.

µ

(a)

µ

(b)

Figure 10: Ion traces (a) and energy distribution of ions (b) from liquid tin droplets ablated by 15 ps pulses for anFC at 30°. Data is corrected for distance between tin target and FC, opening angle of FC and electronic response, asdiscussed 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 graphsfeature a single, very fast peak, with an extremely short rise time (much shorter than for the ps-on-solidcase in Fig. 9(a)). This is naturally reflected in the energy distributions shown in Fig. 10(b), where there isstill significant signal for ion energies >100 keV for the highest pulse energies. The peaks in the TOF tracesdo not show up as distinct peaks in the energy distribution. The energy of the fast peak in the TOF traces

13

will be compared to the fast ions in the other experiments in Sec. (3.2.2). To the best of our knowledge, noexperimental 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 therecorded 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 psexperiments respectively. Nanosecond pulses generally, for similar laser fluence, generate more charge thanpicosecond pulses. Also, generally, more charge is produced from solid targets than on droplets, again takenat similar laser fluence. Apparent in (a) is that most data points follow a power law dependence on thelaser pulse energy, most beautifully apparent for the 10 ns-on-droplet case. This particular case shows apower law behavior that is very similar, especially regarding the apparent power-law energy “offset”, to thebehavior 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 laserenergy it cannot yet be confirmed whether a power law dependence will be reached. The power law beingoffset by a certain constant makes sense physically, as a certain threshold laser pulse energy is required tostart efficient ablation. As total charge yield depends on many variables (e.g. ablation volume, ionizationprocesses, plasma processes, laser-plasma interactions) it is difficult to formulate a model and there is no clearexpectation for the dependence of charge yield on laser energy yet. This is reflected in the available publicliterature, where we were unable to find any clear predictions. As we did not record ablated volumes, wealso cannot make any statements regarding the ionization fraction. There is experimental data on the chargeyield but varying conclusions are drawn from them, with Amoruso et al. [15] describing both a logarithmicand a power law scaling, Toftmann et al. [6] describing a linear scaling and Deuzeman et al. [5] describingan either logarithmic or linear dependence.

µ

(a)

µ

(b)

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

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3.2.2 Fast ion peak

In Sec. (3.1), we observe that picosecond pulses produce faster ions than nanosecond pulses, when comparedat similar laser fluence, both on solid and droplet targets. The fast peaks in the TOF graphs on thenanosecond-on-solid case seem to arrive at a time similar to the single feature found in the nanosecond-on-droplet case. Therefore, we hypothesize that they have similar physical origins. It is instructive to comparesuch features as a function of laser pulse energy. However, these features are not equally apparent, or evenquantifiable, in the energy distributions. Therefore, we infer a “peak energy” corresponding to the maximumof the fast features in the TOF traces (see Fig. 12). From Subfig. (a) we conclude that these apparently similarfeatures have very different absolute energies and scalings thereof with laser pulse energy. Correcting for thedifferent 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.

(a)

(b)

Figure 12: “Peak energy” of charge as inferred from TOF peaks for the fastest peaks visible for the four differentexperimental cases, as a function of laser pulse energy (a) and laser peak intensity (b). Peak intensities are calculatedby dividing peak fluence by FWHM pulse duration. The powers found from the fitting procedure are 1.1(1) for ns-on-solid, 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-on-droplet and ps-on-solid cases. A quick inspection of the TOF graphs Figs. 9 and 10, however, shows thatthe 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 onlythe FC(=30,0) data is considered, as the fast peaks (presumed to be there) are always indistinguishablefrom 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 pulseintensity (and energy) is still quite different. Here too, more detailed analysis is required to make moredefinite statements but this difference in scaling could well be due to plasma “vapor” absorption [5] thatis more relevant for the longer (ps-)pulse length and energies. This should become apparent with futureanalysis 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, wefit power laws to the data. We observe larger powers for the solid target experiments compared the dropletexperiments for both the ps and ns cases. Comparison with literature shows that both Hora et al. [20] andFahler and Krebs [21] find a linear dependence of peak ion energy of fast ions for ns pulses, which is close towhat we find for the ns-on-solid experiment. For ps pulses Hora et al. [20] claim that the ion peak energy is notdependent on laser energy, but this is not in line with observations from either ps experiment. In literature

15

both ambipolar-field diffusion and Coulomb explosion are mentioned as possible causes for the generationof high energy ions [5, 15, 22, 23, 24]. A Coulomb explosion occurs when electrons are removed from theirrespective ions in a tight region, after which the ions repel each other due to Coulomb interactions. Whenthe particles initially are closer together, the acceleration of the ions becomes greater. As such, Coulombexplosions can only occur at surfaces (also at liquid surfaces), while ambipolar-field diffusion can also bemaintained/enhanced when the plasma plume is illuminated by the persisting laser pulse. On the originsof fast ions in the ns cases, Farid et al. [16] claim that the front of the plasma plume is efficiently heatedthrough inverse bremsstrahlung, further accelerating the already energetic ions located there. While Changand Warner [25] also see that laser-plasma interaction further accelerates the plasma expansion, the rateof 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 casesstudied 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 principlesprovide reasonable explanation of the spectra over a broad ion-energy range.

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4 Experiment vs theory

In this chapter, we briefly introduce two theoretical models that describe the expansion of a plasma into thevacuum and compare these models to the four experimental cases studied in this thesis. Those works considerion energy distribution functions in a purely hydrodynamical framework (where the ion velocity is simplyidentified with the macroscopic hydro-velocity of a single-fluid quasi-neutral plasma, given some averagecharge state only) for a particular and well-known type of hydrodynamic self-similar solutions. Because thehydrodynamic solution is not augmented by the coupled solution of the equations of ionization/recombinationkinetics, the resulting hydrodynamic energy distribution function must be compared with the measureddistribution 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 theseparation of electrons from the ions in the quasi-neutral expansion of the plasma cannot be assumed to becomplete 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 measuredenergy distribution of either a single charge state, or of the total charge of all ions captured by a Faraday cupas is done in the following, may be very questionable. Also, the applicability of the hydrodynamic approachmay be very questionable itself.

Generally, ion energy spectra could be adequately inferred from the hydrodynamic density and velocity fieldswhenever the condition regarding typical length scales L λD is fulfilled, where L is a typical plasma lengthscale, and λD is the local value of the Debye radius [2]. The “hard”, high-energy tails of the ion energyspectra, particularly visible in the nanosecond-on-droplet case (see below), may originate from those partsof 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 theinteraction of a solid target with a short laser pulse (∼ ps), yielding the distribution function (“spectrum”)

dN

dE∝ 1√

Ee−√E/E0 (5)

with N the particle count and E0 a measure for the energy in the system ZkBTe with Z the charge state andTe the electron temperature. An example of such a spectrum is plotted in Fig. 13 (gray curve). Assumed isthat 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 interactionssuch as our ps-cases.

The second model, introduced by Murakami et al. [4], considers the hydrodynamic, isothermal expansion ofmass-limited plasmas with different geometries, ranging from infinite-planar to spherical cases. This modelconsiders laser-plasma heating and can thus be applied to cases involving longer pulse lengths such as ourns-cases. Also, this model may be relevant for the 15 ps-long pulses as significant plasma “vapor” absorptionis to be expected for these pulse lengths [5]. The model predict a ion energy distribution given by

dN

dE∝√Eα−2

e−E/E0 , (6)

with α the geometry of the plasma expansion (α = 1 for planar, 2 for cylindrical, 3 for spherical) and E0

similarly defined as above as 12mR

2 with ion mass m and plasma expansion rate R. For α = 3, Eq. (6)reduces to the well-known Maxwellian energy distribution and is the only model function here that canexplain a peaked ion energy distribution, with a maximum at E = 1

2E0. Examples of this spectrum areshown in Fig. 13 (colored curves); one curve is plotted for each of the mentioned values of α. When looking

17

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 ofMurakami’s model converge, because their similar exponential factor dominates there.

α α α

μ

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 anelectron temperature Te in units of eV,

Te = 27(A/Z)1/3λ4/3I2/3L , (7)

assuming unity absorption, where A is the ion mass number at charge state Z, λ is the laser wavelengthnormalized by 1µm, and IL is the laser intensity normalized by 10× 1011 W/cm2. This yield typical temper-atures 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] theauthors arrive at even higher E0 = 2−3 keV for plasma temperatures of 30−50 eV. This serves to give us afeeling 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 presentedabove.

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. Asdescribed above, we expect that the model of Murakami et al. [4] would be best suited to describe these dataand we use Eq. (6) to fit the four functions. The traces for low laser pulse energy have no clear maximumand 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 interpretedusing Eq. (5) In the work of Murakami et al. [4] very similar experimental traces are presented which are verywell fit by Eq. (6). Those experimental traces deal with Xe-LPPs but references to the original experimentalworks are incomplete and/or refer to unpublished work.

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µ

(a)

µ

(b)

Figure 14: Different data sets (points) for the ns-on-solid case (a) and the ns-on-droplet case with fits of Murakami’stheoretical model (dashed), Eq. (6). Fitting parameters are E0, α and the normalization constant. For each fit α is setto 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) eVfor 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) eVfor FC(=30,0), here α = 2 for both FCs. In (b), E0 = 3550(29) eV and 358(4) eV for the higher and lower laserenergy 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 byusing α = 3. This is a reasonable assumption as the relevant length scale of the plasma at these energies islarger 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 isstrongly dependent on the details of the chosen fit procedure (here, and in the following, we choose weightingproportional 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, itshould 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 inthe ion energy distribution seen for the nanosecond-on-droplet case (see below) and we thus expect similarphysical 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 placedat 30°. Visible inspection of the curves immediately makes us discard Mora’s model and the “plateau” visiblefor both cases at the low ion energies, is only described by any measure of accuracy by choosing α = 2 inMurakami’s model. Nevertheless, they are very poorly described by the model which is not at all able toexplain the sharp peak structure at high kinetic energy visible for all but lowest of laser pulse energies. Thispeak structure is followed by a very rapid drop-off toward higher energies that cannot be described by anyof the models. This apparent energy “cutoff” occurs could be compared (in future work) with theory worksdescribing 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 whichwould shed more light on the origins of the ion energy distribution and influence thereof.

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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 FCsplaced at 2° and 30° (see setup). The short nature of the laser-solid interaction would indicate that Mora’smodel, which does not take laser-plasma heating into account, is best suited here. We fit Eq. (5) to the fourdata sets and find reasonable agreement, especially in the case of the FC(=2,0) trace at the higher laser pulseenergy 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).

µ

(a)

µ

(b)

Figure 15: Different data sets (points) for the ps-on-solid case (a) and the ps-on-droplet case with fits of Mora’stheoretical 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 thehigher laser energy data is 400(11) eV for FC(=2,0) and 114(5) eV for FC(=30,0). For the lower laser energy fitsE0 = 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) eVand 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 (whichthus favor small anisotropy factors k) could be more sensitive to the more “thermal” part of the plasmaexpansion and not the fast ion front which typically flows along the surface normal [1, 2, 10]. We note thatthe 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 theFC. Again, most hydrodynamical models (as are discussed in this chapter) would predict an ion spectrumthat 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 placedat 2° and 30° (see setup). We chose here slightly different laser pulse fluence to facilitate comparisons withthe above picosecond-on-solid case which, at a shorter 4.5 ps pulse length, would otherwise have a higherintensity. As before, the short nature of the laser-solid interaction implies that Mora’s model, which doesnot take laser-plasma heating into account, is best suited here. It indeed fits very well to the two showndata sets. The low energy 1 mJ case yields a fit parameter E0 = 254(8) eV, the high energy 10 mJ case gives

20

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 errorsin the deconvolution procedure (see Sec. (2.3) and Appendix III). From simultaneous EUV-spectroscopicinvestigations, we understand that there is a significant ionization degree at least up to Z = 13. At the sametime, Eq. (7) implies a plasma electron temperatures scaling ∼100–1000 eV over the here studied range oflaser intensities IL = 6× 1011–7× 1012 W/cm2, in line with the found values for E0 but only when assumingunity value for Z. This is not in agreement with the EUV-spectral observations. A straightforward analysisis hampered not only by the missing information about the charge state distribution, but also by the non-flatspatial laser beam profile and curved droplet surface which means that we are actually dealing with a largespread in intensities. Nevertheless, we conclude that the shape of the ion energy distributions are very welldescribed 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 thedata for higher laser energy, but fails to do so for the higher ion-energy part. The model poorly describesthe ion energy distributions for the ns-on-droplet case. Mora’s model excellently describes the ps-on-dropletdata, and works in varying degrees to describe data from the ps-on-solid case. In the latter case, especiallytowards lower laser energy and greater angle from the target normal, Mora’s model fails to describe theexperimental results. This shows that the foundations of the theoretical approach to plasma expansion intothe 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 theexperiment and in the theory. In the experimental case, work is currently underway to use an electrostaticanalyzer with the ns-on-solid setup to gather charge state resolved ion data. Concerning theory, we notethat Kelly and Dreyfus [28] and Tallents [29] postulate, rather than derive, Maxwellian functions, one percharge state, to describe plasma expansion. Burdt et al. [30] have done charge-resolved measurements andfit such functions to their results, claiming good agreement between them. However, the underlying physicsis not yet understood fully.

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5 Conclusion

This thesis contains a study of ion energy distributions from laser-produced-plasmas, inferred from experi-mentally obtained time-of-flight Faraday cup traces, for four different experimental cases. These cases werenanosecond-on-solid, nanosecond-on-droplet, picosecond-on-solid, and picosecond-on-droplet. The data col-lection for the nanosecond-on-solid case was also part of this thesis, as well as development of a data analysismethod and application thereof to the data of all four cases. On comparison of the four cases, it is clearthat the ion energy distributions pertaining to them are not at all similar. Of these differences especially thedifference between the solid-target cases and the droplet cases is an important finding, as often the simplersolid-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 picosecond-pulsed experiment on the other. The foremost differences are that, at similar fluence, ps-pulses produce farfewer 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 twotheoretical 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 alwaysdescribe the data, however, indicating that the theory is not complete as of yet. Especially the apparentcutoff at large kinetic energies in the ns-on-droplet case is of particular practical importance and theoreticalinterest and should be further studied. Further comparisons between experimental and theoretical work canbe made when charge state resolved energy distributions are obtained, as additional material exists in whichcharge states are considered individually.

Acknowledgements

I would like to thank the EUV Plasma Dynamics group at ARCNL for hosting the work done for this thesisand for many insightful discussions. I also want to thank the ARCNL group EUV Generation and Imagingfor the use of their laser for studying the ps-on-droplet case. Furthermore I want to thank the AMOLFworkshop and ARCNL technicians for aid and technical support during the experiments and Mikhail Baskofor enlightening communications.

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Appendix I Derivation of data correction equation

Basic laws we can use are U = IR and I = CU .

After applying Kirchhoffs laws on our system we know the following:

URbias= −UFC (8)

UFC = UB + URC+ Uout (9)

Uout = UCC(10)

IRbias+ Iions = IB + IFC (11)

IB = Iout + ICC(12)

IB = IRC(13)

Using these equations, we want to find Iions as a function of system constants and the measured voltage,Uout

Iions = IB + IFC − IRbias (14)

Iout =Uout

Rosc(15)

ICC= CCUCC

(10)= CCUout

IB(12)= CCUout +

Uout

Rosc(16)

IFC = CFCUFC(9)= CFC(UB + URC

+ Uout)

= CFC

(IBCB

+RC IB + Uout

)(16)= CFC

(CCCB

Uout +Uout

CBRosc+RCCCUout +

RCUout

Rosc+ Uout

)(17)

IRbias =URbias

Rbias= − UFC

Rbias

using eq. 17 = − 1

Rbias

(CCCB

Uout +1

RoscCB

∫ t

0

Uout(τ) dτ+

+RCCCUout +RCRosc

Uout + Uout

)(18)

=⇒ Iions(t) =

(1

Rosc+CFC

CB

1

Rosc+

1

Rbias+

1

Rbias

CCCB

+RC

RbiasRosc

)Uout(t)+

+ (CFC + CC +CCCFC

CB+RCCFC

Rosc+RCCCRbias

)Uout(t)+

+ CFCCCRCUout +1

RbiasCBRosc

∫ t

0

Uout(τ) dτ (19)

25

Appendix II System parameters for the different experimental se-tups

Table 1: System parameters for the four experimental cases

solid droplet

ns Pulse energy: 0.8 – 400mJ Pulse energy: 0.5 – 371mJPulse duration: 6ns Pulse duration: 10nsBeam diameter: 90µm× 90µm Beam diameter: 100µ m× 105µ mPeak fluence: 8.717–4358 J/cm2 Peak fluence: 4.2–3119 J/cm2

VS and VFC : -100V, -60V VS and VFC : -100V, -45VRC-time: 104 W×2.46× 10=10 F RC-time: 100W×1.64× 10=10 FFC angles: (-2,0),(0,30),(-30,0) FC angles: 30° and 60°FC distance: 0.73,0.77,0.645 FC distance: 0.366,0.366FC opening angle: 5.3E-5,4.77E-5 FC opening angle: 2.11E-4

Droplet diameter: 45µmEnergy fraction on droplet: 0.72894

ps Pulse energy: 0.1 – 2.5mJ Pulse energy: 0.1 – 10mJPulse duration: 0.5 – 4.5ps Pulse duration: 15 – 105psBeam diameter: 105µm× 95µm Beam diameter: 100µm× 90µmPeak fluence: 0.885–22.12 J/cm2 Peak fluence: 0.981–98.1 J/cm2

VS and VFC : -100V, -30V VS and VFC : -60V, -20VRC-time: 104 W×1.64× 10=10 F RC-time: 104 W×1.64× 10=10 FFC angles: (-2,0),(0,30),(-30,0) FC angles: 30° and 60°FC distance: 0.24, 0.73, 0.26, 0.26 FC distance: 0.366,0.366FC opening angle: 4.91E-4, 5.3E-5, 4.18E-4 FC opening angle: 2.11E-4

Droplet diameter: 30µmEnergy fraction on droplet: 0.5383

26

Appendix III Effect of errors during data analysis

In Fig. 16(a) and (b) respectively energy distributions are shown for which the data is given a wrong offsetor a wrong value for the capacitance was used during correction with Eq. (2). This is to show the effects thatsmall mistakes in those corrections would have on the final ion energy distribution. In (a), correct offsetsthat are given to the data are of the order 0.04 V, so the wrongly shifted data have offsets that deviatefrom that ∼25 %. It is clear that most of the data appears unchanged when a wrong offset is applied. Thethe low-energy ion signal is significantly shifted, sufficiently so to explain the drop in the distribution thatneither theoretical model could explain. This means that great care has to be taken when considering themeasured behavior of low-energy ions. As expected, the noise floor also shifts with shifting offset. In (b),energy distributions are plotted for which, during data analysis, the value of the capacitance CC (as in Fig. 5)is varied by 25 %. This is the primary capacitance responsible for reshaping the data. Again the deviatingplots are very similar to the properly corrected plot. Upon closer inspection it does become clear that thehigh-energy feature shifts slightly with capacitance, also by ∼25 %. That a deviation in CC value results ina similar deviation in the high-energy feature can be readily understood from Eq. (2), where CC is a leadingfactor for both the U and U terms, which dominate the correction at high energy.

µ

(a)

µ

(b)

Figure 16: Energy distribution plots made with different (wrong) offsets to the measured voltage (a) and with different(wrong) values for the capacitance used for correcting the data using Eq. (2) (b). This is to the end of finding out howa mistake during the process of data-correction influences the data.

27

Appendix IV Necessity of data correction

In Fig. 17 ion energy distributions are shown for which the data has not (a) or has (b) been correctedas discussed in Sec. (2.3). Noticeable is that without offset correction, the height of the noise floors differsignificantly from the plots that are offset corrected. In (a), the energy distributions all shift to lower energies,since the data are not corrected for signal delay caused by the “RC-effect”. This also causes some featuresin the distributions to be washed out, especially towards high energy (e.g., the “shoulder” at high energy inthe blue curve). Since the Eq. (2) has not been applied to the data, the traces in (a) are also smoother thanin (b).

µ

(a)

µ

(b)

Figure 17: Energy distribution plots for the four experiments, without (a) and with (b) the corrections discussed inSec. (2.3) applied to the data.

28