Abstract

Ultrashort laser pulses allow for the in-volume processing of glass through non-linear absorption, resulting in permanent material changes and the generation of internal stress. Across the manifold potential applications of this technology, process optimization requires a detailed understanding of the laser–matter interaction. Of particular relevance are the deposition of energy inside the material and the subsequent relaxation processes. In this paper, we investigate the spatio-temporal evolution of free carriers, energy transfer, and the resulting permanent modifications in the volume of glass during and after exposure to femtosecond and picosecond pulses. For this purpose, we employ time-resolved microscopy in order to obtain shadowgraphic and interferometric images that allow relating the transient distributions to the refractive index change profile. Whereas the plasma generation time is given by the pulse duration, the thermal dynamics occur over several microseconds. Among the most notable features is the emergence of a pressure wave due to the sudden increase of temperature and pressure within the interaction volume. We show how the structure of the modifications, including material disruptions as well as local defects, can be directly influenced by a judicious choice of pulse duration, pulse energy, and focus geometry.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Ultrashort laser pulses offer various possibilities for in-volume structuring and modification of transparent materials [1,2]. Due to the high intensities in the focal region, non-linear absorption takes place and causes different modifications, depending on the pulse energy, focusing conditions, and repetition rate. For example, homogeneous refractive index changes can be induced to inscribe optical waveguides for linear [3,4] and non-linear [5] applications, nanogratings can be generated allowing birefringent modifications [6,7], while microexplosions can be triggered for efficient glass cutting [810].

For dicing applications, several attempts have been made to tailor the laser–matter interaction. Notably, spatial beam shaping concepts, such as Bessel-beam [1113] or filament applications [14,15], provide extended interaction volumes. Temporal shaping, e.g., burst [1618] or asymmetric pulses [19], enhances structural changes and disruptions or combined spatio-temporal solutions [20,21] reduce unwanted non-linear effects and improve the energy transfer from the laser pulse to the solid.

Clearly, the multitude of parameters precludes a blind trial-and-error approach for process optimization. Rather, a detailed understanding of the energy deposition processes and the resulting modifications in the material is essential.

In this paper, we analyze the spatio-temporal evolution of the plasma and the subsequent relaxation processes for different pulse durations and pulse energies. From this study, quantitative information about the non-linear propagation dynamics, characteristics of energy transfer from the plasma to the lattice, and crucial temperatures for material processing can be drawn and compared. While the investigations have been performed in alkali-aluminosilicate glass (unhardened Corning Gorilla glass), the main conclusions are valid for a broad range of glass species.

2. EXPERIMENTAL SETUP

Several non-linear effects influence the energy density and extension of the generated plasma. In this context, self-focusing shows a significant influence in this study. The critical power for self-focusing [22,23] can be calculated by

Pcr=3.77λ28πn0n2,
where 3.77 is a coefficient for a Gaussian beam [24], and λ denotes the laser wavelength (1026 nm). Parameters at the denominator are n0=1.5031 [25] as the linear and n2 the non-linear refractive index. The latter quantity has been measured in Corning Gorilla glass (Code 2318) to have a value of n2=4.3·1020m2/W using the Z-scan technique [26]. This is in good comparison to other publications [27,28] Accordingly, the critical threshold for self-focusing is Pcr=2.46·106W. In this work, the pulse peak power used is in the range of 1300Pcr. This regime is typical for current approaches of ultrashort pulse laser cutting of glass.

In order to investigate the energy deposition of an ultrashort laser pulse inside a glass sample, we set up a pump-and-probe experiment for measuring the optical transmission [29,30] and the phase shift using an interferometric approach [3135], as shown in Fig. 1.

 figure: Fig. 1.

Fig. 1. Pump–probe setup with delayable, frequency-doubled (513 nm), 200 fs short probe pulse and pump pulse with 200 fs, 1 ps, 6 ps, or 12 ps pulse duration, respectively. A grating pair serves to recompress the temporally prechirped probe pulse. An additional Wollaston prism with polarizer and analyzer enables interferometric measurements.

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The laser source used is a PHAROS-SP by Light Conversion generating pulses at a wavelength of 1026 nm with adjustable pulse duration from 200 fs to 12 ps [full width at half maximum (FWHM)] and pulse energies up to 1 mJ at 1 kHz repetition rate.

3. SHADOWGRAPHIC IMAGING

Shadowgraphic imaging is the basis for the precise analysis of spatio-temporal plasma evolution and non-linear propagation effects of single laser pulses. Therefore, an external pulse picker selects single pulses and its signal acts as trigger for the camera. Pulses with different durations are generated by changing the internal pulse compressor alignment, resulting in temporally chirped pulses of desired duration. The beam is separated by a beam splitter into a pump and a probe beam. After separation, the pump beam is focused approx. 1 mm below the surface of the glass sample by a microscope objective [numerical aperture (NA) 0.35]. A microscope objective with NA 0.35 was used in our case to achieve high intensities with small focus diameters, with acceptable Rayleigh length (no need for larger NA) and minimized non-linear effects (lower NAs favor filamentation [23]). Note that due to the refractive index mismatch between the planar interfaces, a deeper focusing without correction leads to pronounced spherical aberrations [3638]. As depicted in Fig. 2, the simulated intensity distribution in vacuum (a) will be spread for a focusing depth of 1 mm (b) and drops in intensity. Additionally, another spot behind the focal region will appear. With a further increase of focusing depth ((c)=2mm, (d)=5mm), a gradual increase of the separation between the two spots as well as a further intensity spreading and more spots occur (d).

 figure: Fig. 2.

Fig. 2. Simulated intensity distribution obtained by a microscope objective (NA 0.35) and 1026 nm laser pulse in vacuum (a). Spherical aberrations occur, dependent on the defocusing depth [(b)=1mm, (c)=2mm, (d)=5mm]. The intensity distribution from (d) is widely spread and additional maxima occur (e). Laser beam is incident from the left.

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As glass sample Corning Gorilla glass (unhardened) is used. We specifically choose unhardenend glass, since it is easier to handle. However, this is not expected to cause differences because the main investigation occurs in the volume of the material, far away from the surface. The Gorilla glass (Code 2318) is a composition containing SiO2 (69 mol.%), Na2O (13 mol.%), Al2O3 (8 mol.%), MgO (5 mol.%), K2O (1.7 mol.%), and CaO (0.5 mol.%) [39]. An external grating compressor served to recompress the probe pulses to a duration of 200 fs, which allows sampling of the free carrier dynamics on relevant time scales for the energy transfer to the lattice with high temporal resolution. The achieved durations of the pump-and-probe pulse are measured by an autocorrelation measurement device (APE-pulseCheck) at the later point of interaction.

To observe the interaction region and to investigate the dynamics therein, the probe pulse irradiates the sample perpendicular to the pump pulse and is imaged by a microscope objective (20×NA 0.42) onto a charge-coupled device (CCD)-camera (transverse configuration with plane-parallel alignment of the sample to the probe beam). The spatial resolution dmin of the obtained shadowdgraphic and interferometric images is given by the diffraction limit determined by NA and wavelength of the employed observation microscope, which corresponds to dmin=800nm in our particular case.

The probe pulse is frequency doubled (513 nm) to distinguish it from parasitic background radiation from the pump beam. The probe pulse diameter is approx. 0.5 mm, and its energy is ca. 30 nJ. A bandpass filter in front of the CCD is used to avoid overexposure due to the broadband plasma radiation and scattered pump light. Reference pictures are recorded previous to the irradiation by the pump beam. Through background subtraction, the image quality is enhanced, and disturbing interferences are minimized. The probe pulse can be delayed up to 7 ns with respect to the pump pulse by adjusting the delay line. The starting point t=0ns is defined by the first occurrence of significant signal drop in the shadowgraphic image. At this starting point, the intensity of the probe beam is locally reduced due to absorption, reflection, and scattering by the plasma. Theoretical considerations based on the Drude model [40,41] show that these carriers cause a significant drop of the optical transmission for electron densities >5×1018cm3 for the probe wavelength of 513 nm. After each shot, the sample is moved to a pristine position to avoid cumulative effects inside the material. The measurement technique reported here is based on the reproducibility of the laser–matter interaction. As a prerequisite, the material used has to be homogeneous and invariant for sample translation. In all recorded images, the pump beam is incident from the left and propagates to the right.

A. Spatio-Temporal Analysis of the Dynamics of the Transmission Evolution at High Pulse Energies

At first, we analyze the spatial and temporal dynamics of the transmission changes due to the laser-induced carriers. In particular, we study the plasma formation and decay processes for high energies of the pump pulse. It is expected that this high carrier density regime causes a rather slow electronic dynamic range [42], although this will complicate the interpretation of competing excitation and relaxation phenomena. However, this is a typical processing window for glass cleaving applications [18,43], where specific damage aspects are needed [42].

During the initial ignition, a thin straight plasma channel emerges around the focal region (at z=0μm). This ionization process is mainly triggered by non-linear absorption (tunnel- and multiphoton ionization) [4446]. Subsequently, the plasma seems to grow towards the incoming beam in negative z direction and acquires an elongated shape. In addition, the transmission at z=0μm significantly drops (see Fig. 4), which indicates that the plasma becomes denser, presumably due to both multiphoton and avalanche ionization [4648].

At later sampling times (t>1ps), the high-energy part of the pump pulse couples into the existing plasma and deposits its energy within the beam caustic (indicated by white, dashed lines in Fig. 3, where the simulated caustic is applied to the actual process image), leading to a tear-shaped plasma, following the isointensity shells (see yellow shells in Fig. 3).

 figure: Fig. 3.

Fig. 3. Spatio-temporal plasma evolution and resulting material modifications with a 6 ps and 200 μJ laser pulse, imaged as false-color shadowgraphic representation for the transmitted probe-beam (transmission scale identical to Fig. 5). The generated plasma in the focal region starts to grow towards the incoming beam, following the beam caustic, on a ps timescale. On a later time scale, long-living decay and lattice temperature-associated states can be observed, which relax within several ns.

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In this stage, the transmission within the focal region drops to its minimum value below 7% (see also Fig. 4), again indicating that the plasma becomes denser. At the trailing edge of the pump pulse, the interaction zone is enlarged only minimally due to the reduced intensity. We attribute the pseudo-movement towards the incoming beam to the moving breakdown effect [49,50] as a consequence of the spatial development of exposure of light. Thereby, absorption occurs in front of the focus due to the increasing intensity and space-variant excitation [51]. In the following, we will use the term “plasma front” in terms of boundary between excited plasma and still pristine sample. This plasma front prevents, to some extent, a further injection of energy into the actual focus region, also known as distributed-shielding effect [50,5254]. After approx. 14 ps, no further expansion of the excited area (V7×104μm3) can be observed as the influx of energy ceases towards the end of the pulse, and the existing plasma recombines in accordance with its characteristic lifetime.

 figure: Fig. 4.

Fig. 4. Measured transmission evolution at z=0μm (in the focus) for different pulse durations of the pump pulse.

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For the recombination process, the specific influence of the material used is essential [55]. Measurements in fused silica show a fast relaxation of approx. 150 fs [56,57] mainly due to self-trapping of free electrons, whereas multi-component glasses such as soda lime glass or Al2O3 show decay times of 100 ps [5861]. Furthermore, some glasses such as BK7 exhibit broad absorption bands, which persist over 5 ns [42,62]. Since Gorilla glass is also a multi-component glass with a high content of Na2O and Al2O3, we expect a similar decay time of several 100 ps. Accordingly, no significant change in transmission can be measured at the beginning of this decay phase (see Fig. 4). It remains nearly constant for 100 ps. Within the next 400 ps, only a slight transmission recovery to around 20% in the focal region can be observed. In front of the focus, the transmission recovery occurs faster; compare, e.g., the region between z=40μm and z=100μm (Fig. 3). Investigations in Gorilla glass at lower pump energies [27] show a similar behavior, where a significant transmission rise sets in approx. 25 ps after ignition in front of the focus, also indicating a fast relaxation channel. In our investigations, a significant recovery of the transmission occurs in the former focal region after approx. 500 ps. The low transmission values as well as different decay speeds indicate that various transient states and defects [30,42,6366] are generated.

Subsequent probing in the ns-regime shows a transmission recovery to its initial value in front of the focus. Only at the former focal region a reduced transmission remains. This reduced transmission (70%) is permanent (sampling time 1s) and can be attributed to scattering at disruptions (see Fig. 3). The energy relaxation, however, does not only occur via defect states. Due to a laser-interaction time larger than the electron–phonon coupling time of 1ps [67,68], a significant rise in lattice temperature can be expected. Especially, the strong probe-beam absorption over several 100 ps within the focal region might indicate a superheated liquid phase transformation [6973]. These observations are in good agreement with [42], where two different structural relaxation paths are shown, dependent on the deposited energy.

Moreover, a sampling in this time regime reveals a pressure wave, which leaves the former interaction region after approx. 500 ps and propagates through the sample with an average velocity of 6.0×103m/s±0.6×103m/s, slightly above the velocity of sound in glass (5.5×103m/s). With further propagation, an exponential decrease in the velocity can be detected.

B. Influence of the Pulse Energy

We now study the influence of different pump pulse energies on the plasma evolution. In this case, an exemplary pulse duration of 6 ps (FWHM) is chosen, where general dynamics have been described before. The pulse energy is varied between 25 and 200 μJ, corresponding to P1×Pcr and P10×Pcr, respectively. Representative in situ shadowgraphic images (Fig. 5) are acquired with the pump and probe setup. Here, the maximal plasma expansion is recorded after approx. 14ps±2ps for all pulse energies in accordance with the pulse duration. Without the continued influx of energy, the plasma ceases to expand at this time.

 figure: Fig. 5.

Fig. 5. Pump–probe images for 6 ps pulse duration, NA 0.35, and different pulse energies. A false-color illustration is used for the transmission of the probe pulse after 14 ps (time of maximum plasma expansion).

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Increasing the pulse energy predominantly affects the total size of the plasma extension in front of the focal plane. The teardrop shape can be attributed to the intensity distribution within the beam caustic. The detected transmission of the plasma is almost uniform within the complete area with a value below 7%, which corresponds to the minimum reliable resolution of the transmission due to the limited signal/noise ratio. Only a region approx. 50 μm in front of the focus shows a slightly higher transmission. Additional features can be observed when the pulse energy exceeds a certain value. With pulse energies above 50 μJ, a plasma spot appears near the focus, and above 150 μJ, a second plasma spot behind the focal region is formed. With pulse energies around 200 μJ, this region becomes larger (see Fig. 5).

This behavior is attributed to aberration caused by the refractive index mismatch of the objective and focusing depth inside the glass sample. It is in good agreement with the simulated intensity profile in Fig. 2. This behavior is particularly problematic for applications such as precise cutting or drilling, where a localized energy deposition in the material is of interest, especially when using high-pulse energies.

It should also be noted that the exact shape and position of the secondary features strongly depend on the adjustment of the focusing optics. Imperfect alignment or a tilted glass sample causes astigmatism-related plasma spots at the side of the focal region (Fig. 6) and an asymmetric intensity distribution.

 figure: Fig. 6.

Fig. 6. Simulated intensity distribution (a) with a sample tilt of 2°. The pump–probe image (b) shows a plasma spot generation detached from the focal area and the microscope image (c) of the corresponding asymmetric modification.

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C. Influence of the Pulse Duration

In this section, we investigate the influence of the pulse duration from 200 fs to 12 ps, corresponding to typical pulse durations of industrial ultrashort pulse lasers.

An analysis of 200 fs pulses at 200 μJ pulse energy (P300×Pcr) shows significant differences in the plasma generation and evolution (Fig. 7) compared to the 6 ps case. Most obvious is the propagation of the transmission changes in the direction of the laser beam towards the focus, thus completely opposite to the 6 ps case discussed before. Moreover, the higher pulse peak power leads to self-focusing of the beam before the focal region is reached. Therefore, a spatial splitting of the beam into multiple filaments [23,63,74,75] is observed. These separated filaments propagate independently to the focal area with a velocity of 2×108m/s. They already generate a plasma in front of the focus before a plasma in the focal region is created. Consequently, energy from these filaments is partially used to ignite the plasma and is missing in the focal region. As a result, the recorded images do not show such a homogeneously distributed absorption in comparison to the 6 ps pulse and cover a larger area approx. 1 mm in length. Furthermore, white light emission can be detected at the backside of the sample. Also, as discussed later, the shorter pulse duration leads to a more pronounced multiphoton ionization (MPI) process where the avalanche part is significantly reduced [46,76], and the resulting material modifications differ as well.

 figure: Fig. 7.

Fig. 7. Spatio-temporal plasma evolution with a 200 fs and 200 μJ laser pulse, imaged as false-color presentation for the transmitted probe-beam (transmission color bar identical to Fig. 5). The incoming laser beam breaks down into multiple filaments that propagate independently in the laser direction.

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The analysis of 1 ps pulses at 200 μJ pulse energy (P50×Pcr) indicates that several processes are competing under tight focusing conditions, predicted by previous simulations [38]. Here, an efficient generation of plasma takes place at the leading edge of the pulse, well before self-focusing sets in. It can be seen in Fig. 8 and time step 300 fs that in the focal region as well as in front of the focus, in the form of several weak strings, a transmission change can be observed simultaneously. These strings are formed due to filaments, which grow in length over time within the incoming laser pulse direction. Furthermore, a low intense part of the pump pulse ignites a plasma in the focus, which behaves according to the moving breakdown effect and grows towards the incoming laser beam. Subsequently, the plasma from the filaments and focus merge together. After 2.5ps, the laser pulse ends, and no further growth of the interaction area can be detected. Finally, the plasma extinguishes according to the plasma lifetime. With respect to the main ionization process, the avalanche rate is still reduced compared to the 6 ps case [46], and an unobstructed energy transfer to the focal region is limited due to the filament breakup with subsequent absorption effects at the generated plasma in front of the focus [24,50,52,77].

 figure: Fig. 8.

Fig. 8. Pump–probe shadowgraphic images of different plasma evolution phases during the laser–matter interaction with a 1 ps, 200 μJ laser pulse (transmission color bar as Fig. 5). The incoming beam breaks down into multiple filaments that generate a less dense plasma.

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Experiments with 12 ps pulse duration (FWHM) and 200 μJ pulse energy show a similar behavior of the plasma formation as with 6 ps pulses. In this regime, the main driver for the plasma evolution is the moving breakdown, which lasts 23ps±2ps and ends with no further plasma expansion (exemplary Fig. 9 after 20 ps). This plasma is moving slower (106m/s) and over a shorter distance as in the case of 6 ps pulses (compare scaling in Figs. 9 and 3), strongly linked to the reduced intensities and temporal gradients. However, the extended laser–plasma interaction time leads to an increased plasma density due to a stronger contribution of avalanche ionization [2,34,45,46,67,76,78,79]. The subsequent decay process, in terms of absorption losses, is significantly longer with transmission remaining at values around 10% for more than 1.5 ns (Fig. 4), across the entire interaction region. After approximately 500 ps, a pressure wave emerges from the boundaries of the interaction area, which then propagates through the sample. The laser treatment with 12 ps pulses shows strongly modified regions and photo-thermal disruptions [2,80], which lead to mechanical stress [10] and cracks.

 figure: Fig. 9.

Fig. 9. Pump–probe shadowgraphic image of a single-shot 12 ps, 200 μJ laser pulse after 20 ps (time of maximum plasma expansion) and the resulting modification. The entire interaction area exhibits strong disruptions of the material.

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4. INTERFEROMETRIC IMAGING

The shadowgraphic-based measurements yield information about the transmission losses within the interaction area. This reduction is caused by diverse effects, e.g., different absorbing states [40,63,65,66,81], strong scattering effects around small electron gas geometries [82], complex assumptions (collisional scattering time and optical carrier masses), or a possible enhancement of the total absorption cross section due to a strong photon–phonon coupling [30]. Moreover, the transmission values exhibit a low signal-to-noise ratio and are additionally prone to fluctuations and measuring inaccuracy, overshadowing different relaxation processes. As a consequence, an exact calculation of the corresponding electron densities is hindered, and a clear separation of the different effects becomes challenging.

To overcome these problems and gain access to quantitative information over the entire interaction area, the probe arm of the setup was equipped with an interferometric Nomarski configuration [3133,35]. This particular setup is chosen to accommodate the short coherence length of the probe beam and to avoid instability and alignment issues. The probe beam is polarized before the interaction zone through a polarization filter and rotated via a half-wave plate to 45° with respect to the prism wedge orientation. Afterwards, the pulse is transmitted through a Wollaston prism, leading to two beams separated by 2° and orthogonal polarization. At the image plane, two coherent images are produced, which can interfere in the overlapping region by integrating an additional polarizer (analyzer) at 45° to the prism orientation. This setup generates interference patterns [fringes, see Fig. 10(a)], and the analyzer is fine-adjusted to obtain equal intensity distribution in both images [32]. The laser-induced modifications induce a phase shift Δϕ for the probe beam leading to a corresponding bending of the pattern [see Figs. 10(b) and 10(c)]. The fringe shift error is considered to ±1pixels, and random errors amount to ±10%. In order to determine the sign of the induced phase shift, a fused silica sample with inscribed waveguides was used for calibration purposes. Here, the phase shift is induced by a compression (densification of the material in the laser-treated zone), leading to a positive shift [1,3,83].

 figure: Fig. 10.

Fig. 10. Interference images before (a) laser irradiation and at different time steps (b) and (c) of the plasma evolution inside Gorilla glass with a 6 ps and 200 μJ laser pulse.

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In our case, the phase shift Δϕ results from a transient change of the refractive index Δn [84]:

Δϕ=2πΔndxλProbe,

where dx accounts for the diameter of the modified zone and/or pressure wave, and λProbe represents the probe-beam wavelength. To extract the induced phase shift Δϕ across the entire interaction region, a Fourier reconstruction is used. An additional background image before the laser modification is also recorded [Fig. 10(a)] and processed in the same way to remove unwanted optical aberrations. By subtracting it afterwards from the interaction image, only the changes induced by the pump beam contribute to the measured phase [see Fig. 11(a)]. Assuming rotational symmetry, an inverse-Abel transformation [85] can be applied to obtain the three-dimensional refractive index distribution from the two-dimensional projection. A Fourier algorithm [86,87] was used for increased precision and noise reduction compared to the straight-forward onion method. Here, the tradeoff between required accuracy and computational afford can be easily adapted by changing the maximal computed frequency [87]. A systematic offset error could affect all measured data by about ±15% due to the diameter determination and a small defocus of the image plane.

 figure: Fig. 11.

Fig. 11. Reconstructed phase shift (a), refractive index change (b), and line scan (c) through (b) obtained by a 6 ps and 200 μJ laser pulse inside Gorilla glass after 100 ps.

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However, this error is constant for all consecutive measured data and therefore does not affect the relative changes.

Figure 11 shows an example of this extraction and transformation for data captured approx. 100 ps after plasma ignition. Here, the measured phase shift [Fig. 11(a)] in the focal region equals Δϕ=46rad and in front of it Δϕ=24rad. Due to the different diameters, this corresponds to refractive index changes [Fig. 11(b)] of Δn=0.060.08 in the focus but only Δn0.02 in front of it. Figure 11(c) represents a line scan through this interaction region and clearly illustrates the intensity splitting due to a deeper focusing without correction [38] in good agreement with the simulated intensity distribution (Fig. 2).

Analyzing the reconstructed images, similarities and differences compared to the shadowgraphic data can be seen. At the same sampling time an alike teardrop shape and same elongation dependence with respect to different pulse energies can be observed (Fig. 12). However, while the shadowgraphic images (Fig. 5) show an almost uniform attenuation of the probe pulse over the entire interaction region 14 ps after ignition, the processed images of refractive index change reveal a different behavior, as depicted in Fig. 12.

 figure: Fig. 12.

Fig. 12. Calculated refractive index changes for 6 ps pulse duration, NA 0.35, and different pulse energies after 14ps (time of maximum plasma expansion).

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Notably, there is a strong difference between the refractive index changes obtained within and in front of the focal region (see Fig. 12) attributed to the beam caustic and different intensities reached [88]. Typical values measured in the focal region range from 0.07 to approx. 0.1 for all pulse energies used (25μJ100μJ). Increasing the pulse energy above 100 μJ does not increase the measured refractive index change significantly, but the region containing these maximum values is enlarged. The slightly increased refractive index at the tailing edge of the interaction region could be a measurement artifact or due to a further energy transfer from the trailing edge of the pulse.

Furthermore, the existence of two different relaxation pathways dependent on the energy density [42] can be confirmed. Comparing the refractive index after t=14ps (Fig. 12) and t=100ps (Fig. 11) clearly shows a faster drop in front of the focus (Δn=96%), while the relative differences are smaller in the focal region (Δn=20%).

A. Origin of the Refractive Index Changes

For further analysis, it is instructive to investigate the temporal change in refractive index. The induced change can be attributed mainly to four distinct contributions: the optical Kerr effect, the generated plasma, the thermo-optical (TO) effect [89,90] and finally the influence of trapped electrons or defects:

Δn=ΔnKerr+ΔnPlasma+ΔnTO+ΔnTrap.

The trapping term is related to various products resulting from the relaxation of free electrons into lower states. Typical defects are self-trapped excitons (STEs), which are in the range of 1018cm3 [57,65,81,91] equivalent to Δn1×104, and color centers on the order of Δn1×103 [92]. These refractive index changes are close to the resolution limit of our setup and can thus be neglected.

The contributions of Kerr effect, plasma (free electrons), and temperature change result in refractive index changes with similar orders of magnitude but different signs. However, due to their diverse physical origins, these effects occur on different time scales. During the initial ignition phase (t1ps), only the Kerr effect (positive sign) and free electrons (negative sign) contribute to the refractive index change, which, however, makes a clear separation of their respective contributions impossible. After 1ps, the electron energy is partly transferred to the lattice through electron–phonon coupling [67,68], which leads to an additional contribution due to the TO effect. As the laser pulse continues to supply energy, a combination of all three effects occurs. However, on later time scales of several 100 ps to the ns regime, the refractive index change originates only from the TO effect due to the absence of a laser pulse (hence no Kerr effect) and as this is much longer than the plasma lifetime (no free electrons).

B. Spatio-Temporal Analysis of the Refractive Index Changes at High Pulse Energies

The spatio-temporal evolution of the refractive index change will be analyzed in detail exemplary for a 200 μJ and 6 ps pulse in the following.

The temporal evolution within the focal region is depicted in Fig. 13. Within the first picosecond, a negative refractive index change can be measured. Afterwards, a strong positive increase occurs until 14 ps, followed by an oscillation. In the ns-regime the refractive index slowly starts to decrease. This behavior will be discussed in detail below.

 figure: Fig. 13.

Fig. 13. Time-resolved measurements of the refractive index change in the focal region induced by a single 6 ps and 200 μJ laser pulse. (a) Evolution during and shortly after ignition and (b) development on the ns-time scale.

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During the initial phase (t1ps), only the Kerr effect and free electrons contribute to the change of refractive index. Based on the laser parameters applied and focusing optics, a maximum intensity of 1.17×1014W/cm2 at the peak of the laser pulse can be estimated, which leads to a positive refractive index change of approx. ΔnKerr=0.05 due to the Kerr effect. However, during this very initial phase, these values are not expected to be reached yet.

The measured negative refractive index changes (down to Δn0.01, see Fig. 13) are linked to the generation of free electrons (ΔnPlasma), which can be expressed with [33,35,93,94]

ΔnPlasmane2ncr,

where ne represents the electron density, and ncr=1.06×1021cm3 is the critical electron density for the 1026 nm pump laser. Neglecting any contribution from the Kerr effect allows to obtain a lower boundary for the electron density by assuming that only the free electrons contribute at this early time. From Eq. (4), we can estimate an electron density of approx. 2×1019cm3. The actual free electron density is assumed to be higher than this value due to the contribution of the Kerr effect to the refractive index change. In fact, several measurements showed only positive refractive index changes on the order of 0.010.02 during this phase.

With further laser irradiation (sampling time 114ps), the plasma front moves towards the incoming laser beam. Here, both signs of the refractive index can be observed, e.g., as depicted in Fig. 14, approx. 3 ps after ignition. In the focal region, the former negative refractive index change flips to a positive one. And at the plasma front, a negative shift Δn0.025 is measured, corresponding to a free carrier density of at least 5×1019cm3. Especially, the sign change and strong rise of the refractive index in the focus show that, apart from the increasing Kerr effect, the free electrons begin to transfer energy to the lattice and cause a rise in temperature and pressure. This observation is in contrast to the strong absorption characteristics obtained by the shadowgraphic imaging, which suggest a strong negative refractive index due to free electrons. However, in this time domain, the glass temperature and Kerr contribution also rise, leading to a positive refractive index. This illustrates that multiple effects reduce the transmission of the probe beam, which cannot be resolved due to the limited dynamic range of the CCD camera and leads to a superposition of different processes.

 figure: Fig. 14.

Fig. 14. Refractive index changes during the plasma evolution 3ps after plasma ignition. The color bar was adjusted to visualize the different signs. Also, the scale was adapted for better visualization.

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Approx. 14 ps after plasma ignition (at the end of the laser pulse), no further expansion of the modification zone is observed, and maximum changes in refractive index are reached with values around Δn=0.1 in the focal region. Afterwards, a sharp signal drop is measured, mainly caused by the fast reduction of free carrier density [95], leading to an extinction of the Kerr effect as well as no further temperature rise.

For times t14ps, the refractive index starts to oscillate. This oscillation can be linked to a thermalization between the electrons and ions/atoms [2,95]. This leads to a buildup of high temperature and pressure in the interaction region, which can be described with a thermoelastic model and is observable over several ns [84,9699]. It indicates a non-equilibrium state where a tremendous pressure buildup happens around the unaffected area [10,100]. Here, the pristine surrounding area is still at room temperature because thermal diffusion takes place within the μs-time regime [2].

At t=500ps, the measured refractive index change in the former focal region is Δn=0.08. An additional oscillation [see Fig. 13(b)] after approx. 2.2 ns (width 1.4 ns) can be observed. We assume that this oscillation is caused due to the buildup of a secondary pressure wave and its subsequent relaxation, a so-called transversal acoustic shear wave [42,101,102]. This stress wave leads to efficient glass separation [103] but could not be imaged in our shadowgraphic setup due to the limited time delay and setup configuration. After 4 ns, a refractive index change of 0.06 is reconstructed, and a regime sets in where only slight changes of the refractive index can be measured. Now the temperature slowly reduces due to thermal diffusion. The refractive index further declines, and after 7 ns, Δn equals 0.05. In front of the focus, Δn=0.020.002 is measured at this time, significantly lower than in the focal region.

We assume that on these longer time scales, the lattice temperature is mainly responsible for the change of the refractive index due to the TO effect [104]:

ΔnTO=dndT·ΔT=(n21)(n2+2)6n(αβ)·ΔT.

This temperature-dependent refractive index is a result of the competition between polarizability α (distortion of the electron cloud and molecular interactions) and the thermal expansion coefficient β (density change) [104,105] often described by the Prod’homme temperature derivation [89] of the Lorentz–Lorenz equation. Here, α and β are both temperature dependent and dictate the sign of the refractive index change. This can be either positive or negative, depending on the material composition and temperature.

The temperature-dependent refractive index of glass melting with different glass compositions was analyzed in Ref. [105] based on ellipsometric measurements.

One of the glass compositions investigated in Ref. [105] (12.5%Al2O3, 12.5%Na2O, 75%SiO2) is very similar to the Gorilla glass used in our research. The measurements reveal a negative refractive index change from 400 K to 1450K. Temperatures higher than 1450 K (melting point) yield positive refractive index changes compared to room temperature. The refractive index change shows a positive linear slope of dn/dT=6.4×1061/K up to 1800 K (highest value investigated).

A very rough estimation of the temperatures based on these equilibrium values (and simple linear extrapolations and heat independent constants) using [106]

Δn=π1/2n0(dndT)ΔT,

leads to approx. 6600 K in the focal region after 7 ns. Considering the temperature offset of 1450 K to reach a positive refractive index, changes would further increase the transient temperature estimation. Note that this value is far too high. This indicates a temperature significantly over the melting point, where a state of warm dense matter [107] or superheated liquid phase [69,70,72,73] in the focal point is achieved. For an exact calculation of the temperature and pressure within this area, a multi-physical simulation [108] would be needed, combining electrodynamics [48], heat transfer interaction [109], thermo-elastic considerations [110], hydrodynamic models [111], and molecular interactions (e.g., Young’s modulus changes with the temperature [112]). Furthermore, the presence of excited or trapped carriers needs be considered due to the measured high transmission losses (see Section 3.A). This complex theoretical approach exceeds the experimental character of this paper.

However, in front of the focus, the refractive index increase is significantly smaller. The high transmission values in front of the focus show no further presence of free or trapped carriers; thus, an electronic influence on the refractive index can be neglected, and the measured change is mainly due to the TO effect. According to Eq. (6), a temperature of 2650265K can be estimated in this region and sampling time, which is around the melting temperature.

Further information can be obtained about the released pressure wave (longitudinal wave) after approx. 500 ps, which is a form of energy transfer to reach a state of equilibrium. This pressure wave is leaving the hot interaction area and propagates within unaffected material outside the interaction zone.

Assuming that the measured positive refractive index change (Δn=0.03) from the pressure wave is mainly induced by a density change δρ/ρ0 the refractive index change ΔnΔρ can be described with [97,113]

ΔnΔρ=(n01)(δρρ0).

Here, n0 and ρ0 are the refractive index and density of the pristine glass, respectively, leading to a relative change of density of δρ/ρ0 of 6×103 according to Eq. (7), with the equation [84,113,114]

p=Y3(12ω)(δρρ0),
where Y is Young’s modulus (71.7 GPa at room temperature) and ω the Poisson ratio (0.21) [25]. According to Eq. (8), the pressure of this wave can be estimated to about 250 MPa. Here, the Young’s modulus and Poisson ration values at room temperature can be used due to the propagation within the cold surrounding material.

Sampling after 1 s reveals a final negative refractive index change of Δn=0.02 only in the focal region [see Figs. 15(b) and 15(c). This can be attributed to a less dense material or even small voids and cavities [810,73,115], which are too small to be resolved in our setup (resolution limit dmin800nm). Furthermore, at the boundaries of this region, a higher refractive index change =0.005 can be measured. We assume that this is due either to a heat-affected zone [115] or most likely to a material densification. A reason for this densification can be the rarefaction and/or void generation in the focus [116]. Thus, the displaced material is compressed at the edges of the interaction region. Taking Eq. (8) into account, the measured Δn in the focal region equals a stress of approx. 400 MPa, which is in good agreement with the 500 MPa stress fields observed with similar laser parameters [117119].

 figure: Fig. 15.

Fig. 15. Refractive index change induced by a single 6 ps and 200 μJ laser pulse, 7 ns after ignition (a), remaining modification after 1 s (b), and corresponding line scan from A to B (c). Color bar at (a) is equal to Fig. 12, and (b) was adjusted for better visualization.

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5. MATERIAL RESPONSE AND MEASUREMENTS OF THE FINAL STATE

In this section, we will link the main features observed in the pump-probe measurements to the resulting permanent modifications. Therefore, different ex situ methods are used. A microscope with bright-field illumination is used to identify structural changes. For the investigation of defect states, a spectroscopic analysis with a Lambda 950 Spectrophotometer from PerkinElmer for transmission measurements is used. A photoluminescence (PL) setup was established, including various laser sources, emitting wavelengths at 343 nm, 515 nm, and 633 nm, and an Ocean Optics USB4000-UV-VIS spectrometer. Furthermore, a Raman spectrometer (Renishaw, inVia Raman Spectrometer) with a 532 nm laser source as excitation wavelength is used to determine network changes in the glass structure.

Typical microscopic images of a volume modification can be seen in Fig. 16, where the interaction area is imaged perpendicular to the focusing direction. After material processing with pulses at 6 ps and 25 μJ, the recorded microscopic images show a close correlation between the former plasma interaction area and later detected modifications, where two different types of modifications can be observed [see Fig. 16(a)]. In the focal region, a material disruption [9,120] (crack, void, cavity, or damaged region) with a length of approx. 15 μm can be observed, which roughly matches the Rayleigh length of 17 μm.

 figure: Fig. 16.

Fig. 16. (a) Microscopic image of a single-shot volume modification (pulse energy 25 μJ). A disruption in the focal region and dark colored tear-like shape in front of it can be recognized. (b) Microscopic image of single shot volume modification (pulse energy 200 μJ) with longer disruption in the focal region and broader and longer tear-like formation of color centers and refractive index change in front of the focus.

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Furthermore, in front of the focal area, a discolored, teardrop-shaped, 85 μm long region can be observed. Due to the shading, we conclude that in this region, absorbing point defects [51,63,65,121,122] are generated. For higher pulse energy, e.g., 200 μJ [Fig. 16(b)], the damaged region is increased to 75μm, and also the discolored area has an increased width (25 μm) and length (275 μm).

The difference between the two different modification areas can be linked to the in situ measured refractive index changes in the interaction region [see Figs. 11(b) and 11(c)]. The induced refractive index changes decline towards the laser direction (according to the reached iso-intensities [88]) and strongly differ between the focal point and in front of it. This inhomogeneity yields temperatures significantly over the melting point within the focal region and temperatures around and below the melting point in front of the focal region. The latter temperatures do not reach sufficient values to induce large-scale thermal disruptions, but nevertheless cause modifications of the material structure (e.g., bond breaking or density changes) [24,34,43,51,54,77,123,124].

In order to reveal the content of these discolored modifications, a sample with a large-scale modification area, generated with 200 fs pump pulse duration, was processed and spectroscopically analyzed. As shown in the shadowgraphic pump–probe setup, inducing short pulse durations below 1 ps leads to longer interaction regions (Fig. 7). However, due to the lower lattice temperatures, compared to the longer pulses, the resulting permanent modifications are limited in visibility by the bright-field microscope. Mostly, a weak brown shading in the former interaction region can be measured, indicating the generation of metastable electron-hole modifications that result, e.g., in dangling bonds [e.g., E-centers or non-bridging oxygen hole centers (NBOHCs)] or Frenkel defects [63,65,125127]. Transmission measurements on processed large-scale samples show a strong and broad absorption spectrum (see Fig. 17) with a pronounced peak around 4.1 eV and shoulders around 2 eV and 3 eV. This indicates that various defects (see Fig. 18) are generated due to the laser exposure [65,128,129]. Therefore, it becomes obvious that the resulting absorption curve can be deconvoluted into the contributions from the different defect centers, as depicted in Fig. 18. In order to enable this deconvolution, the different defect centers with their specific positions and FWHM values were used. The corresponding amplitudes were adjusted (best fit) to the measured total absorption spectrum, which results in a good agreement with the convolution curve and measured absorption spectrum.

 figure: Fig. 17.

Fig. 17. Transmission of pristine Corning Gorilla glass in comparison to samples treated with pulses of 200 fs and 200 μJ and corresponding absorption spectra.

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 figure: Fig. 18.

Fig. 18. Convolution of the generated color centers within Corning Gorilla glass due to a low dense plasma.

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By thermal annealing (1 h at specified temperatures) of this sample and subsequent transmission measurements, a reduced absorption can be detected (see Fig. 19). This confirms the occurrence of various defects, which can be annealed at different temperatures. Our experiments show that especially the low-energy part of 2 eV can be annealed with temperatures around 200°C and the shoulder around 3 eV with 300°C, respectively. The clear change in absorption between 150°C and 200°C can be linked to trapped hole centers, H3+ (band peak 1.97 eV, width 0.5eVW1/2, also OHC2 center) and H2+ (band peak 2.68 eV, 1.0eVW1/2, also OHC1 center) [130133], which can be annealed with 177°C [130]. Here, the sodium content inside the glass matrix acts as a precursor for these defects [130]. The bleaching of the absorbance with 300°C points to the annealing of trapped-electron centers E3 (band peak 4.06 eV, 1.16eVW1/2) [130,131], associated with two non-bonded oxygen atoms on a tetrahedron. These trapped-electron centers are thermally stable up to 227°C, where the soda content is the precursor again [130,132]. The presence of two Al-OHC (2.3 eV, 0.9eVW1/2 and 3.2 eV, 1.0eVW1/2) and Al-E’ centers (4.1 eV, 0.214eVW1/2) [134] can also be detected.

 figure: Fig. 19.

Fig. 19. Absorption spectra of irradiated (200 fs, 200 μJ) Corning Gorilla glass after thermal annealing.

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During annealing, the Al-E’ absorption peak at 4.1 eV also decreases in intensity but remains measurable up to 700°C. However, a slight shift of the peak position to 4.5 eV can be observed. This signature can be linked to [AlO4]0 centers, which are reduced and partly transformed to Eδ (stable up to 700 K) and Eγ centers (stable up to 1000 K) [135]. This description is in agreement with the PL signal (see Fig. 20) with a centered emission band around 550 nm upon 325 nm laser excitation, which can be attributed to the Eδ centers [136]. Further laser excitation with 523 nm and 633 nm showed no additional emission band, indicating that NBOHC’s or other PL active defect states were not generated.

 figure: Fig. 20.

Fig. 20. Photoluminescence signal of Eδ centers excited with 325 nm radiation of the laser-treated (200 fs pulse duration) sample.

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In conclusion, aluminum and sodium act as precursors for defects and color centers. The generated electron and hole centers undergo an annealing with lower temperatures compared to the E centers. Nevertheless, these color centers can enhance the absorption of follow-up pulses. Moreover, they could lead to an alteration of the beam propagation and influence the conditions for self-focusing [30] or energy deposition [16,137,138] during multi-pulse experiments [139]. In addition, it is possible that these color centers can be transformed to other defect states to generate waveguides [140] through multiple pulses per spot.

To reveal the structural network changes within the focal region, we applied single shots with 12 ps pulse duration. Here, disruptions similar to the modifications within the focal region generated with 6 ps pulses can be observed over the entire interaction region. Thus, these changes can be better resolved due to less scattering on cracks and larger measurement regions.

Strong features in the Raman spectra (see Fig. 21) are visible at 483, 559, 800, 996 and 1093cm1, corresponding to delocalized vibrations of the T-O-T bridging bonds (where T=Si or Al) [141] of the threefold Si and Al rings, Al-O-Al bridges [142], the ω3 peak correlated to the Si-O-Si network [143,144], (Si,Al)-O vibrations in Q3, and (Si,Al)-O0 (O0-bridging O atoms) in Q4 units [145,146], respectively.

 figure: Fig. 21.

Fig. 21. Raman spectra of a non-modified and 12 ps single pulse structured Gorilla glass sample.

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Due to the laser exposure and subsequent rapid cooling, a slightly reduced concentration of the threefold T-O-T rings and the Al-O-Al bridges is noticed. The constant intensity of the ω3 peak indicates that the relative number of SiO4 tetrahedra is unchanged, in agreement with publications concerning fused silica [128,147,148]. The intensity decreases at 1000 and 1100cm1 indicate that the vibrational oxygen bonds [143] lose their interconnection, in which oxygen is released to the matrix and may form voids or cavities. The fact that no other changes of the Raman signal could be observed is indicative of an otherwise largely intact glass network.

Samples processed with pulse durations below 1 ps do not show measurable structural changes. However, this might be partially due to an annealing of the defects during Raman measurement.

6. CONCLUSION

The controlled and uniform energy deposition is a key factor for efficient and high-quality in-volume modification of glass. Therefore, several spatio-temporal beam-shaping approaches have been employed to improve the process. However, a full optimization and tailoring of the induced structural changes requires a detailed knowledge of the transient absorption and decay characteristics. In this work, a pump–probe setup was used to investigate the fundamental laser–matter interaction between ultrashort laser pulses (200fs12ps) and Corning Gorilla glass to analyze the free-carrier dynamics and to identify the origin of material changes. We observed two main effects that influence the plasma generation: “filamentation” and “moving breakdown.”

Experiments with 200 fs pulses showed that the incoming beam breaks up into single filaments, which propagate separately through the sample. This reduces the available energy and in turn prevents the effective energy transfer into the focal region. Investigations with 1 ps showed a simultaneous development by filamentation and moving breakdown, which merge subsequently. In contrast, longer pulses of a few ps duration showed no spatial beam fragmentation. The time-resolved analysis reveals the moving breakdown characteristic: the plasma starts at the focus, confined in a thin channel, grows mainly towards the incoming laser beam, and subsequently broadens according to the beam caustic. Measuring the transmission through the modification zone reveals an energy dependence of the relaxation pathways. In the focal region, low transmission values can be observed for several ns, indicating long-living states during the relaxation, while in front of the focus, a faster relaxation within several 100 ps occurs.

Quantitative measurements of the time-dependent refractive index were obtained using an interferometric pump–probe microscopy setup. The observed refractive index changes were attributed to three distinct contributions: Kerr effect, free electrons, and the TO effect as the main contributions. Measurements with a pulse duration of 6 ps showed maximum refractive index changes at approx. 14 ps, when the end of the incoming pump pulse is reached. Values around Δn=0.1 in the focal region and Δn0.010.05 in front of the focus were measured. Due to the electron–phonon coupling time of approx. 1 ps, the free electron density decreases, and energy is transferred to the lattice. The resulting increase in temperature leads to a significant increase in the refractive index. Moreover, an oscillation of the refractive index with a period in the range of several ns can be detected. This can be linked to a rapid pressure buildup, where finally a longitudinal pressure wave (pressure 250MPa) emerges and subsequently propagates through the sample. After 4 ns, only slight changes of the refractive index can be detected on a ns time scale. After 7 ns, a temperature significantly above the melting point can be assumed in the focal region, leading to a superheated liquid phase. In front of the focus, temperatures around the melting point are reached. Furthermore, the energy-dependent relaxation by a fast and slower pathway was confirmed. The resulting permanent modifications strongly depend on the pulse duration used. For durations below 1 ps, mainly non-stable point defects are generated. The aluminum and sodium contents in the Gorilla glass matrix act as precursors for various oxygen-hole and Al-E’ centers. In contrast, the use of longer pulse durations (12 ps) results in large disruptions at the former plasma interaction area, mainly through cracks and voids. Raman measurements reveal that the glass matrix remains largely intact but the T-O-T bonds, Al-O-Al bridges, and the vibrations in the Q3 and Q4 units were slightly reduced.

These investigations shed light on the complex processes during in-volume ultrafast laser processing of glasses and pave the way towards systematically and precisely controlling the size and character of the modifications through a judicious choice of spatial and temporal exposure parameters. In order to accommodate the demands of highly sensitive applications, the influence of cumulative effects will be the subject of future investigations.

Funding

ZIM program, Bundesministeriums für Wirtschaft und Energie (BMWi) (AiF Projekt KF3207603CR4); Bundesministerium für Bildung und Forschung (BMBF) (ScULP3T, FKZ: 13N13930); International Research Training Group (GRK 2101); Deutsche Forschungsgemeinschaft (DFG); TRUMPF Laser- und Systemtechnik GmbH (Scholarship for Klaus Bergner).

Acknowledgment

Klaus Bergner acknowledges the TRUMPF Laser- und Systemtechnik GmbH for a scholarship.

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2018 (2)

A. Rudenko, H. Ma, V. P. Veiko, J.-P. Colombier, and T. E. Itina, “On the role of nanopore formation and evolution in multi-pulse laser nanostructuring of glasses,” Appl. Phys. A 124, 63 (2018).
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J. Bonse, T. Seuthe, M. Grehn, M. Eberstein, A. Rosenfeld, and A. Mermillod-Blondin, “Time-resolved microscopy of fs-laser-induced heat flows in glasses,” Appl. Phys. A 124, 60 (2018).
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2017 (5)

2016 (5)

C. Mauclair, A. Mermillod-Blondin, K. Mishchik, J. Bonse, A. Rosenfeld, J.-P. Colombier, and R. Stoian, “Excitation and relaxation dynamics in ultrafast laser irradiated optical glasses,” High Power Laser Sci. Eng. 4, e46 (2016).
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D. Grossmann, M. Reininghaus, C. Kalupka, M. Kumkar, and R. Poprawe, “Transverse pump-probe microscopy of moving breakdown, filamentation and self-organized absorption in alkali aluminosilicate glass using ultrashort pulse laser,” Opt. Express 24, 23221–23231 (2016).
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F. Hendricks, V. Matylitsky, M. Domke, and H. P. Huber, “Time-resolved study of femtosecond laser induced micro-modifications inside transparent brittle materials,” Proc. SPIE 9740, 97401A (2016).
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R. Torchio, F. Occelli, O. Mathon, A. Sollier, E. Lescoute, L. Videau, T. Vinci, A. Benuzzi-Mounaix, J. Headspith, W. Helsby, S. Bland, D. Eakins, D. Chapman, S. Pascarelli, and P. Loubeyre, “Probing local and electronic structure in warm dense matter: single pulse synchrotron x-ray absorption spectroscopy on shocked Fe,” Sci. Rep. 6, 26402 (2016).
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P. K. Velpula, M. K. Bhuyan, F. Courvoisier, H. Zhang, J. P. Colombier, and R. Stoian, “Spatio-temporal dynamics in nondiffractive Bessel ultrafast laser nanoscale volume structuring,” Laser Photon. Rev. 10, 230–244 (2016).
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2015 (5)

M. Domke, D. Felsl, S. Rapp, J. Sotrop, H. P. Huber, and M. Schmidt, “Evidence of pressure waves in confined laser ablation,” J. Laser Micro/Nanoeng. 10, 119–123 (2015).

C. Javaux Léger, K. Mishchik, O. Dematteo-Caulier, S. Skupin, B. Chimier, G. Duchateau, A. Bourgeade, C. Hoenninger, E. Mottay, J. Lopez, and R. Kling, “Effects of burst mode on transparent materials processing,” Proc. SPIE 9351, 93510M (2015).
[Crossref]

I. Alexeev, J. Heberle, K. Cvecek, K. Y. Nagulin, and M. Schmidt, “High speed pump-probe apparatus for observation of transitional effects in ultrafast laser micromachining processes,” Micromachines 6, 1914–1922 (2015).
[Crossref]

A. K. Yadav and P. Singh, “A review of the structures of oxide glasses by Raman spectroscopy,” RSC Adv. 5, 67583–67609 (2015).
[Crossref]

N. M. Bulgakova, V. P. Zhukov, S. V. Sonina, and Y. P. Meshcheryakov, “Modification of transparent materials with ultrashort laser pulses: what is energetically and mechanically meaningful?” J. Appl. Phys. 118, 233108 (2015).
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2014 (8)

M. Kumkar, L. Bauer, S. Russ, M. Wendel, J. Kleiner, D. Grossmann, K. Bergner, and S. Nolte, “Comparison of different processes for separation of glass and crystals using ultrashort pulsed lasers,” Proc. SPIE 8972, 897214 (2014).
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B. Rethfeld, A. Rämer, N. Brouwer, N. Medvedev, and O. Osmani, “Electron dynamics and energy dissipation in highly excited dielectrics,” Nucl. Instrum. Methods Phys. Res. Sect. B 327, 78–88 (2014).
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P. K. Velpula, M. K. Bhuyan, C. Mauclair, J.-P. Colombier, and R. Stoian, “Role of free carriers excited by ultrafast Bessel beams for submicron structuring applications,” Opt. Eng. 53, 076108 (2014).
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R. Kammel, R. Ackermann, J. Thomas, J. Götte, S. Skupin, A. Tünnermann, and S. Nolte, “Enhancing precision in fs-laser material processing by simultaneous spatial and temporal focusing,” Light Sci. Appl. 3, e169 (2014).
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S. Karimelahi, L. Abolghasemi, and P. Herman, “Rapid micromachining of high aspect ratio holes in fused silica glass by high repetition rate picosecond laser,” Appl. Phys. A 114, 91–111 (2014).
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S. Butkus, E. Gaižauskas, D. Paipulas, Ž. Viburys, D. Kaškelyē, M. Barkauskas, A. Alesenkov, and V. Sirutkaitis, “Rapid microfabrication of transparent materials using filamented femtosecond laser pulses,” Appl. Phys. A 114, 81–90 (2014).
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J. Lapointe, M. Gagné, M.-J. Li, and R. Kashyap, “Making smart phones smarter with photonics,” Opt. Express 22, 15473–15483 (2014).
[Crossref]

M. Sakakura, Y. Ishiguro, N. Fukuda, Y. Shimotsuma, and K. Miura, “Modulation of crack generation inside a LiF single crystal by interference of laser induced stress waves,” J. Laser Micro/Nanoeng. 9, 15–18 (2014).
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2013 (4)

T. Deschamps, A. Kassir-Bodon, C. Sonneville, J. Margueritat, C. Martinet, D. De Ligny, A. Mermet, and B. Champagnon, “Permanent densification of compressed silica glass: a Raman-density calibration curve,” J. Phys. Condens. Matter 25, 025402 (2013).
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F. Courvoisier, J. Zhang, M. Bhuyan, M. Jacquot, and J. Dudley, “Applications of femtosecond Bessel beams to laser ablation,” Appl. Phys. A 112, 29–34 (2013).
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C. Corbari, A. Champion, M. Gecevičius, M. Beresna, Y. Bellouard, and P. G. Kazansky, “Femtosecond versus picosecond laser machining of nano-gratings and micro-channels in silica glass,” Opt. Express 21, 3946–3958 (2013).
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M. Lancry, B. Poumellec, J. Canning, K. Cook, J.-C. Poulin, and F. Brisset, “Ultrafast nanoporous silica formation driven by femtosecond laser irradiation,” Laser Photon. Rev. 7, 953–962 (2013).
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2012 (1)

S. Petrescu, M. Constantinescu, E. Anghel, I. Atkinson, M. Olteanu, and M. Zaharescu, “Structural and physico-chemical characterization of some soda lime zinc alumino-silicate glasses,” J. Non-Cryst. Solids 358, 3280–3288 (2012).
[Crossref]

2011 (11)

D. G. Papazoglou and S. Tzortzakis, “Physical mechanisms of fused silica restructuring and densification after femtosecond laser excitation,” Opt. Mater. Express 1, 625–632 (2011).
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Y. Hayasaki, K. Iwata, S. Hasegawa, A. Takita, and S. Juodkazis, “Time-resolved axial-view of the dielectric breakdown under tight focusing in glass,” Opt. Mater. Express 1, 1399–1408 (2011).
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T. Deschamps, C. Martinet, D. de Ligny, J. Bruneel, and B. Champagnon, “Correlation between boson peak and anomalous elastic behavior in GeO2 glass: an in situ Raman scattering study under high-pressure,” J. Chem. Phys. 134, 234503 (2011).
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M. Sakakura, T. Tochio, M. Eida, Y. Shimotsuma, S. Kanehira, M. Nishi, K. Miura, and K. Hirao, “Observation of laser-induced stress waves and mechanism of structural changes inside rock-salt crystals,” Opt. Express 19, 17780–17789 (2011).
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M. Sakakura, M. Terazima, Y. Shimotsuma, K. Miura, and K. Hirao, “Thermal and shock induced modification inside a silica glass by focused femtosecond laser pulse,” J. Appl. Phys. 109, 023503 (2011).
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M. Sakakura, M. Terazima, Y. Shimotsuma, K. Miura, and K. Hirao, “Elastic and thermal dynamics in femtosecond laser-induced structural change inside glasses studied by the transient lens method,” Laser Chem. 2010, 1–15 (2011).
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Y. Hayasaki, M. Isaka, A. Takita, and S. Juodkazis, “Time-resolved interferometry of femtosecond-laser-induced processes under tight focusing and close-to-optical breakdown inside borosilicate glass,” Opt. Express 19, 5725–5734 (2011).
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D. Esser, S. Rezaei, J. Li, P. R. Herman, and J. Gottmann, “Time dynamics of burst-train filamentation assisted femtosecond laser machining in glasses,” Opt. Express 19, 25632–25642 (2011).
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S. M. Eaton, M. L. Ng, R. Osellame, and P. R. Herman, “High refractive index contrast in fused silica waveguides by tightly focused, high-repetition rate femtosecond laser,” J. Non-Cryst. Solids 357, 2387–2391 (2011).
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M. Heinrich, R. Keil, F. Dreisow, A. Tünnermann, A. Szameit, and S. Nolte, “Nonlinear discrete optics in femtosecond laser-written photonic lattices,” Appl. Phys. B 104, 469–480 (2011).
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A. Mermillod-Blondin, C. Mauclair, J. Bonse, R. Stoian, E. Audouard, A. Rosenfeld, and I. V. Hertel, “Time-resolved imaging of laser-induced refractive index changes in transparent media,” Rev. Sci. Instrum. 82, 033703 (2011).
[Crossref]

2010 (4)

M. K. Bhuyan, F. Courvoisier, P. A. Lacourt, M. Jacquot, R. Salut, L. Furfaro, and J. M. Dudley, “High aspect ratio nanochannel machining using single shot femtosecond Bessel beams,” Appl. Phys. Lett. 97, 081102 (2010).
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B. Rethfeld, O. Brenk, N. Medvedev, H. Krutsch, and D. Hoffmann, “Interaction of dielectrics with femtosecond laser pulses: application of kinetic approach and multiple rate equation,” Appl. Phys. A 101, 19–25 (2010).
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N. M. Bulgakova, R. Stoian, and A. Rosenfeld, “Laser-induced modification of transparent crystals and glasses,” Quantum Electron. 40, 966–985 (2010).
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D. Grojo, M. Gertsvolf, S. Lei, T. Barillot, D. Rayner, and P. Corkum, “Exciton-seeded multiphoton ionization in bulk SiO2,” Phys. Rev. B 81, 212301 (2010).
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2009 (2)

D. Ivanov and B. Rethfeld, “The effect of pulse duration on the interplay of electron heat conduction and electron–phonon interaction: photo-mechanical versus photo-thermal damage of metal targets,” Appl. Surf. Sci. 255, 9724–9728 (2009).
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F. Dreisow, M. Ornigotti, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Polychromatic beam splitting by fractional stimulated Raman adiabatic passage,” Appl. Phys. Lett. 95, 261102 (2009).
[Crossref]

2008 (4)

A. Mermillod-Blondin, I. M. Burakov, Y. P. Meshcheryakov, N. M. Bulgakova, E. Audouard, A. Rosenfeld, A. Husakou, I. V. Hertel, and R. Stoian, “Flipping the sign of refractive index changes in ultrafast and temporally shaped laser-irradiated borosilicate crown optical glass at high repetition rates,” Phys. Rev. B 77, 104205 (2008).
[Crossref]

L. Englert, M. Wollenhaupt, L. Haag, C. Sarpe-Tudoran, B. Rethfeld, and T. Baumert, “Material processing of dielectrics with temporally asymmetric shaped femtosecond laser pulses on the nanometer scale,” Appl. Phys. A 92, 749–753 (2008).
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R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2, 219–225 (2008).
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A. Horn, I. Mingareev, A. Werth, M. Kachel, and U. Brenk, “Non-interferometric transient quantitative phase microscopy for ultrafast engineering,” Appl. Phys. A 93, 165–169 (2008).
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2007 (6)

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007).
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I. M. Burakov, N. M. Bulgakova, R. Stoian, A. Mermillod-Blondin, E. Audouard, A. Rosenfeld, A. Husakou, and I. V. Hertel, “Spatial distribution of refractive index variations induced in bulk fused silica by single ultrashort and short laser pulses,” J. Appl. Phys. 101, 043506 (2007).
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D. Wortmann, M. Ramme, and J. Gottmann, “Refractive index modification using fs-laser double pulses,” Opt. Express 15, 10149–10153 (2007).
[Crossref]

A. M. Stoneham, J. Gavartin, A. L. Shluger, A. V. Kimmel, D. M. Ramo, H. M. Rønnow, G. Aeppli, and C. Renner, “Trapping, self-trapping and the polaron family,” J. Phys. Condens. Matter 19, 255208 (2007).
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M. Sakakura, M. Terazima, Y. Shimotsuma, K. Miura, and K. Hirao, “Heating and rapid cooling of bulk glass after photoexcitation by a focused femtosecond laser pulse,” Opt. Express 15, 16800–16807 (2007).
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M. Sakakura, M. Terazima, Y. Shimotsuma, K. Miura, and K. Hirao, “Observation of pressure wave generated by focusing a femtosecond laser pulse inside a glass,” Opt. Express 15, 5674–5686 (2007).
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2006 (9)

S. Juodkazis, K. Nishimura, S. Tanaka, H. Misawa, E. G. Gamaly, B. Luther-Davies, L. Hallo, P. Nicolai, and V. T. Tikhonchuk, “Laser-induced microexplosion confined in the bulk of a sapphire crystal: evidence of multimegabar pressures,” Phys. Rev. Lett. 96, 166101 (2006).
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M. Centurion, Y. Pu, and D. Psaltis, “Holographic capture of femtosecond pulse propagation,” J. Appl. Phys. 100, 063104 (2006).
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S. W. Winkler, I. M. Burakov, R. Stoian, N. M. Bulgakova, A. Husakou, A. Mermillod-Blondin, A. Rosenfeld, D. Ashkenasi, and I. V. Hertel, “Transient response of dielectric materials exposed to ultrafast laser radiation,” Appl. Phys. A 84, 413–422 (2006).
[Crossref]

G. Buscarino, S. Agnello, and F. Gelardi, “Characterization of e δ and triplet point defects in oxygen-deficient amorphous silicon dioxide,” Phys. Rev. B 73, 045208 (2006).
[Crossref]

B. Rethfeld, “Free-electron generation in laser-irradiated dielectrics,” Phys. Rev. B 73, 035101 (2006).
[Crossref]

S. Quan, J. Hong-Bing, L. Yi, Z. Yong-Heng, Y. Hong, and G. Qi-Huang, “Relaxation of dense electron plasma induced by femtosecond laser in dielectric materials,” Chin. Phys. Lett. 23, 189–192 (2006).
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E. G. Gamaly, S. Juodkazis, K. Nishimura, H. Misawa, B. Luther-Davies, L. Hallo, P. Nicolai, and V. T. Tikhonchuk, “Laser-matter interaction in the bulk of a transparent solid: confined microexplosion and void formation,” Phys. Rev. B 73, 214101 (2006).
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S. Juodkazis, H. Misawa, T. Hashimoto, E. G. Gamaly, and B. Luther-Davies, “Laser-induced microexplosion confined in a bulk of silica: formation of nanovoids,” Appl. Phys. Lett. 88, 201909 (2006).
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K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(8), 620–625 (2006).
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2005 (11)

G. Zhu, J. van Howe, M. Durst, W. Zipfel, and C. Xu, “Simultaneous spatial and temporal focusing of femtosecond pulses,” Opt. Express 13, 2153–2159 (2005).
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C. Hnatovsky, R. Taylor, E. Simova, V. Bhardwaj, D. Rayner, and P. Corkum, “High-resolution study of photoinduced modification in fused silica produced by a tightly focused femtosecond laser beam in the presence of aberrations,” J. Appl. Phys. 98, 013517 (2005).
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R. Lian, R. A. Crowell, and I. A. Shkrob, “Solvation and thermalization of electrons generated by above-the-gap (12.4 eV) two-photon ionization of liquid H2O and D2O,” J. Phys. Chem. A 109, 1510–1520 (2005).
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L. Skuja, M. Hirano, H. Hosono, and K. Kajihara, “Defects in oxide glasses,” Phys. Status Solidi C 2, 15–24 (2005).
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Q. Sun, H. Jiang, Y. Liu, Z. Wu, H. Yang, and Q. Gong, “Measurement of the collision time of dense electronic plasma induced by a femtosecond laser in fused silica,” Opt. Lett. 30, 320–322 (2005).
[Crossref]

M. Okuno, N. Zotov, M. Schmücker, and H. Schneider, “Structure of SiO2–Al2O3 glasses: combined x-ray diffraction, IR and Raman studies,” J. Non-Cryst. Solids 351, 1032–1038 (2005).
[Crossref]

W. Reichman, D. Krol, L. Shah, F. Yoshino, A. Arai, S. Eaton, and P. Herman, “A spectroscopic comparison of femtosecond laser modified fused silica using kilohertz and megahertz laser systems,” J. Appl. Phys. 99, 123112 (2005).
[Crossref]

A. Couairon, L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Filamentation and damage in fused silica induced by tightly focused femtosecond laser pulses,” Phys. Rev. B 71, 125435 (2005).
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D. Rayner, A. Naumov, and P. Corkum, “Ultrashort pulse non-linear optical absorption in transparent media,” Opt. Express 13, 3208–3217 (2005).
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S. Onda, W. Watanabe, K. Yamada, K. Itoh, and J. Nishii, “Study of filamentary damage in synthesized silica induced by chirped femtosecond laser pulses,” J. Opt. Soc. Am. B 22, 2437–2443 (2005).
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M. Sakakura and M. Terazima, “Initial temporal and spatial changes of the refractive index induced by focused femtosecond pulsed laser irradiation inside a glass,” Phys. Rev. B 71, 024113 (2005).
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2004 (5)

B. Rethfeld, K. Sokolowski-Tinten, D. Von Der Linde, and S. Anisimov, “Timescales in the response of materials to femtosecond laser excitation,” Appl. Phys. A 79, 767–769 (2004).
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G. Méchain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing multiple femtosecond filaments in air,” Phys. Rev. Lett. 93, 035003 (2004).
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A. Horn, E. Kreutz, and R. Poprawe, “Ultrafast time-resolved photography of femtosecond laser induced modifications in Bk7 glass and fused silica,” Appl. Phys. A 79, 923–925 (2004).
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S. Mao, F. Quere, S. Guizard, X. Mao, R. Russo, G. Petite, and P. Martin, “Dynamics of femtosecond laser interactions with dielectrics,” Appl. Phys. A 79, 1695–1709 (2004).
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S. Nolte, M. Will, J. Burghoff, and A. Tünnermann, “Ultrafast laser processing: new options for three-dimensional photonic structures,” J. Mod. Opt. 51, 2533–2542 (2004).
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2003 (9)

Y. Shimotsuma, P. G. Kazansky, J. Qiu, and K. Hirao, “Self-organized nanogratings in glass irradiated by ultrashort light pulses,” Phys. Rev. Lett. 91, 247405 (2003).
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S. V. Garnov, V. I. Konov, A. A. Malyutin, O. G. Tsar’kova, I. S. Yatskovskii, and F. Dausinger, “Dynamics of plasma production and development in gases and transparent solids irradiated by high-intensity, tightly focused picosecond laser pulses,” Quantum Electron. 33, 758–764 (2003).
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S. V. Garnov, V. I. Konov, A. A. Malyutin, O. G. Tsarkova, I. S. Yatskovsky, and F. Dausinger, “High resolution interferometric diagnostics of plasmas produced by ultrashort laser pulses,” Laser Phys. 13, 386–396 (2003).

T. Yagi, M. Susa, and K. Nagata, “Determination of refractive index and electronic polarisability of oxygen for lithium-silicate melts using ellipsometry,” J. Non-Cryst. Solids 315, 54–62 (2003).
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T. Yagi and M. Susa, “Temperature dependence of the refractive index of Al2O3-Na2O-SiO2 melts: role of electronic polarizability of oxygon controlled by network structure,” Metall. Mater. Trans. B 34, 549–554 (2003).
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J. W. Chan, T. R. Huser, S. H. Risbud, J. S. Hayden, and D. M. Krol, “Waveguide fabrication in phosphate glasses using femtosecond laser pulses,” Appl. Phys. Lett. 82, 2371–2373 (2003).
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V. Bykov, A. Osipov, and V. Anfilogov, “Structure of high-alkali aluminosilicate melts from the high-temperature Raman spectroscopic data,” Glass Phys. Chem. 29, 105–107 (2003).
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2002 (4)

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S. K. Sundaram and E. Mazur, “Inducing and probing non-thermal transitions in semiconductors using femtosecond laser pulses,” Nat. Mater. 1, 217–224 (2002).
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2001 (4)

F. Quéré, S. Guizard, and P. Martin, “Time-resolved study of laser-induced breakdown in dielectrics,” Europhys. Lett. 56, 138–144 (2001).
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2000 (1)

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1999 (3)

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1998 (1)

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1997 (3)

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1996 (5)

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1995 (4)

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1994 (3)

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1993 (1)

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1991 (2)

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1989 (1)

1988 (3)

1987 (1)

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1983 (1)

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1982 (1)

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1979 (1)

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1975 (1)

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1974 (1)

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1969 (1)

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1966 (2)

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1960 (1)

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1956 (1)

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Figures (21)

Fig. 1.
Fig. 1. Pump–probe setup with delayable, frequency-doubled (513 nm), 200 fs short probe pulse and pump pulse with 200 fs, 1 ps, 6 ps, or 12 ps pulse duration, respectively. A grating pair serves to recompress the temporally prechirped probe pulse. An additional Wollaston prism with polarizer and analyzer enables interferometric measurements.
Fig. 2.
Fig. 2. Simulated intensity distribution obtained by a microscope objective (NA 0.35) and 1026 nm laser pulse in vacuum (a). Spherical aberrations occur, dependent on the defocusing depth [ ( b ) = 1 mm , ( c ) = 2 mm , ( d ) = 5 mm ]. The intensity distribution from (d) is widely spread and additional maxima occur (e). Laser beam is incident from the left.
Fig. 3.
Fig. 3. Spatio-temporal plasma evolution and resulting material modifications with a 6 ps and 200 μJ laser pulse, imaged as false-color shadowgraphic representation for the transmitted probe-beam (transmission scale identical to Fig. 5). The generated plasma in the focal region starts to grow towards the incoming beam, following the beam caustic, on a ps timescale. On a later time scale, long-living decay and lattice temperature-associated states can be observed, which relax within several ns.
Fig. 4.
Fig. 4. Measured transmission evolution at z = 0 μm (in the focus) for different pulse durations of the pump pulse.
Fig. 5.
Fig. 5. Pump–probe images for 6 ps pulse duration, NA 0.35, and different pulse energies. A false-color illustration is used for the transmission of the probe pulse after 14 ps (time of maximum plasma expansion).
Fig. 6.
Fig. 6. Simulated intensity distribution (a) with a sample tilt of 2°. The pump–probe image (b) shows a plasma spot generation detached from the focal area and the microscope image (c) of the corresponding asymmetric modification.
Fig. 7.
Fig. 7. Spatio-temporal plasma evolution with a 200 fs and 200 μJ laser pulse, imaged as false-color presentation for the transmitted probe-beam (transmission color bar identical to Fig. 5). The incoming laser beam breaks down into multiple filaments that propagate independently in the laser direction.
Fig. 8.
Fig. 8. Pump–probe shadowgraphic images of different plasma evolution phases during the laser–matter interaction with a 1 ps, 200 μJ laser pulse (transmission color bar as Fig. 5). The incoming beam breaks down into multiple filaments that generate a less dense plasma.
Fig. 9.
Fig. 9. Pump–probe shadowgraphic image of a single-shot 12 ps, 200 μJ laser pulse after 20 ps (time of maximum plasma expansion) and the resulting modification. The entire interaction area exhibits strong disruptions of the material.
Fig. 10.
Fig. 10. Interference images before (a) laser irradiation and at different time steps (b) and (c) of the plasma evolution inside Gorilla glass with a 6 ps and 200 μJ laser pulse.
Fig. 11.
Fig. 11. Reconstructed phase shift (a), refractive index change (b), and line scan (c) through (b) obtained by a 6 ps and 200 μJ laser pulse inside Gorilla glass after 100 ps.
Fig. 12.
Fig. 12. Calculated refractive index changes for 6 ps pulse duration, NA 0.35, and different pulse energies after 14 ps (time of maximum plasma expansion).
Fig. 13.
Fig. 13. Time-resolved measurements of the refractive index change in the focal region induced by a single 6 ps and 200 μJ laser pulse. (a) Evolution during and shortly after ignition and (b) development on the ns-time scale.
Fig. 14.
Fig. 14. Refractive index changes during the plasma evolution 3 ps after plasma ignition. The color bar was adjusted to visualize the different signs. Also, the scale was adapted for better visualization.
Fig. 15.
Fig. 15. Refractive index change induced by a single 6 ps and 200 μJ laser pulse, 7 ns after ignition (a), remaining modification after 1 s (b), and corresponding line scan from A to B (c). Color bar at (a) is equal to Fig. 12, and (b) was adjusted for better visualization.
Fig. 16.
Fig. 16. (a) Microscopic image of a single-shot volume modification (pulse energy 25 μJ). A disruption in the focal region and dark colored tear-like shape in front of it can be recognized. (b) Microscopic image of single shot volume modification (pulse energy 200 μJ) with longer disruption in the focal region and broader and longer tear-like formation of color centers and refractive index change in front of the focus.
Fig. 17.
Fig. 17. Transmission of pristine Corning Gorilla glass in comparison to samples treated with pulses of 200 fs and 200 μJ and corresponding absorption spectra.
Fig. 18.
Fig. 18. Convolution of the generated color centers within Corning Gorilla glass due to a low dense plasma.
Fig. 19.
Fig. 19. Absorption spectra of irradiated (200 fs, 200 μJ) Corning Gorilla glass after thermal annealing.
Fig. 20.
Fig. 20. Photoluminescence signal of E δ centers excited with 325 nm radiation of the laser-treated (200 fs pulse duration) sample.
Fig. 21.
Fig. 21. Raman spectra of a non-modified and 12 ps single pulse structured Gorilla glass sample.

Equations (8)

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P cr = 3.77 λ 2 8 π n 0 n 2 ,
Δ ϕ = 2 π Δ n d x λ Probe ,
Δ n = Δ n Kerr + Δ n Plasma + Δ n TO + Δ n Trap .
Δ n Plasma n e 2 n cr ,
Δ n TO = d n d T · Δ T = ( n 2 1 ) ( n 2 + 2 ) 6 n ( α β ) · Δ T .
Δ n = π 1 / 2 n 0 ( d n d T ) Δ T ,
Δ n Δ ρ = ( n 0 1 ) ( δ ρ ρ 0 ) .
p = Y 3 ( 1 2 ω ) ( δ ρ ρ 0 ) ,

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