Oscillatory regimes of Langmuir probe current in femtosecond laser-produced plasmas: Experimental and theoretical investigations

Accepted Manuscript Oscillatory regimes of Langmuir probe current in femtosecond laser-produced plasmas: Experimental and theoretical investigations

P. Nica, S. Gurlui, M. Agop, C. Focsa PII: DOI: Reference:

S0169-4332(19)30718-4 https://doi.org/10.1016/j.apsusc.2019.03.098 APSUSC 42053

To appear in:

Applied Surface Science

Received date: Revised date: Accepted date:

19 December 2018 7 March 2019 11 March 2019

Please cite this article as: P. Nica, S. Gurlui, M. Agop, et al., Oscillatory regimes of Langmuir probe current in femtosecond laser-produced plasmas: Experimental and theoretical investigations, Applied Surface Science, https://doi.org/10.1016/ j.apsusc.2019.03.098

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ACCEPTED MANUSCRIPT Oscillatory regimes of Langmuir probe current in femtosecond laserproduced plasmas: experimental and theoretical investigations P. Nica1,*, S. Gurlui2, M. Agop1, and C. Focsa3 1

Department of Physics, Technical “Gh. Asachi” University, Iasi, 700050, Romania Faculty of Physics, “Al. I. Cuza” University, 11 Blvd. Carol I, Iasi 700506, Romania 3 Univ. Lille, CNRS, UMR 8523, PhLAM – Physique des Lasers, Atomes et Molécules, CERLA – Centre d’Etudes et de Recherches Lasers et Applications, F-59000 Lille, France *Corresponding author: [email protected] 2

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Abstract We investigate the oscillatory regimes of the Langmuir probe current recorded for the electrical characterization of femtosecond laser-produced plasma. The influence of metallic target biasing, which can perturb the ambipolar electric field generated through the charge separation at early stages of the expansion, is also studied. Two distinct behaviors are evidenced, and they point to the existence of two plasma structures: a fast one consisting in promptly ejected highly charged particles, and a slow “tail” of thermalized particles. Theoretically, a non-differentiable model which involves two scale resolutions is developed. Assuming the principle of scale resolution superposition, the current density at global scale is expressed as the sum of the current density at Coulomb and the thermal scale resolutions, in agreement with the experimental data. Interestingly, the theoretical estimation of oscillation frequencies shows a correspondence with the filling factor hierarchy in the fractional quantum Hall effect.

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1. Introduction The description of the fundamental physical processes involved in ultrafast laser interaction with various targets and in subsequent plasma plume evolution plays a key role in the development and improvement of a high number of applications [1–5]. In the particular case of pulsed laser deposition (PLD) [6–8], one wants to measure the plasma parameters (density, temperature, charge state, expansion velocity space-time distribution, kinetic energy) in order to correlate them with the properties of the deposited material (chemical structure, thickness, crystallization). The investigation methods which are usually employed are divided in optical (fast photography, optical emission and absorption spectroscopies, shadowgraphy, interferometry, laser induced fluorescence) non-intrusive methods, and electrical diagnostics (Langmuir probes, Faraday cups, electrostatic analyzers) which are locally perturbing the plasma to separate various electric charges [9]. Among them, the Langmuir probe (LP) technique, consisting in recording the temporal trace of the current for various biasing voltages of a metallic electrode, is one of the simplest from constructive point of view. However, the interpretation of the experimental results is difficult, although the theoretical description has been the subject of many papers in the past years. When applied to transient plasma, one must consider that its parameters are space-time dependent and the expansion velocity field has a non-uniform space distribution. Even in these circumstances, the LP method remains a useful tool in studying laserproduced plasma [10–12], when special caution is taken, e.g. to avoid perturbing simultaneously recorded optical emission [13]. Moreover, if the LP results are compared/completed with the ones obtained through complementary methods [14], further benefits can be gathered. In some of our previous works [15,16], the LP technique was used to characterize the nanosecond laser ablation plume. An oscillatory/multi-peak feature of the recorded current has been generally observed, for various axial and radial distances from the target, various laser fluences, probe biases and for different target materials. In our attempt to find if this is a possible artifact of the recording electrical circuit, we observed that the “arrival time” of the first peak in the time-of-flight current profile is decreasing with the increase of the laser beam energy/pulse and with the decrease of the target-probe distance [15]. Moreover, at certain distances the probe signal becomes smooth (i.e. non-oscillating), while measuring the target current, oscillations are observed for the ionic part [17]. The most significant confirmation that such oscillations are a genuine feature of expanding plume fluctuation was obtained from the comparison with the optical observations [18]. Several papers reported similar oscillations [13,19,20], still their physical interpretation is a matter of debate (see [18] 1

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for a recent review of the field). More recently, we extended our studies to the oscillatory regimes of femtosecond laser-produced plasma [21,22]. From a theoretical point of view, the usual models employed to study laser-produced plasma dynamics (e.g. hydrodynamic or kinetic models) are based on the assumption of the differentiability of the physical quantities that describe it [23–28]. Since the dynamics of ablation plasma is often characterized by instabilities (implying chaotic behaviors and self-structuring), the differentiable procedures become operable only sequentially. The “real” descriptions of dynamics can thus take benefit of non-differentiable procedures, for example in the frame of Scale Relativity Theory (SRT) of constant or arbitrary fractal dimension. Such approach, developed in our previous works [15,17,29], allowed us to infer the occurrence of Coulombian and thermal type behaviors in the dynamics of ablation plasma, and to obtain various space-time dependencies of plasma parameters, such as temperature, concentration, velocity, etc. In the present paper we report two oscillatory regimes of the LP current recorded for the electrical characterization of femtosecond laser-produced plasmas. The influence of the metallic target biasing (which can perturb the ambipolar electric field generated through the charge separation at early stages of expansion) on these regimes is also investigated. The occurrence of these oscillation modes is discussed in correspondence with the two plasma structures which are now generally accepted as being formed during the expansion process [18]. The fractal model of motion, in the frame of SRT with constant and arbitrary fractal dimension, is used to explain the oscillatory regimes by considering not only the space- and time-evolution of the probe current, but also by introducing its dependence on the scale resolution.

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2. Experimental results The experimental set-up, developed mainly for analytical purposes, is described in detail elsewhere [15,18,21,22,30]. Briefly, the experiments have been performed in a stainless-steel vacuum chamber at 10-5 Torr residual pressure. The Ti:Sa femtosecond laser beam (800 nm, 60 fs, 100 Hz repetition rate, 1.7 mJ/pulse) has been focused by a f = 25 cm lens at quasi-normal incidence onto metallic targets placed in the vacuum chamber on an electrically isolated X-Y-Z- micrometric stage. The estimated spot diameter at the impact point has been ~160 µm. Thus the laser irradiance resulted to be 𝐼𝐿 = 0.15 𝑃W/cm2. The current extracted from the plasma plume was measured by a cylindrical Langmuir probe made of stainless steel with 0.8 mm diameter and 5 mm length. It has been placed perpendicular to the plume expansion direction, on the plume axis at 3.5 mm from the target surface. The LP and the target were biased with stabilized DC power sources, at the voltages 𝑉𝑃 and 𝑉𝑇 , respectively. The transitory signals of the probe have been recorded by a digital 600 MHz, 2.5 GS/s oscilloscope (LeCroy, Wave Surfer 62XS), using 50 impedance. The use of low input impedance is better suited for recording fast phenomena. We note however that the oscillatory regimes can be also observed at high (1M) input impedance, as one can see from the insets in Figures 2 and 3 of Ref. [22]. Moreover, a deconvolution procedure allows extracting the fast response from the highimpedance signals, which then matches the temporal traces recorded with 50 impedance (see Figure 4 of Ref. [22]). In Figure 1a the typical temporal evolution of the LP current recorded by the negatively biased probe, VP=-35V, is given for various values of the positive biasing voltages of an Al target, 𝑉𝑇 . One can observe the oscillatory behavior, similar to what was previously reported for ns-laser ablation in our works [15,18]. The net collected charge (time-integrated) resulted to be positive and significantly increased with the target voltage, as expected from the excess of ions due to electrons repelling by the probe negative potential [22]. Regarding the time at which the maximum probe current is recorded, one observes that it remains almost unchanged by the target bias. This result can be discussed in the assumption of a shifted Maxwell-Boltzmann (SMB) velocity distribution of the expanding cloud [22,31], 𝑓(𝑣) ∝ 𝐴 𝑣 3 exp[−𝑚𝑖 (𝑣 − 𝑣𝐷 )2 /2𝑘𝐵 𝑇𝑖 ] (1) with A a constant, 𝑣 the particle velocity, mi the ion mass, vD the drift velocity, 𝑘𝐵 the Boltzmann constant, and 𝑇𝑖 the ion temperature. Eq. (1) leads to a time-dependence of the probe current in the form, 𝐼(𝑡) ∝ 𝐴 𝑡 −𝑛 exp[−𝑚𝑖 (𝑑/𝑡 − 𝑣𝐷 )2 /2𝑘𝐵 𝑇𝑖 ] (2) 2

ACCEPTED MANUSCRIPT where 𝑑 is the target-probe distance and n a parameter which depends on the charge collector geometry and distance from the target [31,32]. The current reaches the maximum at 𝑡𝑚𝑎𝑥 = 𝜏 (−1 + √1 + 1 𝑑 𝑣𝐷 𝑚𝑖 . 2𝑘𝐵 𝑇𝑖

with 𝜏 = 𝑛

2𝑑⁄𝜏 ) 𝑣𝐷

(3)

Having in view the previous experimental observation, we conclude in this context

that the target biasing is not influencing the drift velocity and plasma temperature, even if it perturbs the charge separation at the initial stages of plasma formation. 7

Al target, VP=-35V

Numerical fit sine damped

VT=+19,5 V

2 1

VT=+35 V

0 -1 0.0

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1.5

Fourier transform of oscillatory part

 =8.3 MHz ' =6.4 MHz

=8.8 MHz

0.06 0.04 0.02

' =6.4 MHz

VT=+19.5 V VT=+35 V

=8.4 MHz

' =6.6 MHz ' =6.4 MHz

VT=+46 V VT=+55 V

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Figure 1: The typical temporal evolution of the current recorded by the negatively biased probe, VP=-35V (a), oscillatory part obtained by subtracting the continuous part (b), its Fourier transform (c), and frequencies obtained for various target biases VT (d), for fs laser-produced Al plasma.

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The current oscillations evidenced in Figure 1a are superimposed on a continuous part. After removing this continuous contribution [33], the oscillations displayed in Figure 1b are isolated, and they can be analyzed in terms of an usual damped oscillator. Then, the total probe signal is well described by the function, 𝐼(𝑡) ∝ 𝐴 𝑡 −𝑛 𝑒𝑥𝑝[−𝑚𝑖 (𝑑/𝑡 − 𝑣𝐷 )2 /2𝑘𝐵 𝑇𝑖 ] + 𝐵 𝑒𝑥𝑝[−𝛾𝑡] 𝑠𝑖𝑛[2𝜋𝜈𝑡 − 𝜑] (4) where the first term is the continuous part as determined by the SMB distribution function (2), and the second one corresponds to the superimposed damped oscillator, with B a constant, 𝛾 the damping coefficient connected with the collisional processes, 𝜈 = √𝜈02 − (𝛾/2𝜋)2 the oscillation frequency, 𝜈0 the undamped resonant frequency connected with the plasma ion frequency, and 𝜑 the phase given by the target-LP distance and the expansion velocity. Let us now discuss the numerical fitting with sine damped function of the extracted oscillatory part, for various potentials applied on the target (VT) – see Figure 1b). We observed a good accuracy for low and high values, e.g. 𝑉𝑇 =+1.6V and 𝑉𝑇 =+35V, resulting 𝜈=8.37 MHz and 𝜈′=6.46 MHz. For intermediary values, e.g. 𝑉𝑇 =+19.5 V, we distinguished two parts: i) at early recording times, (0-0.4) µs, a fast plasma structure oscillating with a frequency of 𝜈 =8.93 MHz, which is similar with the previous value for 𝑉𝑇 =+1.6V; ii) a slow structure for longer times (>0.4 µs), oscillating with a frequency of 𝜈′ =6.74 MHz, which is similar to the previous value for VT=+35V. This is in agreement with the existence of two types of particles, i.e. plasma structures (often reported in literature [31,34– 39]) being formed during the initial stages of the expansion process: a fast (hot) one consisting in highly charged particles ejected at the beginning, in an expansion process characterized by the 3

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acceleration through the electric field given by initial charge separation, and a slow (cold) tail consisting in thermalized particles with low average charge state. Such two-structures are well-known in literature for both fs- and ns-laser ablation, being observed back in the late ‘70s (see e.g. Ref. [40]). Usually an order of magnitude difference exists between their expansion velocities [41]. In our opinion, the influence of metallic target biasing on the oscillatory regimes consists in perturbing the ambipolar electric field generated through the charge separation at early stages of expansion [42,43]. From the previous numerical fittings, also the damping constants were obtained. They resulted to be of the same order of magnitude as the oscillation frequencies and they increase almost linearly with the target voltage, from 𝛾=5.8 MHz at 𝑉𝑇 = 0V to 𝛾=8.4 MHz at 𝑉𝑇 =+64V. A similar behavior was observed for the amplitude 𝐵 of Eq. (4), thus allowing to conclude again the influence of the target voltage on the charge separation and not on the drift velocity or on the plasma temperature. This conclusion is also supported by the connection between 𝛾 and the collision processes. Following a procedure described in [33], the Fast Fourier Transform (FFT) of the probe current was computed, and it confirms the existence of two oscillatory regimes, identified as the two amplitude peaks in the FFT spectra (Figure 1c), which are evidenced differently at various target voltages. Their frequencies are plotted in Figure 1d vs. VT, as they were obtained from the above mentioned damped sine curves and from Fourier transforms, as full and empty marks, respectively. Let us observe that both methods gave similar results, two groups being computed, of average values 𝜈=8.5 MHz and 𝜈′=6.5 MHz, and showing only a weak evolution with the probe biasing. Let us further investigate if the previous features are specific to Al target, or if they are more general, and can be extended to different types of target materials. Thus, for Cu target, using identical experimental conditions and data treatment as previously discussed – see Figures 2a)-d) that are in correspondence with Figures 1a)-d), a similar behavior is observed through the existence of two oscillating structures. At early recording times, the probe current oscillates with a higher frequency (𝜈=10.5 MHz), while at longer times it is reduced to 𝜈′=6.4 MHz (Figure 2a). We note that these are averaged values, obtained both from the numerical damped sine fitting procedure (Figure 2b) and from Fourier transforms (Figure 2c), for various target biasing (Figure 2d), and they correspond to the fast and slow plasma structures.

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=11.0 MHz

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=10.6 MHz  =9.6 MHz

' =6.6 MHz

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Frequency (MHz)

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Figure 2: The temporal evolution of the current recorded by the negatively biased probe, VP=-35V (a), oscillatory part obtained by subtracting the continuous part (b), its Fourier transform (c), and frequencies obtained for various target biases VT (d), for fs laser-produced Cu plasma.

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VT= -1.6V VT= -9.5V VT= -54.2V

1.0

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Fast plasma structure

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Particularly for Cu target, both oscillations are simultaneously observed in the probe temporal traces, regardless of the probe biasing (see Figures 2a-c) used in the present measurements. Such behavior is slightly different when comparing with the case of Al target, where they were recorded only for intermediate values, VT=+(19÷35) V. In our opinion such difference can be explained by the fact that average charge state of Cu plasma is lower than in the case of Al plasma (for details see Ref. [22]). Moreover, Cu ions are heavier and consequently their initial motion in the internal electrical field due to the separating electrons [42,44] is less influenced by the target bias. An additional experimental confirmation of two-oscillating structures results from recording the probe current at negative biases of both the probe (𝑉𝑃 = −30 𝑉) and the Al target (Figure 3). We observed that the positive oscillations corresponding to the fast part, recorded for target potential higher than the probe’s one (e.g. 𝑉𝑇 = −1.6 𝑉 and 𝑉𝑇 = −9.5 𝑉), are mirrored in the negative domain when the target bias becomes with respect to 𝑉𝑃 almost symmetrically lower (e.g. 𝑉𝑇 = −63.6 𝑉 and 𝑉𝑇 = −54.2 𝑉). Meantime, the oscillations corresponding to the slow plasma structure are preserved in the positive current domain, showing that this part is not influenced by the target potential due to the dominating thermal processes characterizing the ejection of these low average charge state particles.

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Figure 3: Temporal evolution of the current recorded by the negatively biased probe, VP=-30V, for various negative Al target potentials VT

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We note that these two oscillatory modes can be evidenced also from a more careful analysis of experimental data reported by us in [33] for ns-laser ablation. In this previous work, we observed that the numerical fitting failed in the final part of the probe temporal trace in the case of Al target, while the superposition of two frequencies was inferred for Cu target. Other authors reported in [20] that the oscillatory behavior manifests differently in the fast and the slow parts of the LP signal, and a range of probe-target distances where oscillations can be recorded was established. When using Ni target [45], the LP temporal trace has a multi-peak structure that can now be considered as a result of an overlapping of two oscillation modes, while for multi-component targets this leads to an even more complex signal [33]. Let us now discuss possible correlations with plasma parameters. In [33] we assumed that the ion equation of motion in the electric field near the probe is x   ei x   2pi x  0 , where f pi   pi / 2 is the plasma ion frequency and the dissipative term is given by the electron-ion collision frequency, 𝜈𝑒𝑖 [46,47] , 6 2 3/ 2 (5a,b) [Hz] f pi  210 z ni / A [Hz] ,  ei ~ 1.5  10 z ne ln Λ / Te with ni [cm-3] the ion density, Te [eV] the electron temperature, A [a.m.u] the atomic mass, ln Λ the Coulomb logarithm, and z the ion charge state. At several millimeters from the target, ion densities of the order of 𝑛𝑖 =1012 cm-3 are deduced from the constants of the electric recording circuit, probe area and expansion velocity (obtained from the time of flight), the agreement with other studies being detailed in [21]. Considering this value for ion density, singly charged ions and Al target, using (5a) it results 𝑓𝑝𝑖 =40.4 MHz, while for Cu target one obtains 𝑓𝑝𝑖 =26.25 MHz. These values are higher than the previous experimental data of 𝜈 or 𝜈′, even when comparing with the undamped resonant frequency, 𝜈0 . Meantime, using (2b) for an electron temperature of 1 eV, similar electron density, 𝑛𝑒 , and ion charge state, 𝑧, and assuming a commonly used value for Coulomb logarithm, ln Λ=10 [48], it 5

ACCEPTED MANUSCRIPT results 𝜈𝑒𝑖 =15 MHz, which is of same order of magnitude as the experimental value of 2𝛾 resulted from the damping parameter. In our opinion, in such comparisons it is difficult to reveal an exact match due to the following reasons: i) plasma density and temperature are space-time dependent at similar scales with current oscillations; ii) Eqs. (2a,b) are deduced for ideal conditions and through (2b) only electron-ion collisions are considered; iii) extraction of the LP signal oscillatory part implies additional errors, as well as the fitting procedure due the high number of parameters to be computed. However, such model remains useful because it can predict a particular oscillation regime and it gives some associated thresholds. It is well known that for 𝛾 > 2𝜋𝜈0 the system becomes overdamped, the oscillation vanishes, and we conclude that their manifestation depends on the characteristic times of the electrostatic (elastic) and collisional processes.

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3. Theoretical aspects In the following, the fractal model of motion, in the form of SRT with arbitrary and constant fractal dimension [15,49] is used to explain theoretically the oscillation modes of an ablation plasma. We mention that the fundamental hypothesis of the model is that the plasma particle motion is taking place on continuous and non-differentiable (fractal) curves, which implies: i) instead of working with a dynamic variable, described through a strictly non-differentiable mathematical function, we will just work with a set of approximations of that function, derived through its averaging at various scale resolutions. Consequently, every dynamic variable acts as the limit of a functions family, the function being non-differentiable for a null scale resolution and differentiable for a nonzero scale resolution [50]; ii) any dynamic variable depends not only on the space and time coordinates, but also on scale resolutions; iii) motions constrained (in a classical differential approach) on continuous and differentiable curves in an Euclidian space, are replaced with free motions, without any constrains, on continuous, non-differentiable curves in a fractal space. Thus, the motion curves have double identity: both geodesics of the fractal space and streamlines of a fractal fluid; iv) for large time resolution scales (with respect to the inverse of the highest Lyapunov exponent [51]), the deterministic trajectories are replaced by a collection of potential routes, so that the concept of “definite positions” is substituted by that of an ensemble of positions having a definite probability density; v) new principles become “operable” in the fractal theory of motion, among them being the principles of scale covariance (the physics laws remain unchanged not only at space-time transformations, but also at those of scale resolutions), the principle of scale resolutions superposition (any dynamic variable at global scale resolution is expressed through a sum of equivalent variables at various local scale resolutions), etc [50]. As an explicit application of scale resolutions superposition principle, we show below that, in the fractal theory of motion in the frame of SRT with arbitrary and constant fractal dimension [49], the oscillation modes of an ablation plasma are the manifestation of the dependence of variables describing the dynamics (particularly of probe current) on the scale resolution. Indeed, since in expansion of LPP we consider two “stages”, the Coulomb repulsive and the thermal one, this is equivalent with the existence of a Coulomb scale resolution and a thermal scale resolution. Thus, any global dynamic variable is expressed through a sum of two local equivalent variables, of Coulomb and thermal types, respectively. Let us first characterize the LPP for each scale resolution: i) Behavior at Coulomb scale resolution: Let us assume that the dynamics of LPP at Coulomb scale resolution is given by the spacetime variations of the electrical component E of the local electromagnetic field induced by charge separation, through the equation [52], 𝜕2 𝑬

or, using Ohm’s law,

𝜕𝒋

𝜇𝜀 𝜕𝑡 2 + 𝜇 𝜕𝑡 = Δ𝑬 − ∇(∇ ∙ 𝑬),

𝒋 = 𝜎𝑬 by the space-time dependence of the current density, j, which is given by the equation: 𝜇𝜀 𝜕2 𝒋 𝜎 𝜕𝑡 2

𝜕𝒋

1

1

(6) (7)

+ 𝜇 𝜕𝑡 = 𝜎 Δ𝒋 − 𝜎 ∇(∇ ∙ 𝒋). (8) where 𝜇,  and  are the magnetic permeability, the electrical permittivity, and the electrical conductivity of the ablation plasma, respectively. 6

ACCEPTED MANUSCRIPT We now consider that in Eq. (5) the variables separation method can be applied (for details see [53]) in the form, ̅ (𝒓) 𝒋(𝒓, 𝑡) = 𝑇̅(𝑡)𝑹 (9) and thus this equation takes the form, 𝜇𝜀 𝑑 2 𝑇̅ 𝜎 𝑑𝑡 2

𝑑𝑇̅

+ 𝜇 𝑑𝑡 + 𝛼𝑇̅ = 0

1 ̅ − 1 ∇(∇ ∙ Δ𝑹 𝜎 𝜎

(10)

̅ ) + 𝛼𝑹 ̅ =0 𝑹

(11)

𝜎 𝜀

= 2𝛾, Ω = √Ω0 2 − 𝛾 2 , Ω0 2 =

𝛼𝜎 𝜇𝜀

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where 𝛼 = 𝑐𝑜𝑛𝑠𝑡. > 0 is a constant (named constant of variable separation in the general theory of mathematical physics equations [53]). The solution of Eq. (10) has the form, 𝑇̅ = 𝑇̅0 𝑒𝑥𝑝(−𝛾𝑡)sin(Ω𝑡 + 𝜑) (12) with , 𝑇̅0 = 𝑐𝑜𝑛𝑠𝑡., 𝜑 = 𝑐𝑜𝑛𝑠𝑡., Ω0 > 𝛾

(13a-f)

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meaning that the current density has a damped oscillator behavior. The value of  constant is generally given through initial and boundary conditions imposed by various symmetries of the ablation plasma expansion. We note that such approach is not singular, and for example diffusion modes can be imposed through the constant for separating the variables in the general theory of diffusion [53].

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ii) Behavior at thermal scale resolution: Let us assume that the dynamics of LPP at thermal scale resolution is given by its fractal behavior. Then, such dynamics is described by the fractal hydrodynamics equations system, which in the one-dimensional case takes the form [29]: 𝜆2

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𝜕𝑡 𝑉𝐷 + 𝑉𝐷 𝜕𝑥 𝑉𝐷 = −𝜕𝑥 [− (𝑑𝑡)(4/𝐷𝐹 )−2 𝜌−1/2 𝜕𝑥 𝜕𝑥 𝜌1/2 ] (14) 2 𝜕𝑡 𝑉𝐷 + 𝜕𝑥 (𝜌𝑉𝐷 ) = 0 (15) The Eqs. (14) and (15) correspond to the specific momentum and state density conservation laws, respectively. The variable 𝑉𝐷 denotes the differential velocity which is not dependent on the scale resolution, 𝑑𝑡, while 𝜌 is the state density that, through the fractal velocity 𝑉𝐹 , 𝑉𝐹 = 𝜆(𝑑𝑡)(2/𝐷𝐹 )−1 𝜕𝑥 𝑙𝑛𝜌 (16) becomes dependent on the scale resolution. 𝜆 is a parameter associated with the fractal – non-fractal transition, 𝐷𝐹 is the fractal dimension of motion path [54], and 𝜆2

𝜆(𝑑𝑡)(2/𝐷𝐹 )−1

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𝑄(𝑥) = − 2 (𝑑𝑡)(4/𝐷𝐹 )−2 𝜌−1/2 𝜕𝑥 𝜕𝑥 𝜌1/2 = −𝑉𝐹 2 − 𝜕𝑥 𝑉𝐹 (17) 2 is the specific fractal potential that induces the specific fractal force: 𝐹(𝑥) = −𝜕𝑥 𝑄 (18) Following the procedure given in [29], the solution of the equation system (14,15), for the initial conditions: 𝑉𝐷 (𝑥, 𝑡 = 0) = 𝑉0 = 𝑐𝑜𝑛𝑠𝑡. (19a) 1 (−𝑥/𝛼)2 𝜌(𝑥, 𝑡 = 0) = 𝛼 𝜋 𝑒 = 𝜌0 (𝑥) (19b) √ and the boundary ones: 𝑉𝐷 (𝑥 = 𝑉0 𝑡, 𝑡) = 𝑉0 , (20a) 𝜌(𝑥 = ∞, 𝑡) = 𝜌(𝑥 = −∞, 𝑡) = 0 (20b) takes the form, 𝜌(𝑥, 𝑡) =

1

∙ 𝑒𝑥𝑝 [−

1/2 𝑡 2 𝜋1/2 [𝛼 2 +𝜆2 (𝑑𝑡)(4/𝐷𝐹 )−2 ( ) ] 𝛼 𝑥𝑡 𝑉0 𝛼 2 +𝜆2 (𝑑𝑡)(4/𝐷𝐹 )−2 2 𝛼 𝑡 2 𝛼 2 +𝜆2 (𝑑𝑡)(4/𝐷𝐹 )−2 ( ) 𝛼

𝑣(𝑥, 𝑡) =

Using these equations, the current density becomes:

7

(𝑥−𝑉0 𝑡)2 𝑡 2 𝛼

𝛼 2 +𝜆2 (𝑑𝑡)(4/𝐷𝐹 )−2 ( )

]

(21) (22)

ACCEPTED MANUSCRIPT 𝑥𝑡

𝑗𝐷 = 𝜌𝑉𝐷 =

𝑉0 𝛼 2 +𝜆2 (𝑑𝑡)(4/𝐷𝐹 )−2 2 𝛼 3/2 𝑡 2 𝜋1/2 [𝛼 2 +𝜆2 (𝑑𝑡)(4/𝐷𝐹 )−2 ( ) ] 𝛼

∙ 𝑒𝑥𝑝 [−

(𝑥−𝑉0 𝑡)2 𝑡 2 𝛼

𝛼 2 +𝜆2 (𝑑𝑡)(4/𝐷𝐹 )−2 ( )

]

(23)

Imposing now in Eq. (23) the constraints: 𝛼2 𝜆(𝑑𝑡)(2/𝐷𝐹 )−1

𝑉0 𝛼2 [ ] (2/𝐷 𝑥 𝜆(𝑑𝑡) 𝐹 )−1

≪𝑡≪

2

,

(24)

it takes the approximate form, 𝑉 𝛼5

1

2

2 𝑥

𝛼

0 𝑗𝐷 ≈ 𝜋1/2 𝜆3 (𝑑𝑡) (6/𝐷𝐹 )−3 𝑡 3 𝑒𝑥𝑝 {− [𝜆(𝑑𝑡)(2/𝐷𝐹 )−1 ] ( 𝑡 − 𝑉0 ) }

(25)

From here, by identifying 2

𝑥 ≡ 𝑑, 𝑉0 ≡ 𝑣𝐷 , [

(

𝜆(𝑑𝑡)

2 )−1 𝐷𝐹

𝑚𝑖 𝐵 𝑇𝑖

] ≡ 2𝑘

it results the time-dependence of the ion current density in the form, 𝑚𝑖 𝑑 ( 𝐵 𝑇𝑖 𝑡

2

− 𝑣𝐷 ) ]

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1

𝑗(𝑡)~ 𝑡 3 𝑒𝑥𝑝 [− 2𝑘

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𝛼

(26a-c))

(27)

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where 𝑇, 𝑣𝐷 , 𝑚𝑖 , 𝑑, and 𝑘𝐵 have the same physical meaning as in (2). This result, identical with the often used one, for example in [31], specifies, indirectly through (26c), that the parameter of the Gaussian (19b) is directly proportional with the fractalization “degree”, 𝜆(𝑑𝑡)(2/𝐷𝐹 )−1, and inversely proportional with the most probable speed, 𝑣𝑝 = (2𝑘𝐵 𝑇𝑖 /𝑚𝑖 )1/2 , i.e. 𝛼 = [𝜆(𝑑𝑡)(2/𝐷𝐹 )−1 ]/𝑣𝑝 .

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iii) Behavior at global scale resolution: Assuming now the principle of the scale resolution superposition, the current density at global scale resolution is expressed as the sum of the current density at Coulomb scale resolution, given by Eq. (9), and the one at thermal scale resolution, given by Eq. (23). It results 𝐵

𝑚𝑖 𝑑 (𝑡 𝑇 𝐵 𝑖

𝑗(𝑡) = 𝐴𝑒𝑥𝑝(−𝛾𝑡) sin(Ω𝑡 + 𝜑) + 𝑡 3 𝑒𝑥𝑝 [− 2𝑘

2

− 𝑣𝐷 ) ]

(28)

D

with 𝐴 and 𝐵 arbitrary constants, which is similar with the experimental result given by (4). Since the thermal energy of the particle is proportional with the “chaos quanta” (for details see [49,50]), (

2

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𝑘𝐵 𝑇𝑖 = 2𝑚𝑖 𝜆(𝑑𝑡) 𝐷𝐹 𝜈 (29) with  the oscillation frequency of ions and, moreover, the coefficient associated with the fractal – non-fractal transition is given by collisions, 𝑘 𝑇 𝜆 = 𝑚𝐵𝜈 𝑖 (30) 𝑖 𝑐

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with 𝜈𝑐 the global frequency characterizing the collision processes, then from (29) and (30) it results the ion oscillation frequency in the form 𝜈 𝜈 = 2𝑐 (𝑑𝑡)1−(2/𝐷𝐹 ) (31) Through the fractalization parameter, 1

𝑓(𝑑𝑡, 𝐷𝐹 ) = 2 (𝑑𝑡)1−(2/𝐷𝐹 ) (32) one can describe “homogeneous” dynamics, characterized by only one global 𝐷𝐹 , that has the same properties at any resolution scale (mono-fractal dynamics). We note that “non-homogeneous” dynamics can also exist, their “scaling” properties being described by various 𝐷𝐹 values which quantify local singularities, so that these dynamics can be characterized only by local values of 𝐷𝐹 parameter (multi-fractal dynamics). For such multi-fractal structures one define dynamics singularity spectrum, 𝐹(𝐷𝐹 ), having the significance of a function for which a given 𝐷𝐹 parameter specifies a particular class of dynamics (for details see [51]). Then, the fractalization parameter (32), written in the form, 1 𝑓[𝑑𝑡, 𝐹(𝐷𝐹 )] = 2 (𝑑𝑡)1−[2/𝐹(𝐷𝐹 )] (33) and connected with continued fraction of type

8

ACCEPTED MANUSCRIPT 𝑓[𝑑𝑡, 𝐹(𝐷𝐹 )] →

1 𝑚+

𝑝1 + 𝑝2 +

(34)

𝛼1 𝛼2 𝛼3 𝛼4 𝑝3 + 𝛼 ⋱ 𝑝𝑛−1 + 𝑛 𝑝𝑛

with

7 3 5 3 5 3

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𝑚 = 1,3,5, …, 𝛼𝑖 = ±1, and 𝑝𝑖 = 2,4,6, … (35a-c) as also results from [55], can mimic various self-similar sets, and thus multi-fractals [56–58]. In such context, through similarity with fractional quantum Hall effect, the role of 𝑓[𝑑𝑡, 𝐹(𝐷𝐹 )] becomes that of a filling factor, 𝜇 = 𝑓[𝑑𝑡, 𝐹(𝐷𝐹 )], and thus (31) written in the form: 𝜈 = 𝑓𝜈𝑐 (36) can describe an hierarchy of Haldane type for oscillation modes of an ablation plasma [59]. We note that the above “scenario” is able to reproduce all quantum Hall effect steps [60] corresponding to experimentally observed fractional filling factors with odd denominators, 2 1 2 2 4 4 5 𝑓 = ; ; ; ; ; and (37) 3

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They correspond to filling factors given in Haldane notation [59] by [3,2], [3], [3,-2], [1,2,-2], [1,2] and [1,4], all derived from m=1 and m=3 hierarchies. For example, one can apply Eq. (36) in case of our experimental data by choosing 𝜈𝑐 = 𝜈 ′ = 6.4 MHz, i.e. corresponding to the slow plasma structure, and by imposing 𝑓 = 4/3; 5/3, to obtain 𝜈(4/3) = 𝜈′ ≈ 8.5 MHz and 𝜈(5/3) = 𝜈′ ≈ 10.6 MHz, which are in agreement with the experimental data for the fast structures of Al and Cu targets, respectively. This interesting analogy with the quantum Hall effect will be subject of further experimental and theoretical investigations in our group.

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4. Conclusions The existence of two types of oscillations for the Langmuir probe current used to characterize the expansion of fs-laser produced plasma was experimentally evidenced. Their appearance is influenced by the external bias applied on the metallic target. Thus, in the case of Al target, for negative probe bias of VP=-35V, we observed oscillations of averaged frequencies 𝜈=8.5 MHz and 𝜈′=6.5 MHz, which were attributed to the two plasma structures ejected through different mechanisms, i.e. a fast (hot) one consisting in highly charged particles ejected at the beginning, in an expansion process characterized by the acceleration through the electric field given by charge separation, and a slow (cold) tail consisting in thermalized particles with low average charge state. The positive low target bias, e.g. VT = +(0÷10) V, determines the presence of a high oscillation frequency, while for VT = +(40÷60) V oscillations of lower frequency are induced, and for intermediate values they are both present in early and late parts of the probe current, respectively. Similar results were obtained when using Cu targets: the probe current initially oscillates with a frequency of 𝜈=10.5 MHz, while later on it becomes 𝜈′=6.4 MHz, the values being averaged over the various target biases. In this case both oscillating regimes are recorded in the probe temporal trace regardless of probe positive biasing, which was tentatively explained through the higher ion mass and lower average charge state. An experimental confirmation of the previous results was obtained when recording the probe current for negative bias by also negatively biasing the target at adjacent values. We observed that the positive oscillations corresponding to the fast part, recorded for target biasing of higher value with respect to the probe, are mirrored in the negative domain when target bias becomes lower. Meantime, the oscillations corresponding to the slow plasma structure are preserved in the positive domain, showing that this part is not influenced by the target potential due to the dominating thermal processes. Theoretically, we developed a model which considers different behaviors at two scale resolutions. At Coulomb scale, the propagation equation of the electrical component of the electromagnetic field led to a behavior of damped oscillator type for current density. Through constraints which are imposed by various symmetries of the expanding plasma, one can get various plasma oscillation modes. At thermal resolution scale, the ion current is given by a shifted MaxwellBoltzmann velocity distribution function. Assuming the principle of scale resolution superposition, the current density at global scale resolution is expressed through the sum of the current density at Coulomb scale resolution and the one at thermal scale resolution, result which is in a good agreement with the experimental data. Finally, an interesting analogy with the quantum Hall effect was noticed and will be subject of further studies. 9

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Acknowledgements This work has been partially supported by the Agence Nationale de la Recherche through the LABEX CEMPI (ANR-11-LABX-0007), as well as by the Ministry of Higher Education and Research, Hauts de France Council and European Regional Development Fund (ERDF) through the Contrat de Projets Etat-Region (CPER Photonics4Society). References

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[47] F. F. Chen, Introduction to Plasma Physics and Controlled Fusion, 2nd ed., Plenum Press, New York, 1984, Vol. 1. [48] P. Mulser, G. Alber, M. Murakami, Revision of the Coulomb logarithm in the ideal plasma, Phys. Plasmas. 21 (2014) 042103. doi:10.1063/1.4870501. [49] I. Merches, M. Agop, Differentiability and Fractality in Dynamics of Physical Systems, World Scientific Publishing, Singapore, 2016. doi:10.1142/9606. [50] L. Nottale, Scale Relativity and Fractal Space-Time: A New Approach to Unifying Relativity and Quantum Mechanics, Imperial College Press, London, 2011. [51] R.C. Hilborn, Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers (2nd Edition), Oxford University Press, Oxford, 2000. doi:10.1093/acprof:oso/9780198507239.001.0001. [52] J.D. Jackson, Classical Electrodynamics, John Wiley & Sons, New York, 1962. [53] V.P. Pikulin, S.I. Pohozaev, Equations in Mathematical Physics, Springer, Basel, 2001. doi:10.1007/978-30348-0268-0. [54] B.B. Mandelbrot, The Fractal Geometry of Nature, WH Freeman and Co., New York, 1983. [55] D. Hensley, Continued Fractions, World Scientific, Singapore, 2006. [56] A. Lakhtakia, R. Messier, V. V. Varadan, V.K. Varadan, Incommensurate Numbers, Continued Fractions, and Fractal Immittances, Zeitschrift Für Naturforsch. A. 43 (1988) 943-955. doi:10.1515/zna-1988-1106. [57] R.M. Corless, G.W. Frank, J. Graham, Chaos and continued fractions, Phys. D Nonlinear Phenom. 46 (1990) 241–253. doi:10.1016/0167-2789(90)90038-Q. [58] R.M. Corless, Continued Fractions and Chaos, Am. Math. Mon. 99 (1992) 203-215. doi:10.2307/2325053. [59] F.D.M. Haldane, Fractional quantization of the hall effect: A hierarchy of incompressible quantum fluid states, Phys. Rev. Lett. 51 (1983) 605–608. doi:10.1103/PhysRevLett.51.605. [60] R.E. Prange, S.M. Girvin (Eds.), The Quantum Hall Effect, Springer, New York, 1990. doi:10.1007/978-14612-3350-3.

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ACCEPTED MANUSCRIPT Figure caption

Figure 1: The typical temporal evolution of the current recorded by the negatively biased probe, VP=-35V (a), oscillatory part obtained by subtracting the continuous part (b), its Fourier transform (c), and frequencies obtained for

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various target biases VT (d), for fs laser-produced Al plasma.

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Figure 2: The temporal evolution of the current recorded by the negatively biased

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probe, VP=-35V (a), oscillatory part obtained by subtracting the continuous part (b), its Fourier transform (c), and frequencies obtained for various target biases

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VT (d), for fs laser-produced Cu plasma.

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Figure 3: Temporal evolution of the current recorded by the negatively biased probe,

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VP=-30V, for various negative Al target potentials VT.

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Graphical abstract

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High-frequency oscillations recorded by a Langmuir Probe in a transient plasma produced by femtosecond laser ablation of metals Evidence of two distinct oscillations regimes, related to different physical processes A fractal model with scale resolution superposition is able to account for experimental data Interesting speculative analogy with the filling factor hierarchy in the fractional quantum Hall effect

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