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Abstract
We show that the inscription velocity of fs-laser written structures in YAG crystals can be significantly improved by the use of MHz repetition rates for the writing process. Using a 10 MHz inscription laser, record high writing velocities up to 100 mm/s are achieved. Also, the selective etching process is accelerated using a diluted mixture of 22% H3PO4 and 24% H2SO4. The diluted mixture enables selective etching of up to 9.6 mm long, 1 µm wide and 18 µm high microchannels in 23 days. The etching parameter D of 11.2 µm2/s is a factor of 3 higher than previously reported and the selectivity is even increased by an order of magnitude.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Femtosecond laser direct writing utilizes nonlinear absorption in transparent dielectrics occurring at the high intensities of strongly focused ultrafast pulses. The high order intensity dependence of the nonlinear process enables to create three-dimensional (3D) micro- and nanometer scale material modifications limited to the focal volume [1–3]. It was shown that such laser induced material modifications exhibit higher etching rates, which enable selective etching of microchannels [4,5]. This opens wide fields of applications ranging from micro-opto-fluidics, liquid jet nozzles, and photonics to fully integrated lab-on-a-chip applications [6–10]. The selective etching process of various structures has been extensively studied, especially for glasses [5,11–13]. Nowadays selective etching of quartz glass even finds industrial applications [14].
Compared to crystals, glasses exhibit inferior mechanical and thermal stability properties. Selective etching of fs-laser modified crystals, particularly of laser gain media, is thus interesting for active devices, which should persevere high pressures and heat. Up to now selective etching has been demonstrated e.g. in quartz, sapphire, YAG, and CaF2 crystals. An overview of the current state of the art is given in Table 1. For comparison the latest values are given for quartz glass, too. Neither for the etching rate nor the selectivity of the process a uniform definition is used in literature, thus only etching depths and width vs. etching times are stated in Table 1. The etching rate for microchannels in e.g. fused silica and YAG is not constant but decreases with etching progress [15,16]. Previously, the etching process was described to be diffusion driven and quantified using the diffusion coefficient D of the Brownian diffusion [17]. In this model, the progress of the etching depth d vs. time t behaves similar to the mean free path of a diffusion particle [18–21] (please note that in [18] the fit parameter was defined as (2D)1/2 to simplify Eq. (1). Here, we use the definition of Einstein [17]).
Table 1. Current state of research on selective etching of fs-laser written structures in crystals and quartz glass. The highest values for YAG are highlighted in bold. Channel depths obtained by etching from both sides are marked with ‘*’
A fast etching progress is crucial for the fabrication of long microchannels. Up to now, the fastest selective etching of fs-laser written structures in YAG was reached using a mixture of 43% H3PO4 and 48% H2SO4 as etching agent at an etching temperature of 105°C, resulting in a diffusion coefficient D of 3.69 µm2/s [18].
Increased etching rates give rise to the fabrication of complex hollow structures consisting of many separately inscribed single lines. An approach to this end is the use of inscription lasers with high pulse repetition rates providing enough spatial overlap even at high writing velocities. In glasses, fs-laser structuring at MHz repetition rates is well investigated, but there are only few reports on fs-laser inscription in crystals utilizing such high repetition rates. Selective etching of sub-micrometer diameter structures in YAG inscribed at 1 MHz writing rate with a writing velocity of 2 mm/s was shown recently [27]. In Yb:CaGdAlO4 crystals fs-laser writing of waveguides was demonstrated at 10 MHz rates and writing velocities up to 100 mm/s [28], but etching was not tested.
Here we present for the first time selective etching of fs-laser written structures inscribed in YAG crystals at 10 MHz repetition rate, enabling writing velocities up to 100 mm/s. Furthermore, we significantly increase the diffusion coefficient D using a diluted mixture of 22% H3PO4 and 24% H2SO4 as etching agent. The drastically reduced fabrication times should facilitate the future fabrication of complex large-scale hollow microstructures inside YAG.
2. Experimental methods
The fs-laser structuring was done as described in [18,28] with two different laser systems: a 1 kHz chirped pulse amplifier fs-laser system of 1 W maximum average power, a center wavelength of 775 nm, a beam parameter product M 2 < 1.1, as well as a Fidelity HP High Energy Yb-doped fiber laser supplied by Coherent Inc. with 10 MHz repetition rate, a maximum average output power of 10 W, a center wavelength of 1040 nm, and a beam parameter product of M2 < 1.3. To investigate the dependence of the selective etching process on the structuring parameters, we inscribed single tracks in up to 9.6 mm long samples in a writing depth of 364 µm below the crystal surface, which was prepared as described in [18]. The writing parameters are listed in Table 2. Selective etching in different crystal orientations ([111] and [100]) was only investigated on tracks written with a lens of 3.1 mm focal length for the 10 MHz setup, because the 10 MHz inscription laser was no longer available after we found improved etching results using the lens with 4.5 mm focal length in [111] orientation.
Table 2. Parameters used for femtosecond laser writing.
Inscribed and selectively etched material modifications were investigated, and the etching progress was tracked by regular inspection with an optical microscope. Different samples were etched in 43%H3PO4 48%H2SO4 at 83°C, 90°C, and 105°C and in 22%H3PO4 24%H2SO4 as well as 43%H3PO4 at 83°C. The etching behavior was evaluated using the characteristic parameter D and selectivity S. The latter is determined by measuring the opening angle of the channels as described in [18], but here we are using a microscope with a higher resolution (Keyence VHX 7000).
3. Results and discussion
3.1 fs-Laser induced material modifications
The mechanisms of fs-laser structuring at MHz repetition rates differ fundamentally from those at kHz rates. The short time interval between the pulses leads to heat accumulation [29] and increases the modified volume [3]. Thus, we observed three regimes of material modification at 10 MHz of which two do not occur at 1 kHz (see Fig. 1). For the writing process we found no visible influence of writing direction and beam polarization.
Fig. 1. Light microscope images of fs-laser written tracks along the [111]-direction with (a) 10 MHz, a focusing lens of 3.1 mm, π-polarization (b) 4.5 mm focal length, σ-polarization, and (c) 1 kHz, a focusing lens of 4.5 mm focal length, π-polarization.
Pulse energies in excess of 0.1 µJ at writing velocities between 30 mm/s and 100 mm/s at 10 MHz or 0.5 µJ at 0.1 mm/s and 1 kHz cause large cracks, which are not further discussed in this work. Modification regime 1 occurs at 10 MHz for writing velocities between 30 mm/s and 100 mm/s and pulse energies between the modification threshold of about 60 nJ and the critical values for cracking. Here, the modifications are smooth and well confined in z-direction similar to the writing regime at 1 kHz. Regime 2 of material modifications appears at lower writing velocities between roughly 10 mm/s and 30 mm/s. In this regime the yz-cross sections are segmented along the z-direction (see Fig. 1(a)), v = 10 mm/s), possibly caused by self-focusing effects [27]. Regime 3 occurs at writing velocities below 10 mm/s. Here, the tracks consist of chains of melted dot-like zones along the writing direction. This phenomenon was also observed in fused silica and is attributed to heat accumulation [30,31]. Similar modifications are observed at low pulse energies near the modification threshold for all writing velocities at 10 MHz. However, here the effect is most likely caused by the modification not being continuous.
Material modifications of regime 1 were obtained at both writing rates, and at 10 MHz with both focusing lenses. The investigated ranges of the structuring parameters for all regimes are summarized in Table 3.
Table 3. Structuring regimes for fs-laser writing within the investigated parameter range. The regime 1 is best suited for fs-laser writing and selective etching.
The different properties of the MHz and the kHz laser systems led to slightly broader track cross sections for the written structures at 10 MHz under the given focusing conditions. In regime 1, structures written at 10 MHz measured 15 µm to 40 µm in height and 2 µm to 3 µm in width, compared to values between 6 µm and 30 µm and 1 µm and 2 µm, respectively, for structures written at 1 kHz. The smallest track widths match the values expected from the focal diameter well while the track height is, depending on the writing parameters, up to eight times smaller than the value calculated in [18] with equations from [32].
3.2 Selectively etched microchannels
The etching behavior of structures of regime 1 written at 1 kHz and 10 MHz was similar. All channels showed the shapes expected at the respective etching temperature [18] (cf. Figure 2). At 83°C, the cross section of the channels resembled the fs-laser modification. At higher temperatures, the entrance of the channels etched along [111] direction became parallelogram shaped, whereas for those etched along [100] rhombic facets were formed. We attribute this to a higher etching rate along the crystallographic axes of the cubic lattice, which are marked with “a” in Fig. 2(d).
Fig. 2. Optical microscope images of selectively etched microchannels. Writing and etching parameters are denoted in each image. The unetched parts of the written tracks in (e) are marked with red arrows. The lower two channels were etched completely through the sample. The white scale holds for a) – d) and the black for e).
Actually, the longest channels also exhibited the smallest channel cross sections of all channels in this work. They were 1 µm in width and 18 µm in height at a length of 9.6 mm, etched from two sides (Fig. 2(e)). These channels represent the longest selectively etched microchannels in crystals, to the best of our knowledge, and their length was limited by the sample size. They were obtained by selective etching of tracks written with 1 kHz, 300 nJ, and 0.1 mm/s using a σ-polarized laser beam with the diluted etching agent 23%H3PO4 24%H2SO4 for 23 days. Moreover, to reach a 1 µm narrow, 9.6 mm long channel, a selectivity between etching modified and unmodified material of at least 4800 is required, though we estimate it to be much higher and regard the width to be mainly limited by the inscription focus diameter.
3.3 Selectivity
The selectivity S is the channel length divided by half of the difference of the widths measured at the channels entrance and its end (cf. [18]). The longitudinal section of the channels recorded with a Keyence VHX 7000 microscope has a resolution of 85 nm/px which theoretically enables to determine selectivities of up to 104. However, for very parallel channels (S > 1000), due to difficulties to focus into the center of the longitudinal cross section, the root-mean-square deviation can become very large (> 50%). Nevertheless, the data presented here prove very high parallelism of the channels and thus a strongly selective etching process.
The selectivity decreases significantly for higher etching temperatures (cf. Figure 2(d)). Thus, we did not evaluate it quantitatively for temperatures above 83°C. Figure 3(a)) shows the selectivities vs. the pulse energy for tracks written at 10 MHz with different orientations and polarizations and etched in 43%H3PO4 48%H2SO4 at 83°C. For an f = 3.1 mm focusing lens, a selectivity of 1200 ± 400 was obtained for pulse energies between 70 nJ and 93 nJ and 100 mm/s writing velocity, independent of the sample orientation and inscription beam polarization (cf. Figure 3(a)). This is similar to the values previously obtained for tracks inscribed at 1 kHz [19]. Even higher selectivities of 2000 ± 300 were obtained using a lens of 4.5 mm focal length and writing velocities in the range of 30 mm/s to 100 mm/s at a pulse energy of 74 nJ (cf. Figure 3(b)).
Fig. 3. a) Selectivity vs. pulse energy for different writing beam polarizations and sample orientations. (b) Selectivity vs. writing velocity. All samples were inscribed at 10 MHz and etched in 43%H3PO4 48%H2SO4 at 83°C.
In contrast to the tracks inscribed at 10 MHz, we found a strong dependence on the polarization of the inscription laser for tracks inscribed at 1 kHz. The selectivity was increased by up to an order of magnitude for tracks inscribed with σ-polarized light (S > 5000 vs. S = 810 in π-polarization for 1 kHz and 23%H3PO4 24%H2SO4 at 83°C). For regime 1 tracks inscribed at both 1 kHz and 10 MHz and etched with 23%H3PO4 24%H2SO4 at 83°C, we even found selectivities well in excess of 5000, on the same order as for selective etching of sapphire [24].
3.4 Selective etching progress
All tracks of regime 1 written with 10 MHz showed a similar etching behavior of etched depth d vs. etching time t as observed for 1 kHz written tracks in [18], which could be characterized using the diffusion coefficient D under all investigated etching conditions [18,19]. The orientation and the writing laser beam polarization did not influence the etching rate significantly, which is somewhat surprising given the different facet shapes of channels etched along [100] and [111] direction (cf. sec. 3.2 and 3.3, Fig. 2(d)). The tracks of the writing regime 3 inscribed at 10 MHz are etched significantly slower, similar to tracks written at 1 kHz with fast writing velocities (> 2 mm/s). Thus, we do not regard regime 3 structures suitable for selective etching. The same holds for regime 2 structures which suffer from segmentation in vertical direction, i.e. along the z-axis (see Sec. 3.1).
Using 43%H3PO4 48%H2SO4 at 83°C as etching agent, except for the lowest inscription velocities we found no significant dependence of D on the writing velocity for 10 MHz-tracks of regime 1 (see Fig. 4(a)). This is opposed to the results observed in 1 kHz-tracks [18] and enables very high writing velocities up to 100 mm/s, only a factor of 2 below the record for fs-laser inscription in glasses [33], without significantly increasing the required etching time.
Fig. 4. a) Diffusion coefficient D vs. writing velocity v for tracks written with 10 MHz and σ-polarized inscription laser along the [111] direction of YAG for etching at 83°C in 43%H3PO4 48%H2SO4 and two different focusing lenses. b) Etched depth vs. etching time for regime 1 tracks written with 10 MHz and
1 kHz with a focusing lens of f = 4.5 mm and different etching agents etched at 83°C.
Tracks inscribed at 10 MHz with f = 4.5 mm at 30 mm/s – 100 mm/s show a higher diffusion coefficient D than those inscribed with f = 3.1 mm (cf. Figure 4(a)). It amounts to (2.8 ± 0.2) µm2/s, compared to (1.9 ± 0.1) µm2/s for f = 3.1 mm and is also higher than the values obtained for 1 kHz-tracks of regime 1 (D = (2.2 ± 0.2) µm2/s). We did not find D to increase significantly for etching temperatures up to 105°C, while the selectivity was drastically reduced in this case.
Tracks written at 1 kHz and 0.3 µJ with 0.1 mm/s did not follow the normal diffusion-driven etching progress model when etched with 22%H3PO4 24%H2SO4 at 83°C (cf. Figure 4(b), red curves), but could instead be fitted by a superdiffusion-driven model [34]:
In contrast to a diffusion-driven etching process with α = 1, here α amounts to 1.3 and 1.5 and the diffusion coefficients are Dα = 0.15 µm2/s1.3 and 0.02 µm2/s1.5 for the two red curves in Fig. 4(b). The case of 1 < α < 2 is attributed to a motion between the normal Brownian and ballistic diffusion, called superdiffusion [35], for which deriving the microscopic origin is not straightforward.
Extrapolation of the superdiffusion fits for the 1-kHz-structures shown in red in Fig. 4(b) yields a total etching time of 12 days for the 9.6 mm long structure. We experimentally achieved this length after 23 days of etching. We attribute this to difficulties of maintaining all conditions (temperature and acid concentration) stable over this long time.
The 10 MHz-tracks inscribed with f = 4.5 mm, σ-polarization and structuring parameters of regime 1 show the diffusion-driven square-root dependence with α = 1 with a very high diffusion coefficient D of 11.2 µm2/s when etched with 22%H3PO4 24%H2SO4 at 83°C.
Figure 5 shows the selectivity values vs. diffusion constant for etching at 83°C. The highest selectivity and fastest etching progress are achieved using 22%H3PO4 24%H2SO4 as etching agent. The use of 10 MHz inscription rate yields the highest selectivity, and diffusion parameters within the normal diffusion regime. Nevertheless, considering the anomalous diffusion in case of 1 kHz, similar etching progress can be reached with both 1 kHz and 10 MHz, while the highest selectivity was reached with 1 kHz. Despite the limited available data, an advantage for σ-polarized inscription lasers is suggested in terms of selectivity.
Fig. 5. Figure of merit: selectivity S plotted vs. diffusion coefficient D in YAG etched at 83°C along the [111] direction for different writing and etching parameters.
4. Conclusion
For the first time, to the best of our knowledge, a 10 MHz pulse repetition rate fs-laser was used to inscribe tracks into YAG crystals. Our results show that in a wide range of parameters, structures with properties suitable for highly selective etching with reduced etching times can be inscribed at very fast writing velocities utilizing MHz fs-laser systems.
The repetition rate of 10 MHz enables record-high writing velocities up to 100 mm/s. This is only a factor of 2 below the velocity used in industrial fs-laser-writing of glass [14] and nearly an order of magnitude faster than previous values observed in YAG [26]. The diffusion parameter D quantifying the etching progress and the selectivity S measured on the channels’ opening angle were used to characterize the selective etching process for all investigated etching and writing parameters. Our results feature diffusion parameters up to D = 11.2 µm2/s and selectivities on the order of 1000 for inscription rates of 10 MHz, as well as a superdiffusion behavior for structures written at 1 kHz by using a mixture of diluted phosphoric and sulfuric acid, 22%H3PO4 24%H2SO4, with a lower viscosity than higher concentrated acid mixtures.
This improved etching agent enabled the fabrication of highly parallel 9.6 mm long microchannels with a cross-section of 1 × 18 µm2 in structures inscribed at 1 kHz in 23 days. The etching process of such a structure is predicted to reduce to 11 days for structures inscribed at 10 MHz (D = 11.2 µm2/s). This would represent a remarkable improvement by a factor of 5 compared to previous results and facilitates the fabrication of large-scale complex 3D microstructures within the volume of YAG crystals in reasonable fabrication times.
Funding
Deutsche Forschungsgemeinschaft (501100001659 DFG - EXC 2056).
Acknowledgment
This work is supported by the Cluster of Excellence ‘Advanced Imaging of Matter’ of the Deutsche Forschungsgemeinschaft (DFG) - EXC 2056 - project ID 390715994. The authors thank Prof. Dr. Günter Huber for the fruitful discussions on this work and his instructive comments on the manuscript.
Disclosures
The authors declare no conflicts of interest.
Data availability
All data generated or analyzed during this study are available from the corresponding author upon request.
References
1. K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond laser,” Opt. Lett. 21(21), 1729–1731 (1996). [CrossRef]
2. E. N. Glezer, M. Milosavljevic, L. Huang, R. J. Finlay, T. H. Her, J. P. Callan, and E. Mazur, “Three-dimensional optical storage inside transparent materials,” Opt. Lett. 21(24), 2023–2025 (1996). [CrossRef]
3. C. B. Schaffer, A. Brodeur, and E. Mazur, “Laser-induced breakdown and damage in bulk transparent materials induced by tightly focused femtosecond laser pulses,” Meas. Sci. Technol. 12(11), 1784–1794 (2001). [CrossRef]
4. W. Hansen, S. Janson, and H. Helvajian, “Direct-write UV-laser microfabrication of 3D structures in lithium aluminosilicate glass,” Proc. SPIE 2991, 104–112 (1997). [CrossRef]
5. Y. Kondo, J. Qiu, T. Mitsuyu, K. Hirao, and T. Yoko, “Three-dimensional microdrilling of glass by multiphoton process and chemical etching,” Jpn. J. Appl. Phys. 38, L1146–L1148 (1999). [CrossRef]
6. C. Ross, D. G. MacLachlan, D. Choudhury, and R. R. Thomson, “Towards optical quality micro-optic fabrication by direct laser writing and chemical etching,” Proc. SPIE 10094, 100940V (2017). [CrossRef]
7. R. Osellame, H. J. W. M. Hoekstra, G. Cerullo, and M. Pollnau, “Femtosecond laser microstructuring: An enabling tool for optofluidic lab-on-chips,” Laser Photonics Rev. 5(3), 442–463 (2011). [CrossRef]
8. D. Choudhury, A. Arriola, J. R. Allington-Smith, C. Cunningham, and R. R. Thomson, “Towards freeform microlens arrays for near infrared astronomical instruments,” Proc. SPIE 9151, 915146 (2014). [CrossRef]
9. V. K. Jagannadh, M. D. Mackenzie, P. Pal, A. K. Kar, and S. S. Gorthi, “Slanted channel microfluidic chip for 3D fluorescence imaging of cells in flow,” Opt. Express 24(19), 22144–22158 (2016). [CrossRef]
10. K. Sugioka and Y. Cheng, “Femtosecond laser three-dimensional micro- and nanofabrication,” Appl. Phys. Rev. 1(4), 041303 (2014). [CrossRef]
11. S. Kiyama, S. Matsuo, S. Hashimoto, and Y. Morihira, “Examination of etching agent and etching mechanism on femtosecond laser microfabrication of channels inside vitreous silica substrates,” J. Phys. Chem. C 113(27), 11560–11566 (2009). [CrossRef]
12. M. Hermans and F. Riedel, “Selective, laser-induced etching of fused silica at high scan-speeds using KOH,” J. Laser Micro/Nanoeng. 9(2), 126–131 (2014). [CrossRef]
13. F. Chen and J. R. de Aldana Vazques, “Direct femtosecond laser writing of optical waveguides in dielectrics,” in Laser Micro-Nano- Manufacturing and 3D Microprinting, A. Hu, ed. (Springer, 2020).
14. J. Gottmann, M. Hermans, N. Repiev, and J. Ortmann, “Selective laser-induced etching of 3D precision quartz glass components for microfluidic applications—up-scaling of complexity and speed,” Micromachines 8(4), 110 (2017). [CrossRef]
15. F. Venturini, W. Navarrini, G. Resnati, P. Metrangolo, R. Martinez Vazquez, R. Osellame, and G. Cerullo, “Selective iterative etching of fused silica with gaseous hydrofluoric acid,” J. Phys. Chem. C 114(43), 18712–18716 (2010). [CrossRef]
16. J. Siebenmorgen, K. Petermann, G. Huber, K. Rademaker, S. Nolte, and A. Tünnermann, “Femtosecond laser written stress-induced Nd:Y3Al5O12 (Nd:YAG) channel waveguide laser,” Appl. Phys. B 97(2), 251–255 (2009). [CrossRef]
17. A. Einstein, “Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen,” Ann. Phys. 322(8), 549–560 (1905). [CrossRef]
18. K. Hasse, G. Huber, and C. Kränkel, “Selective etching of fs-laser inscribed high aspect ratio microstructures in YAG,” Opt. Mater. Express 9(9), 3627 (2019). [CrossRef]
19. K. Hasse, “Femtosekundenlaserstrukturierung und selektives Ätzen von Laserkristallen für Laseranwendungen,” Dissertation, Universität Hamburg (2020).
20. J. Lv, B. Hong, Y. Tan, F. Chen, J. R. V. de Aldana, and G. P. Wang, “Mid-infrared waveguiding in three-dimensional microstructured optical waveguides fabricated by femtosecond-laser writing and phosphoric acid etching,” Photonics Res. 8(3), 257 (2020). [CrossRef]
21. S. S. Fedotov, L. N. Butvina, and A. G. Okhrimchuk, “Plastic deformation as nature of femtosecond laser writing in YAG crystal,” Sci. Rep. 10(1), 19385 (2020). [CrossRef]
22. S. Matsuo, Y. Tabuchi, T. Okada, S. Juodkazis, and H. Misawa, “Femtosecond laser assisted etching of quartz: microstructuring from inside,” Appl. Phys. A 84(1-2), 99–102 (2006). [CrossRef]
23. S. Matsuo, K. Iwasa, T. Tomita, S. Hashimoto, and T. Okada, “Femtosecond laser-assisted etching of fluoride crystals,” J. Laser Micro Nanoeng. 6(3), 245–248 (2011). [CrossRef]
24. S. Juodkazis, H. Misawa, T. Ebisui, R. Waki, S. Matsuo, and T. Okada, “Control over the crystalline state of sapphire,” Adv. Mater. 18(11), 1361–1364 (2006). [CrossRef]
25. M. Hörstmann-Jungemann, J. Gottmann, and M. Keggenhoff, “3D-microstructuring of sapphire using fs-Laser irradiation and selective etching,” JLMN-Journal of Laser Micro/Nanoengineering 5(2), 1005–1011 (2010). [CrossRef]
26. D. Choudhury, A. Ródenas, L. Paterson, F. Diaz, D. Jaque, and A. K. Kar, “Three-dimensional microstructuring of yttrium aluminum garnet crystals for laser active optofluidic applications,” Appl. Phys. Lett. 103(4), 041101 (2013). [CrossRef]
27. A. Ródenas, M. Gu, G. Corrielli, P. Paiè, S. John, A. K. Kar, and R. Osellame, “Three-dimensional femtosecond laser nanolithography of crystals,” Nat. Photonics 13(2), 105–109 (2019). [CrossRef]
28. K. Hasse and C. Kränkel, “MHz-repetition rate fs-laser-inscribed crystalline waveguide lasers inscribed at 100 mm/s,” Opt. Express 28(8), 12011–12019 (2020). [CrossRef]
29. R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008). [CrossRef]
30. J. Qi, Z. Wang, J. Xu, Z. Lin, X. Li, W. Chu, and Y. Cheng, “Femtosecond laser induced selective etching in fused silica: optimization of the inscription conditions with a high-repetition-rate laser source,” Opt. Express 26(23), 29669–29678 (2018). [CrossRef]
31. S. M. Eaton, H. Zhang, P. R. Herman, F. Yoshino, L. Shah, J. Bovatsek, and A. Y. Arai, “Heat accumulation effects in femtosecond laser-written waveguides with variable repetition rate,” Opt. Express 13(12), 4708 (2005). [CrossRef]
32. Q. Sun, H. Jiang, Y. Liu, Y. Zhou, H. Yang, and Q. Gong, “Effect of spherical aberration on the propagation of a tightly focused femtosecond laser pulse inside fused silica,” J. Opt. A: Pure Appl. Opt. 7(11), 655–659 (2005). [CrossRef]
33. M. Malinauskas, A. Žukauskas, S. Hasegawa, Y. Hayasaki, V. Mizeikis, R. Buividas, and S. Juodkazis, “Ultrafast laser processing of materials: from science to industry,” Light: Sci. Appl. 5(8), e16133 (2016). [CrossRef]
34. M. F. Shlesinger, G. M. Zaslavsky, and J. Klafter, “Strange kinetics,” Nature 363(6424), 31–37 (1993). [CrossRef]
35. E. Awad and R. Metzler, “Crossover dynamics from superdiffusion to subdiffusion: models and solutions,” Fract. Calc. Appl. Anal. 23(1), 55–102 (2020). [CrossRef]
