Frequency dependence of nonlinear absorption in fused silica measured with picosecond pulses at various focal lengths

: Nonlinear absorption is the key process to generate laser-induced in-volume modiﬁcations in transparent dielectrics such as waveguides or three-dimensional data matrix codes. We present a comprehensive parameter study about nonlinear absorption in fused silica using a picosecond laser at various focal lengths. Beginning at a focal length of 100 mm, we measure a strong frequency dependence of the saturation absorption. Reducing the focal length results in a decrease of the saturation absorption. After passing a threshold focal length, the saturation absorption increases drastically and the frequency dependence starts to decrease. At the ﬁnal focal length of 6 mm we measure almost no frequency dependence. In order to explain our measurements, we used the theory of optical breakdown and ﬁlamentation. Nonlinear absorption measurement can become a promising tool for better process control during the generation of in-volume modiﬁcations in transparent dielectrics.


Introduction
In-volume modifications of transparent dielectrics became significantly relevant during the last decade due to promising applications such as integrated optics [1], microfluidic chips [2] or display cutting for mobile devices [3].
The fundamental process to generate in-volume modifications is nonlinear absorption (NLA). Extensive research has been carried out to understand the complex interaction between dielectrics and laser radiation. Two subprocesses called photoionization and avalanche ionization are used to describe the theory of NLA. By focusing ultra-short laser pulses with a microscope objective inside bulk dielectrics, very high intensity emerges in the focal area, which leads to the generation of free charge carriers in the conduction band. This process is called photoionization. After the generation of first charge carriers in the conduction band, the second process called avalanche ionization starts. Through impact ionization, further electrons can be lifted in the conduction band until a plasma is formed [4,5]. The absorbed laser energy in the free electrons is transferred to the lattice and causes permanent modifications in the bulk material [6].
Depending on laser parameters, focusing lenses and bulk material, the intensity of the plasma varies. A strong plasma leads to the formation of optical breakdown (OB) whereas a weak plasma forces the effect of filamentation (FIL) [7]. Figure 1 shows the differences between both modes of plasma formation and the resulting in-volume modifications.
By using focusing lenses with low NA, the intensity I in the focal area is not high enough to form a strong plasma. The additional effect of spherical aberration (see Fig. 1) enlarges the focal length and reduces the intensity further [8]. Only a weak plasma emerges which leads Fig. 1. By changing the numerical aperture (NA) of focusing lenses, plasma formation inside the bulk material can be influenced. A weak plasma due to low NA leads to Filamentation (FIL) whereas strong plasma leads to optical breakdown (OB). The different modes of plasma formation can be distinguished by the resulting in-volume modification (indicated by arrows).
to defocusing. After reaching the critical power P Crit , FIL occurs due to self-focusing by the Kerr effect [9]. The cycle of focusing and defocusing repeats until the intensity is insufficient for plasma formation and forms a long filament. By using focusing lenses with high NA or microscope objectives, the intensity in the focal area becomes very high. This leads to the formation of a strong plasma with very high absorption. Almost the whole laser energy is absorbed in the focal area. A drop shaped modification emerges with an inner and an outer zone. The inner zone shows the electrical damage due to the plasma formation and the outer zone shows the thermal damage (melting) by heat transfer [10] (see Fig. 7 right hand side).
In this study the influence of the focal length parameter f on the in-volume modification was investigated. By varying the focal length, the transition from FIL to OB was observed.

Experimental setup
Soda-lime glass microscope slides with a thickness of 1.2 mm and a size of 76 × 26 mm (DIN ISO 8037-1) were used as samples. High sample quality and the need for a large number of studies of the interaction of laser and soda-lime glass led to this choice. Figure 2 illustrates the setup to measure the NLA of soda-lime samples. A beam expander was used to adjust the collimated beam according to the different focusing optics. CNC controlled axes moved the samples through the focal area. All experiments were performed with an axis feed rate of 20 mms −1 . The transmitted laser radiation was measured with a power meter. Table 1 displays the parameters of the used laser system "Duetto" from Time Bandwidth Products AG. Pulse repetition frequency (PRF) was only set to 50, 100 or 200 kHz.  Table 2 displays the parameters of all used focusing optics and the resulting beam parameters. Microscope objectives (MO) were used to generate focal diameters below 4 µm. For larger focal diameters, plano-convex spherical focusing lenses (FL) were used. The focal diameter 2w 0 was calculated with where M 2 is the beam quality factor, λ is the wavelength, f is the focal length and D 0 is the input beam diameter which was adjusted according to the entrance aperture of the different focusing optics. Rayleigh length z R was calculated afterwards with

Nonlinear absorption measurement
The NLA measurement was performed with the setup shown in Fig. 2. A sample was moved with a constant speed v Axis through the focal area and the transmitted power was measured with the power meter and compared to the measured power without sample. The experimental absorption A Exp was then calculated with from [11] where E Trans is the transmitted pulse energy through the sample, E Pulse is the pulse energy without sample and R is the Fresnel reflectivity which was calculated with where n SLG is the refractive index of soda-lime glass with value of 1.52 ± 0.01 and n Air is the refractive index of Air with value of 1. This equals a Fresnel reflectivity of 0.043, which means that 4.3% of the energy will be reflected by the sample from one surface. Based on the measurements with pico-and nanosecond pulses by Nahen et al. in [12], scattering and reflection from the laser-induced plasma were neglected. They found that scattering and reflection amount to only a few percent of the incident laser energy. For picosecond pulses the influence of reflection and scattering was even lower. Therefore, the approximation of experimental absorption by A Exp ≈ 1 -R -T is justified (here, T stands for transmission). Figure 3 illustrates the experimental absorption A Exp of soda-lime glass with focusing lens VII with a focal length f of 100 mm at various pulse energies E Pulse . It can be seen that the slope as well as the saturation absorption depend on the PRF. By increasing the PRF, saturation absorption becomes lower, but the slope becomes steeper. The laser-induced modifications which have been produced with those parameters had the form of filaments. Figure 4 shows the experimental absorption A Exp of soda-lime glass with focusing lenses III to VI with a focal length f of 40, 50, 60 and 75 mm.
All results show exactly the same behavior as described for Fig. 3. The only effect of a reduction of the focal length f is a slightly decreasing saturation absorption for all PRF. Again, all laser-induced modifications with these parameters had the form of filaments. Figure 5 shows the experimental absorption A Exp of soda-lime glass with focusing lenses I and II with a focal length f of 25 and 30 mm. For PRF of 50 and 100 kHz the results correspond to the previous observations. The saturation absorption is further decreasing, and the slope of the curves becomes almost identical. For the PRF of 200 kHz a new effect occurs. A knee (marked with an arrow) can be observed in the curve and the saturation absorption becomes higher than the value for 100 kHz. A comparison with the laser-induced modifications produced with these parameters (see Fig. 7) showed a transition from FIL to OB exactly in the knee of the curve between 22.1 and 23.1 µJ. By further reducing the focal length f, only OB can be produced in the sample. Figure 6 shows the result of the NLA measurement with microscope objectives with a focal length f of 6 and 15 mm. The typical result of OB is a very high saturation absorption with similar values for all PRF over 70 percent. PRF dependency of the curve slope is also present during OB, but not directly from the beginning.  Figure 8 presents a concentration of all previous graphs by plotting the saturation absorption A Sat for all PRF over the different focal lengths f. The form of the laser-induced modification (FIL or OB) at A Sat is described by the shape of the data point. Figure 8 indicates directly the transition from FIL to OB by the jump in the saturation absorption between 15 and 25 mm. Even the change from FIL to OB at a certain focal length between different PRF can be detected by analyzing the deviation from the fitted functions. The increase of A Sat for larger focal lengths is visible as well.

Influence of the velocity on absorption
A first measurement to analyze the influence of the feed rate v Axis was performed with a microscope objective with focal length f of 6 mm. Axis feed rates were set to 20, 40 and 60 mms −1 and the  PRF was 50 kHz. For these parameters, the distance between two pulses are 0.4, 0.8 and 1.2 µm. With a calculated focal diameter 2w 0 of 2 µm the pulse overlap becomes noticeably smaller by increasing v Axis . Figure 9 illustrates the results of this measurement. Increasing the axis feed rate leads to a reduction of the saturation absorption. This effect can be explained by the higher density of free charge carriers, induced by the previous pulse, at lower velocities. A reduction of the velocity below 20 mms −1 leads to strong crack formation in the bulk material due to thermally induced stress.

Conclusions
The theory of plasma strength explaining the effects of FIL and OB coincides very well with the results obtained in this study. Larger focal lengths lead to weak plasma formation and FIL whereas small focal lengths lead to a strong plasma and OB.
With the laser parameters used, the threshold focal length for transition from FIL to OB lies around 20 mm. At focal lengths slightly above the threshold, the transition from FIL to OB can occur just by increasing the pulse energy. This effect was only observed for high PRF and could be an indication for thermal accumulation effects.
First measurements of the NLA at various axis feed rates showed a reduction of the saturation absorption at higher feed rates. This effect can be explained through the decreasing density of remaining free charge carriers, induced by the previous pulse, at higher velocities.
We were able to show that the NLA measurement can be used to distinguish between FIL and OB without analyzing the in-volume modification. NLA measurement during the production of in-volume modifications can be a promising tool for process control.