We investigate the effect of saturation in the propagation of ablation channels performed in fused silica with many incident femtosecond pulses and laser fluence slightly above the ultrafast ablation threshold. A 110 fs Ti:Sapphire laser system is used in the experiments and the results are compared with theoretical predictions performed with a numerical model developed by the authors. Diffraction of the incoming pulses at the entrance of the channel as well as reflections at the walls of the channel play a crucial role in the progress of the crater as it is shown by means of the numerical results. The effect of the pulse duration in the shape of the ablation channel is also investigated.
©2006 Optical Society of America
Femtosecond laser ablation of dielectrics has become increasingly important in the last few years owing to the large amount of practical applications in materials processing. When intense ultra-short laser pulses are focused in a transparent dielectric, strong-field ionization (multi-photon, tunneling or barrier suppression) and subsequent avalanche ionization lead to the generation of a free-electron plasma. When the density of the free electrons exceeds a certain threshold, enough energy is absorbed to produce macroscopic ablation [1, 2, 3]. The largest density of ionized electrons is reached only in the central part of the focal region due to the highly non-linear nature of the strong-field ionization process. Moreover the ultra-short irradiation time strongly reduces the existence of thermal effects  that appear when picosecond or nanosecond pulses are employed, making femtosecond laser irradiation an excellent tool for the micro-structuring of dielectrics.
In many practical applications, irradiation with many incident laser pulses is commonly used. For instance, the fabrication of microchannels in dielectrics requires the irradiation with many pulses in order to get the desired channel depth [5, 6, 7]. In other cases this technique is used to get precise control of the shape and depth of the ablation craters [8, 9]. In these situations, the fabrication process has to be carried out mainly using a trial-error procedure.
In this work we investigate the effect of saturation in the growth of ablation channels performed with many incident femtosecond laser pulses in fused silica. The laser fluence in our studies is taken to be slightly above the ultra-fast ablation threshold for the material. The experimental results performed with a 110 fs laser system are compared with numerical simulations obtained with a theoretical model developed by the authors that provides good insight in the process of channel growing. The paper is organized as follows. Section 2 is devoted to present the femtosecond laser system and the experimental setup as well as some ablation results. The numerical model is described in Section 3 and the results are shown and discussed in Section 4. Finally, Section 5 is devoted to conclusions.
The femtosecond laser system consists on a mode-locked Ti:Sapphire oscillator and a regenerative amplifier, producing 110 fs (FWHM) laser pulses at a repetition rate 1 KHz and central wavelength 796 nm. The pulses were focused using a f=50 mm achromatic doublet lens. Prior to focusing, the beam was shaped by an iris in order to get the desired spot size at the focus. In order to facilitate an unambiguous comparison of the ablation results for different iris sizes, we kept constant the peak fluence at the focus around 7 J/cm2. This value is two times the measured ablation threshold fluence  for fused silica (Fth ≃ 3.5 J/cm2) with our femtosecond pulses.
We performed many ablation channels varying the number of incident laser pulses in a fused silica sample working at a low (1 Hz) repetition rate. The depth of the ablation channels in terms of the number of incident laser pulses is shown in Fig. 1 a) for two different sizes of the spot at the focus. In the first case (blue dots) the iris diameter was set to 4 mm. With this choice the computed FWHM of the central peak of the Airy figure at the focus (intensity) was ≃ 10 μm. In the second case (red dots) the iris diameter was set to 5 mm leading to a FWHM of the Airy figure of ≃ 8.2 μm. The depths were measured with an optical microscope operating in transmission mode and 1μm accuracy. The evolution of the channels depth has two well-differentiated regimes: a linear growth during some tens of incident pulses followed by a saturation. Once the saturation regime is reached the depth and shape of the channels remain nearly unchanged. The saturation of the channels performed with a larger size of the spot (red dots) occurs at a larger number of incident pulses leading to deeper channels, as it can be seen in the figure.
It is well known that the saturation of the channels is directly related to the Rayleigh range of the focusing lens. Of course, this is the case when working with laser fluences much larger than the ablation threshold [5, 6]. However, when the laser fluence is slightly above the threshold (just like in our experiments) the Rayleigh range is not the main limiting factor in the ablation depth. Note that in our studies the Rayleigh range (ZR = πw 2 0/λ) is greater than 100 μm for both spot sizes because of the diaphragm, and the saturation occurs at a much smaller depth. In these cases the main limiting factor in the growth of the ablation crater is the diffraction of the incident pulse at the entrance of the channel as it will be shown with our numerical simulations in the following sections. Note that the fluence of the incident pulse exceeds the ablation threshold only in the central part of the beam and thus the diameter of the ablated crater strongly limits the propagation inside the channel.
3. Numerical model
We investigate theoretically this effect with a numerical model. Our theoretical model is based on the direct numerical integration of the scalar wave-equation for the electric-field component of the incident laser pulse in cylindrical coordinates:
where P is the dipole density of the medium. The dielectric is described as an ensemble of classical harmonic oscillators so that the dipole density is computed from the following equation of motion:
The oscillator parameters (Ω0, nB and γ) are chosen in order to reproduce the optical properties of the material (refractive index, group velocity dispersion and absorption). The free-electron generation (strong-field ionization and impact ionization) and recombination are included through the equation
Laser-energy absorption is taken into account, phenomenologically, through the damping constant of the harmonic oscillators, γ, that is assumed to depend on the free electron density. For ultra-short pulses (femtoseconds) the energy transfer time from electrons subsystem to lattice ions is significantly larger than the pulse duration and can be neglected [11, 12, 13]. After the interaction with each one of the pulses the material will be ablated in the regions where the free-electron density equals or exceeds some critical electron density . The details of the model are obviated in the present work with the aim of brevity but can be found in  where very good agreement between numerical results and experiments for the overall evolution of ablation channels performed in fused silica.
Due to the enormous computational requirements for the simulations we employ pulses of 50 fs duration. For this pulse duration, the computed ablation threshold with our model is Fth ≃ 2.6 J/cm2. Thus, we take a peak fluence F 0 = 5.2 J/cm2 for the incident pulses (two times the threshold) in order to get a clear comparison with the experiments. The beam shape is taken Gaussian for simplicity as well as the temporal profile:
with τ = 50 fs. The beam size, ρ 0, is taken to fit the experimental situation. The wavelength is λ = 800 nm.
4. Results and discussion
In Fig. 1 b) results for the evolution of the ablation depth are given. The two regimes in the evolution of the channels can be clearly seen in the simulations and the penetration depth for which the channel growth saturates is very well reproduced. Note, however, that in the experiments the number of incident pulses required to reach the saturation regime is larger (a three factor approximately) than in the simulations: this is the effect of a very large penetration depth chosen in our simulations in order to reduce the very large run times.
In Figs. 2 a) and 2 b) we show pictures of the shape of the experimental saturated ablation channels for both spot sizes and the parameters described above taken with a differential interference contrast (DIC) microscope in side view. In Figs. 2 c) and 2 d) we show films for the evolution of the ablation channels for both sets of parameters. Scale labels are in microns and each frame corresponds to an incident laser pulse. The horizontal axis is the propagation direction (z-axis) and the plane z = 0 is the air-dielectric boundary of the sample.
In the first few pulses the channels reveal the well-known concave shape. However after some more pulses the change in the geometry of the air-dielectric boundary strongly affects the propagation of the laser field. Thus the shape of the ablation channel changes adopting a trumpet-like structure. The agreement between the experimental result and the numerical predictions for the saturated shapes is very good. It should be noted that the diameter of the channel entrance becomes larger with increasing the number of pulses as can be seen in the simulations. This effect will be discussed later but it is very interesting to point out here that it appears even in a simplified model like ours that does take into account neither incubation effects [16, 17] nor shot-to-shot instabilities.
In Fig. 3 we present movies for the propagation of the 1st, 15th, and 40th pulses with the 8.2 μm spot size. The panels on the left show the squared magnitude of the electric field (in logarithmic scale and arbitrary units) at several time steps during the propagation. The panels on the right show the free-electron density generated by multi-photon ionization and impact ionization in the sample. The scale is logarithmic as indicated in the figure (in m-3).
During the irradiation with the 1st pulse (Fig. 3 a)) the flat air-dielectric boundary does not distort the pulse (maxima and minima of the field are planes). As the pulse enters the material the free electron plasma begins to be created. In the points of the dielectric where the fluence gets the largest values, the free electron density exceeds the critical density (ncrit = 1.75×1027 m-3). Thus the absorption and reflectivity of the material grows in this area leading to a “hole” in the tailing part of the pulse. This metallic behavior of the ionized part of the dielectric has been studied in detail in several works (see for instance [11, 12] and the experimental work ). This thin layer limits the energy density that propagates inside the material.
Once the channel has grown due to ablation from several pulses, (see Fig. 3 b)), the complex geometry of the boundary modifies the energy-density distribution. As the pulse enters in the previously created channel (in white) multiple reflections at the boundaries lead to a maxima and minima distribution due to the overlapping of the waves. Inside the material there is overlapping among different waves: the incident wave which is refracted at the walls of the channel after suffering diffraction at the entrance, the waves propagated through the flat air-dielectric boundary and the waves refracted at the walls due to multiple reflections. All these contributions are responsible for the generation of an irregular pattern of ionized electrons that shows a spike structure aside the channel. There is also a high electron density, larger than the critical density, at the tip of the hole and at certain areas around the hole that are responsible for the growth of the channel depth. In particular this is the explanation of the increase in the entrance diameter that can be noticed in the movies of Fig. 2.
Fig. 3 b) should be compared to Fig. 3 c) where we show the propagation of the pulse in a saturated ablation channel. In this case the spikes structures in the ionized electron density appear but, however, the ionization at the tip of the hole is negligible. This is caused by a destructive interference among the several waves that overlap in this area that give rise to a minimum of laser fluence. Moreover there is neither appreciable ionization at the neighborhood of the entrance of the channel and thus the diameter of the hole does not grow further: under this circumstances, the shape of the channel remains unchangeable by the incoming radiation.
A possible explanation of the shape and depth of the saturated ablation channels could be the self-accommodation of the channel to the diffraction pattern caused by the space-limitation of the incident pulse at the entrance of the crater. In order to get some insight into this idea, we compute the diffraction pattern (in vacuum) generated by an aperture equal to the entrance diameter of the saturated channel shown in Fig. 2 c) by numerically solving the scalar wave-equation (1) and integrating in time the squared magnitude of the electric field:
The resulting energy distribution is depicted in Fig. 4 in linear gray scale. Some resemblance can be found to the saturated ablation channel: for instance, the trumpet shape shown in Fig. 2 c) is very similar to the minima structure of the pattern. However, the saturation depth of the channel computed with our model (≃ 12 μm) can not be successfully explained in terms of diffraction: note that this value lies approximately in a maximum of the diffraction pattern and thus further growth of the channel should be expected. The clear reason for these discrepancies is the modification of the diffraction pattern due to the effect of multiple reflections at the walls of the crater. Thus a simple calculation like the diffraction pattern is not of great usefulness by itself in order to describe the main features of the channel, provided that reflections need to be taken into account.
Finally, we investigate the effect of the pulse duration in the evolution of the ablation channels. We compare the results obtained with our model for two pulse durations, 40 fs and 20 fs, and the same beam size. In order to facilitate an unambiguous comparison of the results we take a peak fluence equal to two times the fluence threshold computed for each time duration: F 0(40fs) = 2Fth(40fs) = 4.80 J/cm2 and F 0(20fs) = 2Fth(20fs) = 3.62 J/cm2. The computed depth (dots) and diameter (dashed line) of the entrance of the channel are shown in Fig. 5. During the first few incident pulses, the behavior is nearly the same for both pulse durations. However, the entrance diameter for the 20 fs pulses stops growing leading to a faster saturation in the penetration depth. The shapes of the saturated channels appear also in Fig. 5. The different behavior for different pulse duration can be understood as follows. In the case of a long pulse, reflection at the walls and diffraction give rise to interference among the waves. These interference process generates patterns of maxima and minima of the laser fluence. In some points of space where exist maxima of the field it happens that the ionization can be large enough to generate the critical density of electrons and the subsequent ablation. However, in the case of a very short laser pulse this interference process does not occur. The short temporal duration avoids the spatio-temporal overlapping of the multiple waves.
Optical damage areas
The numerical model that we employ for our simulations does not account for incubation effects. Once the incident pulse has irradiated the material generating a large amount of ionized electrons, the areas where the free-electron density equals or exceeds the critical value are removed. However, the areas with large free-electron density but smaller than the critical density are assumed to cause no kind of change in the material and that the electrons recombine before the incidence of the following pulse. Obviously, this is not true as it has been shown in previous works (see for instance ) and some structural modifications appear in the material.
One of the effects of the large free-electron densities reached inside the material can be seen in the DIC images in Fig. 2. Note that some thin spikes appear emerging from the edges of the channels at a small angle with respect to the forward direction (horizontal axis). These structures are laser-induced damage areas similar to those observed in the filamentary propagation of intense femtosecond pulses in fused silica  and appear also with different sets of laser parameters (see for instance figure 1 in  or figure 5 in ). They are well explained in terms of ionized electrons (see Fig. 3). Large densities appear near the air-dielectric boundary that are responsible for the growth of the channel, as previously discussed, but there are also some areas inside the material with an important amount of ionization but not large enough to produce ultrafast ablation. In these areas, the cumulative effect of ionization during several laser pulses gives rise to the existence of optical damage structures.
In this work we have investigated the effect of saturation in the growth of ablation channels performed in fused silica with many incident femtosecond laser pulses. The fluence of the laser pulses was taken slightly above the ultra-fast ablation threshold for fused silica. Experimental results were compared to numerical simulations obtained with a numerical model developed by the authors and good agreement was found. The saturation of the ablation channels noticed in the experiments was well reproduced in the simulations and it can be understood in terms of diffraction of the incoming pulse at the entrance of the channel. Moreover, we have investigated the effect of the pulse duration in the shape of the ablation channel. Our simulations predict that the shorter the laser pulse the smaller is the saturated shape for the same laser fluence.
This work has been partially supported by the Spanish Ministerio de Ciencia y Tecnología (FEDER funds, grant BFM2002-00033) and by the Junta de Castilla y León and Unión Euro-pea (FSE, grant SA026A05). We thank C.R. Vázquez de Aldana for DIC microscopy images. Numerical simulations have been performed in the Beowulf linux cluster at the Universidad de Salamanca. Experimental work was carried out in the Terawatt laser facility Servicio Láser at the Universidad de Salamanca.
References and links
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