Temporal Airy pulses for controlled high aspect ratio nanomachining of dielectrics

NADINE GÖTTE, THOMAS WINKLER, TAMARA MEINL, THOMAS KUSSEROW, BASTIAN ZIELINSKI, CRISTIAN SARPE, ARNE SENFTLEBEN, HARTMUT HILLMER, AND THOMAS BAUMERT* University of Kassel, Institute of Physics and CINSaT, Heinrich-Plett-Str. 40, 34132 Kassel, Germany University of Kassel, Institute of Nanostructure Technologies and Analytics and CINSaT, Heinrich-Plett-Str. 40, 34132 Kassel, Germany *Corresponding author: baumert@physik.uni‐kassel.de


INTRODUCTION
Dielectric materials are transparent from the IR over the visible into the UV spectral region and are therefore ideally suited for nanophotonic devices in the very same spectral range.Thus, machining methods in the few hundred nanometer range and below are necessary.However, the structuring of nonconductive dielectric materials by conventional processing techniques like electron beam lithography or focused ion beam (FIB) milling [1] suffers from the charging of the material's surface [2].The implementation of ultrashort laser pulses as a direct patterning method is an attractive alternative, as this can overcome charging effects and also provide very fast processing times.Based on the high nonlinearity of the laser-material interaction, the enhanced energy confinement, and limited heat diffusion, ultrashort laser pulses have become the instrument of choice for dielectric structuring on a micrometer or even nanometer scale [3] as well as for applications in nanosurgery [4,5].
Primary processes induced by ultrafast laser pulses involve nonlinear electronic excitation on the femtosecond time scale and energy transfer to the lattice on the picosecond time scale, followed by phase transitions like ablation or melting.Thus, temporal laser control can facilitate synchronization between the light and the material response and therefore lead to an efficient coupling of the laser energy [6,7].When restricting the excitation window to the femtosecond regime in order to confine heat diffusion, the generation of a sufficient energy density within the electronic system is the key parameter to initiate the abovementioned phase transitions.The transient free carrier density plays a fundamental role in determining the optical properties in addition to various propagation and relaxation mechanisms.Depending on intensity, free carriers are initially promoted into the conduction band by strong field excitation, i.e., multiphoton ionization (MPI) or tunnel ionization.These free electrons can further absorb energy by inverse bremsstrahlung and subsequently induce carrier multiplication by collisions.This process is often termed avalanche ionization (AI).A detailed overview of laser excitation in dielectrics can be found in Balling and Schou [8] and references therein.Optical damage thresholds were used as experimental evidence for exceeding a certain critical electron density after the laser interaction, and the regulatory effect of pulse duration has been investigated over the last two decades [9,10].Studies of transient electron densities range from intensities below [11,12] up to well above the breakdown threshold [13][14][15], confirming the described scenario.Optical damage thresholds were also used to demonstrate control of MPI and AI [16].Furthermore, it has been demonstrated that direct structuring of dielectric materials with temporal Airy pulses (TAPs) generates reproducible surface structures reaching the sub-100-nm regime [17], which is an order of magnitude below the diffraction limit.Here, a first experimental indication was given that TAPs can create deeper structures as compared to bandwidth-limited pulses.Other approaches to creating high aspect ratio channels in dielectrics basically utilize spatial beam propagation based on self-focusing effects or Bessel beam generation and will be discussed later.
In this contribution we demonstrate the creation of high aspect ratio channels in fused silica by specifically temporally tailored femtosecond laser pulses in a power regime where selffocusing effects can be neglected.The findings are supported by rate equation simulations.

IDENTIFICATION OF THE TECHNOLOGICAL REQUIREMENTS
The physical picture behind our temporal pulse tailoring is quite intuitive: ablation of dielectrics is often attributed to a critical electron density of a pure electron-hole plasma in the range of 10 21 electrons∕cm 3 rather than a critical energy density W within the electronic system.In the Drude model, the corresponding plasma frequency ω pl ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi e 2 N CB ∕m e ϵ 0 p is in the near-IR, and laser light at 785 nm will start to be strongly absorbed, i.e., the penetration depth into the material will strongly decrease with increasing electron density.A special feature regarding the conduction band electron dynamics in fused silica is a fast mechanism leading to a rapid reduction of the electrons into a self-trapped exciton (STE) state, settled on a time scale of 150 fs [11].The decrease in conduction band electron density goes hand in hand with the decrease in the Drude absorption.However, STEs can serve as an additional source for the creation of free electrons on the time scale of the laser interaction and eventually contribute to the energy transfer to the lattice on time scales of a few hundreds of picoseconds [18,19].
This means that the STEs play an important role for all pulses longer than 150 fs.To take advantage of this, an optimized pulse sequence consists of a short and intense pulse followed by a long, low-intensity part.The short pulse is intense enough to excite a significant number of electrons into the conduction band, well below the critical electron density.This allows the laser pulse to penetrate deeply into the material.Shortly after the excitation, the electrons are stored in STEs and are ready for a re-excitation by lower intensities, as those states are located within the bandgap.This is realized by the low-intensity part of the ideal pulse sequence.This re-excitation process might be repeated several times until the energy density for ablation is reached.In order to limit effects due to heat diffusion, the overall pulse sequence should be in the range of a few picoseconds.Temporal pulse shapes that mimic the optimal behavior can be synthesized with the help of spectral phase modulation [20].In a Taylor expansion of the spectral phase, the third term φ 3 leads to third-order dispersion (TOD).Mathematically, the temporal pulse envelope is described by the product of an Airy function and an exponential decay [21].This is why we refer to these pulses as TAPs.Pulses with positive φ 3 start with an intense main pulse followed by a pulse sequence with decreasing intensity [see the upper panel of Fig. 2(d)].
Increasing φ 3 enlarges the pulse duration as well as the temporal asymmetry, whereas the instantaneous frequency remains constant.In contrast, modulation with the second-order term φ 2 leads to group delay dispersion (GDD), which is characterized by a temporally symmetric intensity profile [see the upper panel of Fig. 2(e)] and a linear sweep of the instantaneous frequency.An increase in φ 2 also leads to an elongated pulse duration.A detailed mathematical description can be found in Wollenhaupt et al. [21,22].
In our experiments, we focus linearly polarized femtosecond laser pulses with a central wavelength λ 785 nm under ambient conditions with a long distance microscope objective (NA 0.5) above, on, and below a fused silica surface [Figs. 1  and 2(a)] with a bandgap of E gap;VB 9 eV [23].Three different pulse shapes are used: bandwidth-limited pulses of 30 fs duration, TAPs with increasing TOD, where the statistical pulse duration [21] increases from 260 fs for φ 3 1 × 10 5 fs 3 via 770 fs for φ 3 3 × 10 5 fs 3 to 1.5 ps for φ 3 6 × 10 5 fs 3 , and temporally symmetric pulses with a duration of about 1.4 ps (φ 2 1.5 × 10 4 fs 2 ), matching the statistical pulse duration of the longest TAP.For each pulse shape, the fluence is 2.5 times above the corresponding damage threshold (2.5 × E thr ).In addition, ablation structures resulting from the longest TAPs are compared to structures of a bandwidth-limited pulse at the same fluence.The results for selected pulse shapes are displayed and compared to simulations in Fig. 2. The upper panels show the relative intensity profile of each pulse with respect to the bandwidth-limited pulse at a fluence of 2.5 × E thr [Fig.2(b)].In the simulations the excitation and recombination dynamics in the direction of laser propagation (z-axis) are calculated via rate equations, taking the electron density in the conduction, valence, and trapping bands as well as the total electron density into account (for details, see Supplement 1).

EXPERIMENTAL SETUP
The experimental setup is sketched in Fig. 1.Laser pulses of 30 fs duration (FWHM) at a central wavelength of 785 nm are provided by an amplified Ti:sapphire laser system (Femtolasers Femtopower Pro).Single pulse selection is achieved via the Pockels cell of the amplifier.Before entering the material processing platform, the pulses are guided through our high-precision home-built pulse shaper [24].The pulse shaper is used in a "phase-only modulation" mode, and the overall dispersion of optical elements up to the sample surface is coarsely compensated with the help of a commercial acousto-optic pulse shaper (Dazzler) contained in the amplifier and fine adjusted with the home-built pulse shaper prior to applying the required phase functions.Temporal pulse characterization of selected pulse shapes was performed via cross-correlation measurements directly in the interaction region.The linearly polarized and temporally shaped laser pulses are focused on a UV-grade fused silica sample (Spectrosil 2000) using a microscope objective (Zeiss LD Epiplan 50×/0.50NA) with a 7 mm working distance.The 1∕e 2 intensity spot radius was measured using a knife-edge method, and the obtained value of 1 μm was applied in the simulations.The calculated Rayleigh length for the measured spot radius is 4 μm.The sample is translated by a three-axis piezo table for each pulse exposure.The focal position is changed in steps of Δz 1 μm for fixed pulse energy and pulse shape.After laser processing and cleaning, the samples are analyzed by cross-section cutting with FIB followed by scanning electron microscopy imaging.

MACHINING RESULTS AND HIGH ASPECT RATIO FORMATION
The machining results for bandwidth-limited pulses at 100 nJ 2.5 × E thr pulse energy are displayed in Fig. 2(b).Shallow ablation structures are created for all focal positions.When the pulse energy is increased to 250 nJ, the hole depth stagnates, as can be seen in Fig. 2(c).
This observation changes when fused silica is processed with shaped laser pulses.When the temporal pulse shape is tailored to the abovementioned asymmetric [Fig.2(d)] and symmetric [Fig.2(e)] intensity distributions with pulse lengths of 1.5 and 1.4 ps, respectively, hole structures with a huge aspect ratio (hole depth to channel width) up to 30:1 are observed.The hole depth of each pulse shape and the focusing position are well reproduced in the simulations.We observe that the deeper the laser focus is located inside the bulk material, the longer the ablation structure is.
For asymmetric pulses, the channel width of the ablation structures is in the range of 200-350 nm, and therefore well below the 785 nm wavelength.A decrease in the temporal asymmetry, accompanied by a shorter temporal pulse duration, leads to a decrease in the hole depth and aspect ratio, as presented in Figs.3(a   For symmetric pulses, the same depth is reached; however, larger diameters and irregularities are observed.The mechanisms responsible for the irregularities have not been identified so far.These pulses seem to be less robust with respect to high aspect ratio machining.
Figure 3 summarizes the hole depths in the experiment and the simulation and the aspect ratio achieved with respect to the focus position for increasing the temporal asymmetry and the symmetric pulse profile.In addition, cross-correlations of temporal pulse shapes, measured directly in the interaction region, are displayed.

DISCUSSION
Bandwidth-limited pulses have a significantly higher peak intensity than elongated temporally shaped pulses for the same pulse energy.Even when setting the same relative pulse energies with respect to E thr , the bandwidth-limited pulse still has a peak intensity approximately 3 times higher than that of a TAP with φ 3 6 × 10 5 fs 3 [Figs.2(b) and 2(d)].At these high laser intensities, air close to the target will be excited, since the ionization potential of air E gap;air 14.6 eV is only 1.6 × E gap;CB of fused silica.This leads to a significant absorption [25] of the laser pulse before it reaches the surface of the sample, limiting the energy input into the material.In our simulations, we included absorption in air for bandwidth-limited pulses based on results from Liu et al. [26].In addition, bandwidth-limited laser pulses of 30 fs create a very high conduction band electron density within the first few hundreds of nanometers that will lead to a rapid and strong absorption and reflection of the laser pulse, suppressing energy deposition beyond this distance.As the maximum temporal intensity of the pulse has passed, the conduction band electron density N CB t strongly decreases due to the fast trapping mechanism of 150 fs [11] into exciton states (E gap;TB 5.2 eV [23]), as illustrated in Fig. 4(b).Free carrier absorption in air and inside the sample due to a high conduction band electron density is one mechanism leading to stagnation of the hole depth.Stagnation from absorption due to strong field excitation [27] is not considered in our simulations.
In comparison to bandwidth-limited laser pulses, temporally shaped pulses-either symmetrically or asymmetrically-interact with the material in a different fashion.Their low peak intensities avoid absorption in air as well as strong absorption close to the surface.Thus, these pulses are able to provide more energy for the excitation in the depth of the material, which might be sufficient for material ablation.The temporal structure of the asymmetric pulse is furthermore responsible for the characteristic interplay of trapping and re-excitation of electrons ending in an oscillating behavior of N CB t [Fig.4(c)], showing that there is still an observable energy deposition after 2.5 ps.The maximum N CB t is much lower than for bandwidth-limited pulses, even though the same deposited energy of W 1.4 kJ cm 3 is accomplished at the end of the pulse [Fig.4(a)].As listed in Table S1 of Supplement 1, it is noteworthy within this context that 52% (36% for the symmetric and less than 1% for the bandwidth-limited pulse) of the energy required for ablation is discharged into the material by the re-excitation of STEs via MPI and mainly AI.As a consequence, the energy deposition takes place in a confined region, because the reexcited STEs are localized close to a single SiO 2 complex [11].
Regarding the subwavelength dimensions of the hole structures, we note that not all critically excited material can be ejected during the ablation process.This is supported by the rims formed close to the craters.The rims are not significantly larger for temporally shaped laser pulses than for holes generated by bandwidthlimited pulses, as shown in Fig. 2. As the excited material is nearly completely enclosed by the unexcited material, high pressure inside the material can lead to the formation of super dense material phases arising during the excitation and ablation process and ending in densified material close to the channel crater.This was recently observed in the case of void formations in fused silica [28] and silicon [29].In addition, it was recently found that laser excitation in water by TAP pulses is distributed over a strongly reduced area and greater depth compared to ultrashort bandwidth-limited laser pulses [30].
For temporally symmetrically shaped pulses, the ablation threshold is approximately 1.8 times higher compared to the that of asymmetrically shaped ones (178 versus 100 nJ; see Table S1 of Supplement 1) due to the even lower peak intensity.In this case, almost the first half of the pulse is not intense enough to excite a significant number of initial seed electrons for AI, as indicated by the shaded area in Fig. 4(d).In addition, Fig. 4(a) shows the accumulated energy density over time, which starts to increase after time zero, showing that the second half of the pulse mainly provides the energy for ablation.For higher pulse energies, the onset of absorption will be shifted toward earlier parts of the pulse, but a significant fraction of the pulse will still be unused for excitation.In the end, the absorbed part of the temporally symmetric pulse is a temporally asymmetric shape.As mentioned above, this is reflected in the very similar temporal dependences of the accumulated energy density for both shaped pulses, as shown in Fig. 4(a).
Regarding the slightly larger channel width, we assume that the continuous heating of the conduction band electrons results in a high average kinetic energy, causing a distribution of electrons over a larger volume.The consequence is supposed to be an extended affected zone of deposited energy and therefore an increased channel width, resulting in a smaller aspect ratio for temporally symmetrically shaped pulses.This is supported by Fig. 4(d).On one hand, the maximum N CB t is clearly larger than for TAPs, and no oscillations within N CB t can be observed.On the other hand, the maximum N CB t is noticeably smaller than for bandwidth-limited pulses, due to the smaller peak intensity and the electron trapping.As mentioned above, the re-excitation of STEs contributes noticeably to W with about 36% (Table S1 of Supplement 1), which is attributed to the elongation of the pulse in time.
In the end, all this also manifests in a high penetration depth [Fig.3(d)] for temporally symmetric shaped pulses, albeit with much lower efficiency and smaller aspect ratios.
A direct comparison of the experiment and simulation in Fig. 2 shows that on one hand the hole depths are in very good agreement, while on the other hand the channel width could be less well reproduced.This is related to the obvious limitations of the simulation, which includes neither the energy transfer to the lattice nor subsequent phase transitions, as discussed above.
To estimate the limits of the presented temporal pulse shaping technique, we simulated the maximum depth in the simulations for focusing at the surface.We increased the power for the TAP with φ 3 6 × 10 5 fs 3 to 33.3 × E thr (Table S3 of Supplement 1), reaching the critical power for self-focusing.The simulated ablation depth of 15.5 μm is more than 4 times larger than the Rayleigh range.In terms of the temporal asymmetry and pulse duration, we increased the TOD parameter to φ 3 2 × 10 6 fs 3 , with a corresponding pulse duration of 5.1 ps.The simulated hole depth for 2.5 × E thr is around 4.7 μm, which is only slightly deeper than for φ 3 6 × 10 5 fs 3 .However, the higher the temporal asymmetry and pulse duration, the more energy it takes until the critical power for self-focusing is reached compared to a low TOD parameter and shorter pulse durations.Furthermore, the simulations show that the channel diameter is nearly constant over the Rayleigh length.Thus, focusing below the surface can double the range, as seen in Fig. 2(d).
Note that similar observations have been reported using pulse durations of about 0.5 ps [31].However, as discussed above, a large fraction of the pulse energy does not contribute to the ablation process for temporally symmetric laser pulses.
Other approaches [32][33][34][35][36] in which the resulting dimensions of the structures are comparable to ours apply powers well above the critical power with similar numerical apertures, and thus nonlinear propagation effects become appreciable, like self-focusing and filamentation [37].In these investigations the laser pulses are either focused into the bulk with multipulse irradiation [32] or onto the surface with single laser pulses [33].In the latter case, spherical aberrations are also used to elongate the focal region in the propagation direction.However, for similar hole depths and aspect ratios, pulse energies approximately 1 order of magnitude higher are required.A second method for high aspect channel generation is the utilization of spatial pulse shaping techniques, leading to nondivergent Bessel beams, which may exceed the critical power for self-focusing.Long and thin conical profiles with a very high aspect ratio of a secondary crater in borosilicate glass were observed due to spontaneous reshaping of the incoming Gaussian pulse into a Gaussian-Bessel pulse [34], and nanochannel arrays with Bessel beams, focused onto the backside of the material generated via a virtual axicon, were presented as well [35].It was also demonstrated that temporal stretching of the laser pulse can be beneficial for energy deposition into the material with Bessel beams [36].
Indeed, we show that temporally asymmetric pulse shaping can be utilized for the generation of nanochannels as a temporal analog to spatial pulse shaping like Bessel beams.However, the important point is that for TAP, much less pulse energy is required and all resulting high aspect structures are generated with single laser pulses well below the threshold for self-focusing.
In general, we expect that temporally shaped pulses will also create deeper structures in dielectric materials without a fast trapping mechanisms, as this simple "seed and heat" picture is valid in those materials as well [17,30,31,38].
Just as in the case of fused silica, the earlier part of an elongated pulse will create an electron density well below the damage threshold, leading to moderate absorption.Since the remaining part of the pulse will increase the conduction band electron density step by step, the parts with low intensity at the end of the pulse are affected most from the absorption.In any case, we expect that TAPs will be the most efficient choice for the generation of nanochannels, because nearly the entire pulse contributes to the excitation and nearly no energy is wasted compared to long temporally symmetric laser pulses.The latter aspect could be important in laser machining with competing absorption processes.
Within this context we note that in sapphire, a dielectric material with no distinctive trapping bands, TAPs are able to generate structures with a higher aspect ratio than bandwidth-limited pulses, as was recently demonstrated by Hernandez-Rueda et al. [38].In that study, the resulting ablation structures were evaluated by atomic force microscopy (AFM).However, in the case of a high aspect ratio and narrow ablation structures with dimensions close to the AFM tip, the resolution of that technique is limited and might not be able to indicate the real depth.To verify this prediction, postmortem analysis providing full spatial information is a fitting approach.

SUMMARY AND CONCLUSION
We demonstrated the efficient generation of uniform high aspect ratio ablation structures in fused silica utilizing single-shot TAPs in a regime where the chosen intensity was well below the threshold for self-focusing and thus filamentation.Neither spatial beam shaping methods nor nondivergent beam propagation techniques were applied.
TAPs consist of a short and intense pulse where the peak intensity is well below the intensity needed to create a critical electron density and therefore penetrate deep into the material.The energy density needed for ablation is reached with the trailing part of the laser pulse, making use of AI and a continuous re-excitation of STEs.Heat diffusion is limited by restricting the overall pulse duration to a few picoseconds.Under moderate focusing conditions (NA 0.5), this results in high aspect ratio ablation structures (∼30∶1) with channel widths in the range of a few hundred nanometers.Similar ablation structures with lower aspect ratios were obtained for temporally comparable but symmetrically shaped pulses; however, approximately twice as much pulse energy was required.Furthermore, we controlled the structure depth by tuning the asymmetry, i.e., the pulse duration, of the temporally shaped laser pulses: the more pronounced the asymmetry and the longer the pulse duration, the deeper the structures.
An adapted simplified rate equation model (see Supplement 1) has been developed as supporting evidence.Numerical simulations and experimental results show good agreement with respect to hole depth.
We argued that similar mechanisms could be exploited in other dielectric materials as well.Hence, our results enable a fast processing technique for tailored design in several optical and photonic applications as well as for implementations in nanosurgery.

Funding. Deutsche Forschungsgemeinschaft (DFG) (SPP1327).
Temporal Airy pulses for controlled high aspect ratio nanomachining of dielectrics: supplementary material NADINE GÖTTE,1 THOMAS WINKLER, 1 TAMARA MEINL,2 THOMAS KUSSEROW, 2 BASTIAN ZIELINSKI, 1 CRISTIAN SARPE, 1 ARNE SENFTLEBEN, 1 HARTMUT HILLMER, 2 THOMAS BAUMERT, 1,* Where CB VB TB tot ( ), ( ), ( ), N t N t N t N are the time-dependent electron density in the conduction, valence, trapping band and the total (excitable) valence electron density; ( ) , , are the multiphoton (avalanche) coefficients for excitation from the valence band and trapping band into the conduction band, respectively; T 150 fs t = is the trapping time in fused silica [1] and is the retarded time [2], [3] for avalanche ionization with corresponding to the number of required photons with an energy of ph E for impact ionization [4], [5] and s,fix 1 fs τ = is a fixed electron scattering time used to calculate the retarded time [6].This takes into account that a conduction band electron requires a finite time to reach the critical energy for impact ionization [5], [3] due to several single photon absorption and scattering events.Relaxation processes from the trapping to the valence band take place on a much longer timescale of a few hundreds of picoseconds [7], [8] and are thus neglected.
As we concentrate our studies to an excitation regime, where tunnel ionization has only a minor contribution, it is not included in our simulations.Furthermore the ponderomotive force, which causes an increase in bandgap and photonicity, is also neglected in our model.

Intensity evolution of the laser pulse
in ( , , ) I t r z is the time-and radial-dependent intensity that penetrates into the material and is calculated by Supplementary Material 0 I is the incident intensity at material's surface, where absorption in air for bandwidth-limited pulses is already subtracted.Absorption in air is calculated in the same way as the absorption inside the sample, however the excitation modelling is based on results from Liu et al. [9].( , ) 1 was calculated using the Fresnel equations at normal incidence in combination with the Drude model.The refractive index of air is air 1 n = and the complex refractive index exc n  of the excited fused silica sample is defined in equation (S5).For bandwidth limited laser pulses, however, the refractive index of air is also affected by the excitation above the sample, leading to an electron density dependent expression, as in the case of fused silica.The depth-dependent laser intensity is calculated by the following differential equation in propagation direction [10]: The first two summands (absorption due to multiphoton ionization) are neglected as it is small in comparison to free carrier absorption in the case of the shaped laser pulses.The second term describes the Beer-Lambert absorption where 2 Im n c ω α′ = ⋅  is the absorption coefficient and Imn  the imaginary part of the complex refractive index.The complex refractive index in turn, changes as the material gets excited exc n n →   due to the decreasing valence band electrons.This is described by the dielectric function ( ) ε ω of the excited material [11] for the laser angular frequency ω based on the Drude-Model: ) The first two terms describe the contribution of the valence band electrons to the dielectric function, where C M β is the Clausius-Mossotti parameter [11] chosen such that the refractive index of the unexcited material is Parameters for beam waist and pulse duration are taken from experimental data.

Determination of the parameters for simulation
One set of parameters for excitation coefficients of MPI σ and AI α for excitation from the valence ( ) and trapping band ( ) into the conduction band was extracted from the measured data.To that end, the parameters for all pulse shapes were fitted such that the simulated threshold energies matched the measured ones most closely (Table S1).
Table S1: Threshold values for material ablation (experiment, simulation and quotient), calculated Keldysh parameter γ [11], [15] for the peak intensity at This value takes into account the accumulated energy which is deposited into the material due to the excitation of electrons from the valence and trapping band into the conduction band via the different excitation mechanisms (MPI and AI), respectively [14].The energy density deposition value in our model is equivalent to the commonly used criterion in which the conduction band electron density CB ( ) N t exceeds a value of 21 -3 1 0 cm .All parameters used are summarized in Table S2.

Discussion of nonlinear propagation effects
Nonlinear propagation effects like plasma defocusing are neglected as well as self-focusing and filamentation, as the longest temporal Airy pulses with third order dispersion of × , which is one order of magnitude below the critical power for self-focusing (see Table S3).The critical power cr P for self-focusing of Gaussian beams is given by [22]: Where n is the linear and 2 n is the nonlinear index of refraction for fused silica.Values for 2 n are reported to be in the range from 3.54 10 cm W − × [23], [20], [17], [24].
For a wavelength λ of 785 nm and the largest value for 2 n of

cm
Even at an applied pulse energy of 250 nJ, where the incoming power of

MW P =
exceeds the critical for filamentation by a factor of approx. 2 (see Table S3), no filament-like structures could be observed by applying bandwidth-limited pulses (compare Figure 2).
Table S3: Calculated ratio of pulse power to critical power for self-focusing and filamentation, respectively of different pulse shapes and intensities.The critical fil P power for filamentation is calculated by )-3(c).

Fig. 1 .
Fig.1.Schematic of the experimental setup for generating high aspect ratio nanochannels in fused silica together with a scanning electron micrograph of a created high aspect ratio channel (lower right).The measured spot diameter of 2 μm is indicated by the red arrows on the sample.

Fig. 2 .
Fig. 2. Machining results for (b), (c) bandwidth-limited pulses, (d) TAPs with φ 3 6 × 10 5 fs 3 , and (e) temporally symmetric pulses with φ 2 1.5 × 10 4 fs 2 at three different focus positions with respect to the surface of the fused silica sample as indicated in (a).For each focus position, scanning electron microscope images of cross-sections obtained by FIB milling (left) are compared to simulations (right).Experiments in (b), (d), and (e) were performed at approximately 2.5 × E thr for the corresponding pulse shape.Results for bandwidth-limited pulses approximately 6 times above their threshold and corresponding to the same pulse energy as used for shaped pulses in (d) are displayed in (c) showing shallow structures.The peak intensities of the applied pulses in the pulse pictograms are normalized to the simulated bandwidth-limited pulse shown in (b).The dark shaded areas close to the hole in the scanning electron microscope images are attributed to densified material after material processing.Vertical white lines in the middle column in (d) and (e) are FIB milling artifacts, and horizontal white bars in (d) and (e) indicate the measured hole depth.

Fig. 3 .
Fig. 3. Experimental hole depth (blue squares), simulated hole depth (solid blue line), and measured aspect ratio (green diamonds) with respect to focus position (minus/plus: focus below/above surface) for four different pulse shapes.(a)-(c) TAP shapes with decreasing TOD parameter and correspondingly decreasing statistical pulse duration [(a) φ 3 6 × 10 5 fs 3 ; 1.5 ps, (b) φ 3 3 × 10 5 fs 3 ; 770 fs, (c) φ 3 1 × 10 5 fs 3 ; 260 fs], where the pulse duration in (a) corresponds to the data in Fig. 2(d).(d) Temporally symmetric pulse shape with φ 2 1.5 × 10 4 fs 2 (1.4 ps pulse duration) corresponding to the data shown in Fig. 2(e).Each set was recorded with a pulse energy 2.5 × E thr .All analyzed pulse shapes are identified by cross-correlation measurements and included as insets in (a)-(d).All cross-correlations are normalized in intensity.The time axes in (a)-(c) are shown for a range of 1 ps, whereas it is 4 ps in (d).Error bars stem from an inaccuracy of 0.5 μm for the focus position and from postmortem hole depth analysis.The aspect ratio is calculated as the hole depth divided by the full channel width at half depth.

Fig. 4 .
Fig. 4. (a) Cumulative simulated time-dependent total energy density W transferred to the fused silica sample for bandwidth-limited pulses (dotted line), TAPs with a TOD of φ 3 6 × 10 5 fs 3 (solid line), and temporally symmetric pulses with φ 2 1.5 × 10 4 fs 2 (dashed line), each at their corresponding ablation threshold E thr .For all pulse shapes the same accumulated energy density of W cr 1.4 kJ cm 3 has been transferred at the end of the laser pulse.(b)-(d) Simulated time-dependent conduction band electron density (black line) in the first layer (1 nm) below the surface in the center of the laser pulse for (b) bandwidth-limited pulses, (c) TAPs with φ 3 6 × 10 5 fs 3 , and (d) GDD-shaped laser pulses with φ 2 1.5 × 10 4 fs 2 , each at their corresponding ablation threshold.The intensities of the temporal pulse shapes (red shaded, right y-axis) are normalized to the bandwidth-limited pulse in (b).Note the different plot range for (b) for conduction band electron density and the temporal close-up of the pulse shape.The black shaded area in (d) shows the part of the intensity that does not contribute to the ablation energy density.
radius at a distance z to the beam waist w0 (measured to be 1 µm)

m
term accounts for the change of the optical properties by the free electrons via the plasma frequency as the electron charge and its corresponding effective conduction band mass, 0 ε is the vacuum permittivity.Scattering time calculation for s τ were performed similar toJiang and Tsai   [12],[10].A constant term for electron-phonon ( e-p 1 fs τ = )scattering and an electron density-dependent contribution and was employed:

× and the fraction of the intensity for which 1 γ
> (MPI is dominant) for each pulse shape.The last two columns indicate the fractional energy contribution for excitation from valence to conduction band (VB  CB) and trapping to conduction band (TB  CB) with respect to the whole excitation energy, respectively.BWL = bandwidth-limited, TOD = third order dispersion, GDD = second order dispersion.the material as an ablation criterion.
ϕ = + × used have a power of only 0.131 MW P = for the main pulse at thr 2.5 E Listing of simulation parameters, which are used for numerical simulations of SiO2.VB = valence band, CB = conduction band, TB = trapping band.
and Bergé[21].For temporal Airy pulses, only the contributions of the main pulse (energy main E and power main P ) to cr P and fil P are considered and listed below.main I corresponds to the peak intensity of the main pulse.BWL = bandwidth-limited, TOD = third order dispersion, GDD = second order dispersion.