Ripple formation with intense Gaussian femtosecond laser pulses close to the damage threshold

The formation of laser-induced periodic surface structures (LIPSS or ripples) is a topic that has been investigated for almost 60 years. More recently with the advent of ultrashort laser pulses this subject has regained interest, in particular, due to interaction regimes that have not been present so far. Consequently a lot of work has been done in that field, especially with comprehensive experimental and theoretical investigations of the scaling of ripple parameters on laser pulse duration, wavelength, applied fluence, shot number and so on. However, there are still a lot of questions. The present work addresses an important issue on that subject. In particular, ripple formation is investigated at high laser intensity, namely at an intensity sufficiently large to generate a femtosecond-laser induced plasma. Thus ripple formation occurs close to damage threshold. Experimental results and theoretical discussion of ripple formation and the interrelation to laser pulse energy deposition, energy transport and sample damage originating from the optical interaction and additional thermal effects, respectively, are discussed. Most important, a reduction of ripple formation threshold with laser intensity and fluence, respectively, has been observed which is associated by a super-linear increase of the ripple area. The scaling of this reduction with laser fluence obtained from theoretical estimates is in good agreement with the experimental data.


Introduction
A lot of theoretical and experimental studies of the processes of condensed matter surface behaviour when irradiated with a train of polarized laser pulses were devoted to metal surfaces [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Specifically, models based on surface plasmon polaritons (SPP) and waveguide modes excitation and their experimental verification for long laser pulses (with a duration larger than the electron-phonon relaxation time) have been reviewed in [1,2]. The extension of the SPP model for ultrashort laser pulses was suggested and verified in [3]. At present this model well describes the grating formation with ripple periods λ r less, but still close to the wavelength λ L of laser radiation. It describes the orientation of the grating vector k r with respect to the laser pulse electric field vector E and is appreciated for condensed matter with different physical properties. The applicability of this model ranges from cw to femtosecond pulse duration and from ultraviolet to sub-millimeter wavelength.
Reference [7] provides a review with details of the mechanisms of the periodic surface structure formation. The process of grating structure 'writing' (fixation) on the material surfaces for long laser pulses is reviewed in [1,2] and that for the ultrashort pulses in [4,7]. For low energy density regimes the most frequently realized generation mechanism is thermocapillary, which was considered in detail in [8] and more schematically in [9,10]. For higher energy densities, namely for a fluence F 0.9 J cm −2 (and a pulse duration of approximately 100 fs), Coulomb mechanisms become important at the metal-vacuum boundary. Then the electron emission from the metal surface is large enough to generate an opaque plasma layer near to the surface. Moreover, the strong electrostatic field which is induced dynamically by the laser pulse, supplies spatially selective metal ions ejection from the irradiated surface area [11].
Depending on the experimental conditions the ratio λ r /λ L may change. Such changes have been observed in experiments (slow variations of λ r values result also from pulse to pulse changes) and variations of the local micro/nano-geometry and of the local dielectric permittivity (via the SPP's dispersion relation)) are described by nonlinear models [5,6]. As a consequence, from experiments a different behaviour of dλ r /dF for the low and the high fluence (F) regime, respectively, was observed and, in particular a change of the sign of this derivative [12,13,16,17]. For the high energy density regime (i.e. large fluence) the sign of the derivative is positive, which is the indication of the opaque plasma layer formation as mentioned above [12,18]. For the particular case of gold and silver, this sign for was interpreted via the behaviour of a solid state metal plasma [14,15].
In addition, energy transfer may play a significant role. The existence of a lateral energy transfer caused by (hot) electron diffusion in a thin gold film irradiated by a femtosecond laser pulse was discussed in [20]. An initially rapid diffusion of (hot) electrons during first few picoseconds was followed by a slower diffusion at longer times. The ratio of the coefficients of the (hot) electron diffusion to the phonon-limited thermal diffusion was approximately 100 (deduced from experiments).
Laser-induced micro and nano-grating structures (i.e. ripples) find a lot of applications. Particularly they are suggested for surface friction reduction, for increased surface absorptivity and an emissivity change, for biological molecules and liquid crystals orientations, for the creation of antibacterial surfaces, etc [19]. Another issue is the potential application in experiments performed at high laser intensity where the structured targets could be generated in situ and used for improved coupling, e.g. for X-ray or high energy particle generation [18,21,22].
Albeit the large amount of experimental and theoretical investigations on laser-induced periodic micro and nanostructure formation, there are still a lot of questions, especially for structure generation with femtosecond laser pulses. Additionally motivated by the possibility of more tailored applications, this drives further research in this field.
The present work contributes to this process with the investigation of nanostructure formation (ripples) and morphology changes (cellular damaged surface) generated with linear polarized femtosecond laser pulses. These were focused to a rather high laser intensity (and thus large fluence), namely to a value close to the damage threshold. We may mention that here the applied intensity is 3 orders of magnitude larger than that, e.g. applied in [20] which takes influence on the related interaction. The threshold for the formation process has been studied at different laser intensities and fluences, respectively, and for laser pulse trains with various shot numbers. The results clearly show that there exists a transitional area between the ripple zone and the neighboured cellular zones with hierarchical orthogonal gratings. Two regimes of ripples formation have been considered which differ for different regimes of supplied laser energy density. In case of large intensity and fluence it has been shown that there is a threshold energy density change for ripple formation. Part of the energy is associated with the lateral energy transfer from the adjacent cellular zone where most of the laser energy is dumped.

Experimental setup and analysis
The experiments were carried out with a Ti:sapphire chirped pulse amplification laser system coupled to an advanced setup for micro and nano structuring applications [18,23]. The system operated at a wavelength of λ L = 775 nm and delivered linearly polarized pulses with a duration of τ L = 153 fs (FWHM). The pulses were focused with an achromatic lens of f = 200 mm focal length to a focal spot with an average diameter of d L = 50 μm (±10%; FWHM), with a slightly elliptical shape (long and short axis of d long = 56 μm and d short = 44 μm, respectively, see figure 2 in section 3; d L = (d long ·d short ) 1/2 ). D L is the corresponding 1/e 2 -width (≈1.7·d L ). The radial spatial intensity distribution was Gaussian.
The measurements were restricted to a range of fluences close to the damage threshold of F dam ≈ 0.8 J cm −2 (this value has been obtained for the present conditions, see [18]), namely between fluences at the central peak F 0 ≈ 0.5 J cm −2 and F 0 ≈ 1.5 J cm −2 . Thus the corresponding peak intensity I 0 was always larger than 10 12 W cm −2 (at the spatial central peak and averaged over the τ L ).
Fluences much above F dam were avoided due to the well-known associated effects, in particular, the contamination and potentially the damage of the focusing optics resulting from target debris. Moreover, this prevents of a significant burr on target which would make the deduction of the diameter D dam of the damage region difficult.
The flat polished Copper targets were irradiated at normal incidence and mounted on motorized xyz-and rotation holder. In order to have well defined conditions, the present investigations concentrated on ripple formation in vacuum (10 − 4 mbar, 10 −2 Pa). Thus chemical reactions that would have accompanied the formation process in ambient gases or air were excluded.
The targets were irradiated with a fixed fluence F 0 and a fixed number of shots N on the same target surface at a laser repetition rate of 1 kHz. Before any change of F 0 and N were applied, the target was shifted laterally to a new position with an unaffected surface. While keeping the target always at the same focal position, the fluence was changed by changing the laser energy by means of calibrated attenuation filters. This procedure is essential, because in contrast to other measurements where the fluence is changed by a change of the focal position, in the present experiments the target stayed always in far field where the interaction conditions are well defined. Thus the laser light was keept always purely s-polarized and contributions from p-polarizations were avoided, which would have present in focal scan experiments when the target is moved outside the focal position (note: outside the focal position, the fluence is reduced, but p-polarized components get better absorbed and the mixture of different polarization components leads to a seemingly lower damage threshold).
Before analysis of the irradiated targets, they were cleaned in an ultrasonic bath. Examination was made with an optical light microscope (LM; Leica DM4000 B/M) and a scanning electron microscope (SEM; Zeiss EVO ® MA10).
Depending on the fluence region, namely (i) F 0 < F th , where F th is the threshold for ripple formation, (ii) F th < F 0 < F dam , (iii) F th < F dam < F 0 and (iv) F 0 well above F dam , four different situations may be discriminated. The first one is not of interest here and the other 3 are displayed in figure 1, where (a) to (c) correspond to (ii) to (iv).
The diameter of the ripple region D th (average diameter of the slightly elliptical profile) could be easily obtained from the SEM images, where one could observe that the boundary between the ripple region and the unaffected region is extremely sharp and thus well detectable (see figures 3(c) and (d) in section 3). In a similar way the diameter of the damage crater D dam could be determined from the images captured with the optical microscope and the SEM, respectively (note that D dam is also equivalent to the inner diameter of the ripple region in figures 1(b) and (c)).

Experimental results
For very low fluence and intensity (F 0 < 0.5 J cm −2 ; I 0 < 2.5·10 12 W cm −2 ) and a small number of shots (a couple of 10 or less), there is just an onset of ripple formation with a very inhomogeneous ripple distribution and even for large N the threshold fluence for ripple formation is not much reduced, namely F th ≈ 0.4 J cm −2 (2·10 12 W cm −2 ; see discussion below). Ripple formation starts preferentially at positions where surface scratches are present, in particular, such ones with an orientation perpendicular to the electric field. However, it also starts if there is a small angular mismatch (see (2a). This onset is consistent with the well-known descriptions of ripple formation as a multi-shot effect [1,4,7]. At this near threshold fluence the laser pulse train produces isolated areas as local tracks of SPP propagation in the direction E (see figure 2(a)). The typical grating period caused by interference of the incident laser radiation with excited the SPP is λ r = λ L /η λ L , where k r || E, k r is the SPP's wave-vector and η is the real part of dielectric permittivity of the copper-vacuum boundary for SPP.
For higher values of F 0 and I 0 , respectively, and a couple of 10 or a couple of 100 shots, ripples are generated rather homogenously in a disk-shaped region (cf figure 2(b)) and with the same characteristics as those in the previous case. If F 0 is increased, but is still below F dam , D th is increased as well. Both conditions correspond to the situation of figure 1(a).
When F 0 comes close to F dam the ripple height is increased (see later) and further increase leads to surface damage (cf figures 2(d) and 3(a)) with a break up or even an explosion (figure 3(b); see also [24][25][26][27]). In that case the centre of the laser spot is covered by a damage crater with a diameter D dam , with a ring-shaped region around (compare figure 1(b)). This region can be well discriminated from the region where ripples are generated.
In the following we will term these regions 'ripple region' and 'damage region', respectively. In the intermediate area between both regions one can observe the formation of additional coarse and coexisting structures with orthogonal (anomalous) orientation with a period = λ L (cf SEM image in figure 3(d)). The ridges and valleys may lead to the excitation and propagation of further localized SPP (edge and channel,  shows the ablation region of (a) in detail. The ripple region at the border to the unaffected zone for a sample irradiated below (F 0 = 0.6 J cm −2 , N = 140) and above the damage threshold (F 0 = 1.3 J cm −2 , N = 160) is shown in (c) and (d), respectively. respectively) and their mutual interference to the observed structure. In consequence that will lead to an increased absorption at the metal surface [27]. With increasing F 0 the period's bifurcation cascade [6] induces a large surface absorptivity and results in a zone with strong damage. Deformed cellular structures with a lateral distance in between of roughly 1 μm (or more; this is of the order of λ L ) are well seen and also randomly distributed and chaotically oriented nano-tips (figure 3(b)). All this leads to an additional and efficient absorption of the laser radiation energy. Hence this zone (black one in figure 1) also possesses the enhanced surface absorptivity.
It is obvious that there is a lower and an upper limit for ripple formation, with the corresponding fluences F min and F max , respectively, which defines the ripple region (cf figure 1). Below F min there is no ripple formation. Thus from figure 1(a) it can be seen that the lowest possible fluence within the ripple region is always given by F min = F th . The maximum available fluence is F max = F 0 . This is illustrated in figure 1(a) and figures 2(a) and (b). In figures 1(b) and 2(c) F min is the same as in (a). However, the maximum possible value now is limited by damage, i.e. F max = F dam < F 0 .
For pulses with a Gaussian profile as in the present experiment, F dam can be determined directly from D dam (a careful analysis has taken into account the different directions within the ellipse). Figure 4(a) displays the result of the data evaluation for the thresholds for ripple formation and damage, respectively. All thresholds are provided for s-polarized pulse irradiated at normal incidence (see section 2). For low peak fluence, F th shows a weak dependence on N. For the present case the scaling can be described by figure 4(a); the errors represent the confidence interval related to the fit of the experimental data). This scaling does not significantly depend on F 0 . We will not discuss this relation more deeply because this scaling is not much different to the observations by other groups [18,23,28,29]. Furthermore, from figure 4(a) it can be seen that F dam has a similar scaling on N but the parameters differ. The scaling coefficient and the single-shot damage threshold are given by Г d ≈ −0.2 (±25%) and F d1 = 2 J cm −2 (±30%), respectively.
The negative values of Г in equations (3) and (4), indicate a pulse induced change of the material or surface property and are the result of accumulation effects. These lead, e.g., to a damage threshold reduction due to incubation as has been observed with ns pulses [2,30] and with fs-pulses [31], [32], respectively.
The single shot ablation threshold F d1 deduced within the present work is in good agreement with the common analysis of ablation depth d abl (including incubation effects). In case of ablation with one shot at . Note that the intention of the present work is not a detailed damage study, and thus the low number of data points used for the fit in (b) is regarded to be sufficient. modest fluence and under the assumption of linear absorption, according to Lambert-Beer's law the ablation depth for a pulse irradiated after n proceeding ones is given by (this follows from, e.g. [23], [33]). Of course this requires that F > F dam (n), otherwise d abl,n = 0 (with F dam (n) from equation (4)). Λ is the laser pulse penetration depth. For ablation with N multiple shots d abl,n has to be summed from n = 0 to N−1 (for single shot ablation N = 1).
A fit of the present experimental data according to this simple model is plotted in figure 4(b). The axis intersection (i.e. d abl = 0) corresponds to the averaged reduced damage threshold due to incubation, which is consistent with F d1 = 2 J cm −2 for single shot ablation. The slope corresponds to Λ = 80 nm. We would like to remark that the results obtained from the present crater diameter measurements and those of the crater depth do agree very well and there is also agreement with the observation of the onset of damage in various SEM images. This note is crucial because there is doubtful work in the literature where a disagreement in the results is reported and thus additional 'free parameters' are introduced which however, are not an integral part of the related model (see equations (1), (4) and (5)).
The obtained threshold fluence also agrees with the result of [31], although it may be noted that the value of [31] is slightly lower. This has been expected because of the focal scan method applied by that group which leads to a value that may be identified with the seemingly lower damage threshold discussed in section 2. The threshold of the present work also agrees with the value of [32] when corrected for incubation as discussed above (F d1 = 2.4 J cm −2 ).
We would also like to remark, that although equation (3) looks quite similar to equation (4), within the experimental error Г d and Г r have notably different values and F th (N) decreases less with N when compared to F dam (N) (see figure 4(a)). The different scalings of F dam and F th , respectively, may have been expected. As material damage and ripple formation result from different physical processes, one cannot expect that the Γ-coefficients and the single-shot damage threshold fluences are the same.
Up to now, the discussion of F th has been restricted to a peak fluence (and intensity) below damage threshold (compare to figures 2(a) and (b)). In the following we will extend this to situations such as displayed in figure 2(d). Figure 5(a) shows the dependence of the radius of the ripple area D th /2 on the peak fluence of the laser pulse F 0 . The experimental data are shown as the symbols where the symbol size represents the experimental error. The applied number of shots is provided in the inset. The blue solid line represents a fit according to equation (2) with F th = 0.5 J cm −2 = const which is in good agreement with the experimental data below F 0 ≈ 1 J cm −2 . However, above this value, the radius becomes significantly larger and cannot be described by the same fit. This larger radius can be identified with D TH /2 displayed in figure 1(c).   (2). It may be seen that for low peak fluences and intensities the ripple formation threshold is rather constant. F th (N) is nearly the same for all peak fluences up to F 0 ≈ 0.9 J cm −2 which is slightly larger than F dam (N). However, for larger values of F 0 there is a significantly larger radius (i.e. D TH /2 instead of D th /2) which can be attributed to a significant reduction of the threshold for ripple formation. The dotted blue curves in The interesting point here is that those 'large' fluences F 0 are present only in the damage region, but not in the region where the ripples are generated. As discussed above, the maximum fluence within the ripple region is always F max = F dam which is obviously below F 0 . But nevertheless the experimental data clearly show that an increase of the fluence in one region (namely an increase of F 0 in the damage region) takes influence on the threshold in another region (i.e. on F th in the ripple region). This situation is illustrated in figure 1: for F 0 < 0.9 J cm −2 ≈ F dam , F th is the same (cf figures 1(a) and (b)), but is reduced to F TH < F th when F 0 clearly exceeds F dam (cf figure 1(c)).
One may also compare the ripple period λ r and the ripple height h for different experimental conditions. For F 0 < F dam the dependence of the ripple period λ r on the shot number and on the fluence was investigated by several groups [12,16,18,24,26,34]. Those works have shown that λ r has only a very weak dependence on N and F 0 , respectively, within the range under discussion. But a repetition of details is not subject of the present work.
Instead, here we concentrate on a comparison of ripples obtained with F 0 below and well above F dam . In particular, we may compare the ripple period λ r obtained for F 0 = 0.6 J cm −2 and F 0 = 1.5 J cm −2 , respectively, and find a significant longer period for a fluence above the damage threshold (see Table 1). For the lower fluence h ≈ 70 nm and h does not differ significantly for N = 80 and N = 120. For F 0 well above F dam and N = 80, h is somewhat larger and possibly even larger for N = 120 when compared to the low fluence case. Just for comparison we would like to note that the depth of the irregular structures in the damage region (cf figure 3(b)) is roughly 0.4 μm.

General
The present experiments were carried out at 150 fs pulses with relative high intensity in focus (I 0 > 10 12 W cm −2 ) when compared to usual ripple formation studies. All applied intensities were well above plasma threshold for metal targets. Consequently there was always a significant influence of the generated free electrons. Hence ripple formation was governed by plasma physics and can be described in terms surface plasma waves (SPP).
Following the discussion in [18], which partly bases on the model of Sakabe et al as the first step [12], the process of ripple formation for the present conditions of copper targets irradiated with intense fs-pulses can be described as follows. First, via a parametric process (such as in [35,36]), a fs-laser pulse induces a plasma wave on the surface. Then during the surface plasma wave excitation and propagation, ions become enriched locally. Thus they experience a strong Coulomb repulsion until the peak of the next electron wave arrives at that position. Hence, those spatially localized ion clouds Coulomb-explode and expand to vacuum [12,37]. Consequently, third, a thin layer is ablated thus giving rise to the formation of periodic grating structures, which can be regarded as an imprint of a 'grating' according to the interspacing of the regions where Coulombexplosion and thus ablation occurs [38,39]. If the laser fluence is large enough, once such structures are formed by the first pulses of a pulse train, an enhancement process might take place for the subsequent pulses within the pulse train. The rise of the heights of imprinted structures is due to the positive feedback between the laser radiation absorption and modulation depth of the resonant structures. The electric field is enhanced near the initially imprinted structures, and it leads to further ablation of the surface, which results in further deepening of Table 1. Ripple period λ r and ripple height h for low and high fluence and intensity, respectively. The '±values' include the uncertainty in the analysis and take into account some small local variations. A strong change of λ r within the ripple region has not been observed.
the structures [11]. Nevertheless, related to this general description, a more detailed discussion of ripple formation in the 'low' and 'large fluence regime', respectively, has to be made. This is subject of the following discussion.

Low fluence regime
For the low fluence case (i.e. F 0 ≈ 0.4 to 0.6 J cm −2 , i.e. I 0 ≈ 2 to 3·10 12 W cm −2 ; cf figure 1(a)) ripple formation starts in the vicinity of the laser peak where the gradient of the fluence and intensity, respectively, is not very large (or zero exactly at the peak; cf figure 1(a)). Due to the high intensity, plasma physics already plays a role, but the formation of an optically opaque layer due to a generated near-surface plasma does not yet occur [3]. Actually, the behaviour of λ r in the range under discussion indicates the formation of structures with the participation of SPP at the metal-vacuum interface where a low concentration of electrons is emitted from the metal surface. We assume that the formation of nanostructures (i.e. ripples) in this range is the result of the combined action of thermocapillary and evaporative mechanisms in the regions where local melts occur: as surface baths of melt are produced under the non-homogeneous laser melting of the metal, and because of a temperature gradient, a surface force arises inside the temperature bath tangentially to the surface. Thus a liquid flow is generated which is proportional to the surface tension temperature gradient (see, for instance, [8,40]). The metal is squeezed from the local melt baths by surface tension force and may partially return back on time scales up to the melt crystallization. At the same time, there is a small increase in ripple period with an increase in F 0 that can be connected with both, the radiative damping of the SPP on the structures of increasing height [5] and the dielectric permittivity behaviour [41]. In the considered range, with an increase of F 0 , also a significant change of the dielectric permittivity of copper [41] and an almost linear increase in the depth of the residual surface relief up to values of h~40 to 70 nm is observed (see also [16,18]). One has to remind, that the remnant relief height may be smaller than the dynamical one due to partial backflow of the metal as discussed above.
The change in the dielectric permittivity of copper is associated with an increase of the electron-ion collision frequency ν ei which depends on the electron temperature T e ∝ F 0 1/2 (this is valid for temperatures below the Fermi temperature T F ) [11], [41], [42]. The growth of ν ei reduces the attenuation length L of the SPP at the end of the fluence and intensity interval under discussion (i.e. F 0 ≈ 0.6 J cm −2 ; I 0 ≈ 3·10 12 W cm −2 ). Therefore, with an increase of F 0 and I 0 , respectively, the increase in efficiency of the input via SPP due to an increase in the height of the relief (see [18]) is compensated by a decrease in L, which as a result does not lead to a noticeable transfer of SPP's energy from the initial zone of structure formation. Consequently there is no unexpected change in the diameter of the ripple region D th . Because of the relatively low temperature T e = T F and because of both, the growth of ν ei and that of the thermal conductivity κ e is modest, the transfer of energy beyond the irradiated area at the considered time intervals is not significantly affected (from our calculations in this range T e < 10 eV; the Fermi temperature for copper is 7 eV). We would like to note that also the gradient of the spatial laser intensity profile and that of the related T e profile is not very large (see discussion above). Therefore, the energy density corresponding to the threshold fluence for ripple formation in the discussed fluence range remains practically constant (cf figure 2).
This conclusion is also confirmed by an estimate based on the relation F th ∝ E/A r and the following scalings. The electron temperature T e in the absorbing volume is proportional to the energy E of the laser pulse. The electron's or heat diffusion length l scales as where τ ei = ν ei −1 is the average time between the electron-ion collisions, m e the electron mass, D diff the diffusion coefficient and t the typical time of hot electron diffusion, which usually is below 1 ps. A r ∝ l 2 and τ ei = const in the considered low energy density range [11]. Form these scalings it follows that F th ∝ T e /l 2 which is approximately const with respect to T e .

Increased fluence regime
For increased energy density, i.e. F 0 ≈ 0.6 to 0.9 J cm −2 , and I 0 ≈ 3 to 4.5·10 12 W cm −2 (cf figure 1(b)), respectively, the electron density n e becomes close to the critical one which results in a change in the mechanism of structure formation and an increase of λ r with an increase in F 0 (cf table 1). This is in contrast to the observations for the low fluence case where it was found that dλ r /dF 0 is approximately constant (established for copper for the low power density regime, see [12]. But for the present intermediate fluence regime this is in agreement with the positive sign of dλ r /dF 0 reported in [12] and verifications for other metals, including tungsten, titanium, platinum etc [13]. In this regime where a plasma layer becomes to be present, the plasma optical properties impose that λ r begins to change with F 0 .
It may be interesting to note that the transition of the 'low fluence' to the 'large fluence' regime may be consistent with the transition of the 'optical' to the 'thermal regime' discussed by Nolte et al [33], however with the new aspects discussed in the following.

Large fluence regime
For even larger fluence and intensity as discussed before, namely for F 0 > 0.9 J cm −2 (i.e. I 0 > 4.5·10 12 W cm −2 ; cf figure 1(c)), due to the further increased absorbed amount of laser pulse energy a significant opaque plasma layer is formed. In this regime T e ∝ F 0 1/2 and T e > 20 eV [43,44]. Moreover, the gradients of the fluence, and of the intensity and of T e are much larger than before. The appearance of the opaque plasma layer signifies the mechanism of the structures formation change. This change is confirmed by following.
First, the transition is characterized by a definite threshold energy density which has been estimated [11] with a value of approximately 1 J cm −2 . Second, there is large decrease of the experimental value of the real part of the dielectric permittivity η which (partially) reflects the dispersion relation of the considered systems (in our simple case the period of grating is given by relation λ r =λ L /η; see section 3). The values of η can be deduced from table 1. For the low energy density regime discussed in section 4.2, η ≈ 1.25 and this roughly reflects the dispersion relation of a copper-vacuum boundary for SPP's. For the present higher energy density regime η ≈ 1.05. This value cannot be related to the copper-vacuum boundary because it cannot be explained well by a change of the optical parameters of copper. Instead, the onset of a plasma layer, namely a plasma layer-vacuum boundary is indicated. This layer is characterised by an electron-ion collision frequency ν ei ∝ T e −3/2 [11] which according to the scaling of T e on F 0 decreases with fluence at the plasma-vacuum interface (compare also [45]).
Consequently η is decreased and thus there is a clear tendency that λ r approaches λ L (the decrease of the dielectric permittivity follows directly from Drude's model with ν ei /ω L as the damping factor; ω L is the angular frequency of the laser radiation) This is in agreement with theoretical predictions [46] for the regime T e ? T F (for the present work this corresponds to 20 eV ? 7 eV) and experimental observations for a number of metals in the high F 0 regime [12][13][14]. Also the positive sign of dλ r /dF 0 is typical for this case. In addition, thirdly, the transition to new an optical system, i.e. a vacuum-plasma layer-metal, results in a drop of F th with F 0 as observed in the experiment ( figure 5(b)).
As thermal effects are important, this is significant. In this high energy density range the dependence of τ ei on T e changes. As has been mentioned above, τ ei ∝ T e 3/2 [11] and thus from equation (6) one can see that the diffusion coefficient scales as D diff ∝ T e 5/2 and the diffusion length as l ∝ T e 5/4 . Hence, e.g., for F 0 ≈ 1.5 J cm −2 , the thermal conductivity κ e = C e (T e )·n e ·D diff increases significantly (C e is the heat capacity). In this case the threshold energy density scales as follows: F th ∝ E/A r ∝ T e /T e 5/2 = T e −3/2 . Then according to T e ∝ F 0 1/2 , one may expect a scaling of F th ∝ F 0 −3/4 which means that the ripple formation threshold decreases with F 0 . For F 0 ≈ 1.5 J cm −2 , with equation (6) one obtains l ≈ 5 μm which well agrees with the increase of the ripple radius by (D TH /2 − D th /2) ≈ 5 μm (see figure 5(a)). On the other hand there might be an energy transfer by propagating electromagnetic surface excitations, namely SPP's, which transfer their energy mainly in the vacuum. For normal incidence of laser radiation the SPP's are excited and propagated in two opposite directions governed by the electric field with the directions discussed in section 3. In the area of the first ring of ripples (area in between D dam and D th , see figure 1(c)) an amplification and dissipation of SPP's occurs at rather large intensity and a rather large intensity gradient with a propagation outward from the first ring boundary. If this would be dominant, F th would be proportional to E/L 2 ∝ T e /τ ei which in the end would lead to a propagation of the excited SPP with a longer propagation path at the interface L ≈ 1.2·λ L (for F 0 = 1.5 J cm −2 ) and to a scaling F th ∝ F 0 −1/4 . The energy dissipation would occur on a spatial scale length of 2 to 3 times L. However, although this process may contribute to a reduction of F th in the range under discussion, it cannot be the dominant one. First, the scaling of F th on F 0 is different when compared to the experimental data (see figure 5(b)) and second the increase from D th /2 to D TH /2 (approximately 5 μm difference) is significantly larger than 2 to 3 times L (less than 3 μm). Moreover, third, this would by in contradiction to the experimental observations where an asymmetric increase of the ripple region has not been observed (note that SPP have a directionality as discussed above and in section 3).
Hence, in the large fluence regime the declining dependence F th (F 0 ) results mostly from the heat transfer by nonlinear thermal diffusion of hot electrons. This effect, as the dominant one, increases with F 0 . It might be supplemented by an energy transfer out the ripple area by SPP's propagating along the vacuum-plasma layer boundary. However this is limited by the finite length of their propagation length which even at large F 0 is below the observed increase of the ripple ring region.
Thus one may conclude that in the frame of the considered model the dependence F th (F 0 ) and, in particular, the decrease of the threshold with fluence, is mainly due to the nonlinearity of D diff and l as functions of T e . The supplied energy E increases linearly with irradiated fluence and the ripple area A r increases super-linear. Thus for the discussed fluence range of approximately 1 to 1.5 J cm −2 , the reduction of F th with F 0 −3/4 correlates qualitatively with the experimental observation (see dotted blue line in figure 5(b)). This confirms the important role of enhanced lateral heat transport by (hot) electron diffusion from the central region of the laser spot (located within D dam and/or D th , respectively) to the neighboured region (>D th ). This present result confirms also the conclusion of [20] where a tendency of an increased (fast) electron diffusion coefficient related to increased electron energies with increasing F 0 has been discussed. However, we have to note that in [20] F 0 1.5 mJ cm −2 which is 3 orders of magnitude less than the typical metal surface damage threshold by pump radiation (1.5 J cm −2 ) and much less than the applied fluence within the present work which is close to F dam . Furthermore the maximum electron energies in [20] were rather low, namely 0.14 eV.

Summary
In summary, within the present work nanostructure generation with ultrashort laser pulses has been investigated in the regime of rather high laser intensity. At these conditions with the intensity (and fluence) close to the damage threshold, the intensity is sufficiently large to generate a laser induced plasma, in particular, an opaque surface plasma layer. Consequently plasma physics plays a major role for the whole interaction process. Experiments were performed with a train of linearly polarized 150 fs pulses which were focused on flat solid copper samples at normal incidence. The experimental results with an increase of the intensity from I 0 = 2 to 8·10 12 W cm −2 with the corresponding peak fluences F 0 ≈ 0.5 to 1.5 J cm −2 have been explained by a change of the transient interaction regime from laser pulse absorption by a metal solid state plasma to one by an opaque near surface plasma layer. The absorption process is then followed by the resonant nanostructure formation via the Coulomb mechanism with surface plasmon-polariton excitation and propagation. Here the particular influence of the laser pulse on the optical properties of the surface region play an important role as they take influence on the SPP's dispersion relation and its attenuation length. In the present work, we have considered this influence for a low and a high intensity/fluence interaction regime, where the regimes are discriminated by the damage threshold.
In particular, we have discovered experimentally that for the high intensity regime the process of plasma formation is followed by a considerable lateral energy transfer. According to theoretical estimations, the energy transfer is mainly due to nonlinear hot electron diffusion which may be supplemented by an energy transfer by non-equilibrium surface plasmon-polaritons excited at and propagating along the plasma-vacuum boundary. This conclusion extends the results of previous works on low-energy electron transport, now to the much higher hot electron energy. This takes influence on the ripple formation.
Another important novel aspect is the observed reduction of the ripple formation threshold at high intensity or peak fluence, respectively. This is associated with a noticeable super-linear increase of the ripple area. As the intensity in the ripple region always has an upper limit which may be well below that at the central peak of the laser spot, the experimental observation of the laser-induced ripple threshold lowering is caused mainly by rapid electronic lateral diffusion from the spot centre. The scaling of this process, i.e. F th ∝ F 0 −3/4 , is in good agreement with the experimental data.
A further issue associated with the electronic transport is the smooth increase of the nanostructure period with laser fluence. In the plasma formation regime this is due to the decrease of the hot electron's collision frequency and the increase of the plasma density which is typical for most metals.
All together the present experimental work with its theoretical estimates provides an important contribution for nanostructure generation with femtosecond laser pulses at rather high laser intensity, where the term 'high' has to be regarded with respect to experiments related to ripple formation. In that sense it may also stimulate further theoretical work.