Abstract

We report on a systematic investigation of selective etching of fs-laser inscribed microstructures in Y3Al5O12 (YAG). The resulting microchannels are up to 8.9 mm long and exhibit cross sections from below 10 µm to more than 100 µm. Aspect ratios of up to 593 were achieved. Investigations with different structuring and etching parameters revealed that the etching process is mainly diffusion determined. The etching depth depends on the square root of time, similar to the well-known Brownian motion. In addition, we could enhance the etching diffusion constant by a factor of two, reducing the time to etch the longest channel by an order of magnitude, using a 1:1 mixture of sulfuric and phosphoric acid instead of pure phosphoric acid. The observed fundamental time dependence in conjunction with diffusion coefficients up to 160 µm/h1/2 makes the etching behavior highly predictable and paves the way toward arbitrary three-dimensional micro- and nanostructuring over long distances in crystalline materials.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Corrections

2 January 2020: A typographical correction was made to Eq. (3).

1. Introduction

Ultrashort-pulse laser writing is a versatile high precision fabrication technique for various kinds of micro- and nanostructuring of different materials. In particular in dielectrics, modification techniques such as surface micromachining or micro- and nanostructuring inside the material, e.g. selective etching of inscribed structures, open a wide range of applications in micro-integrated optics, photonics, microfluidics or micromechanics [15].

For writing structures inside transparent media, ultrashort laser pulses are strongly focused into the material. This causes intensity dependent nonlinear absorption resulting in a modification of the material. The etching rate of the fs-laser modified material in a suitable etching agent is remarkably increased compared to the unmodified material. Thus, highly selective etching of the laser-written structures is possible.

Selective etching of fs-laser modified transparent dielectrics has been demonstrated first in photomachinable glass [6]. Successively it developed into a well-established three-dimensional microfabrication technique mostly applied in glasses, using potassium hydroxide base (KOH) or hydrofluoric acid (HF) as etching agents [4,6]. This method was also applied to sapphire [79] using HF as etching agent. However, in sapphire the etching progress was slow and the longest channels reported are limited to 1 mm by the selectivity of the etching process. In contrast, YAG has a similar hardness but can be selectively etched using less dangerous phosphoric acid [911] as the etching agent. Since these first reports, channels fabricated by this method enabled very high aspect ratios at µm-scale diameters. In sapphire crystals aspect ratios in excess of 103 have been reached [7,8], the values in glass approached 350 [12,13]. The longest selectively etched channels were about 1 mm in sapphire and 10 mm in glass. The longest channel currently reported in YAG was 4 mm long with cross-sectional dimensions in the sub-micrometer range [9].

Aside from the fabrication of three-dimensional hollow microstructures inside passive crystalline materials such as sapphire and YAG, selective etching is also a versatile technique for structuring laser active media. Ytterbium-doped YAG is the most common laser crystal for fs-laser inscribed waveguide lasers and amplifiers [10,14], but also inscription into other materials was reported [1517]. In combination with fs-laser inscribed waveguide lasers and/or amplifiers, selective etching of three-dimensional inscribed structures paves the way for integrated lab-on-a-chip applications of micro-optical and microfluidic devices.

Here we investigate the selective etching process of fs-laser inscribed structures in the undoped laser host material YAG (Y3Al5O12) in detail. Such a parameter study is crucial for future applications which require long channels or extensive micro-structuring such as jet nozzles for external or internal mixing of different substances for the time- resolved observation of chemical processes [18] or lab-on-a-chip structures for cell separation [19]. The design and feasibility study of such larger structures require proper initial knowledge of the fabrication time and parameters, but corresponding data was not available for YAG up to now.

We found the etching behavior to be highly predictable with the etching depth depending on the square root of time. Thus, the time needed to etch a structure of a given length could be calculated before etching. Additionally, we obtain the longest microchannels in YAG reported up to now of up to 8.9 mm in length. The use of a mixture of phosphoric and sulfuric acid at elevated temperatures instead of pure phosphoric acid as etching agent increases the etching rate by a factor of ≈ 2. Furthermore, we quantified the selectivity of the etching process by the taper of the etched structures estimating the reliability of this measurement carefully. Our aim is to better understand the selective etching process physically and make it predictable and reproducible.

2. Experimental methods

2.1 fs-laser structuring: method and parameters

To inscribe the structures, we used a CLARK-MXR CPA-2010 chirped-pulse amplification femtosecond laser system. It has a pulse duration of ≈ 150 fs centered at 775 nm wavelength at a repetition rate of 1 kHz with pulse energies up to 1 mJ and a Gaussian beam profile with a beam quality of M2 < 1.1.

A sketch of the inscription setup is shown in Fig. 1. The pulses were sent through a circular mode-cleaning aperture with a diameter of 0.6 mm positioned directly behind the amplifier exit. The pulse energy was adjusted by a combination of a half-wave plate and a polarizing beam splitter to energies between 0.18 and 5.4 µJ, before sending the pulses onto the focusing lens. Since the laser light was not collimated perfectly, the beam diameter increases at ∼ 2 m distance from the aperture to the focusing lens, so that the lens is fully illuminated. Sample and focusing lens were mounted on motorized, computer-controlled air bearing Aerotech ABL10050 translation stages with a positioning precision of 2 nm and a translation velocity up to 100 mm/s. At our fixed repetition rate of 1 kHz the maximum velocity for a continuous structure inscription into the sample is limited by the spatial overlap of the pulses to ≈ 2 mm/s, depending on the focus diameter. The incident sample surface was polished to spectroscopic quality to prevent scattering and distortions of the beam. We adjusted the sample surface perpendicular to the beam by means of back reflection using a two-axis tilt stage.

 figure: Fig. 1.

Fig. 1. Sketch of the fs-laser structuring setup. The red arrows indicate the E-field vectors of the laser beams used for structuring with respect to the orientation of the inscribed tracks.

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To investigate the dependency of the etching behavior on the structuring parameters, tracks were inscribed with different parameters. To inscribe single tracks in the xy-plane (transversal writing scheme) the sample was moved in x-direction (see Fig. 1) with velocities between 25 µm/s and 2000 µm/s. The tracks were written in a depth of ≈ 360 µm below the surface in [100] and [111] oriented crystals. Both polarizations of the laser beam parallel (π) and perpendicular (σ) to the writing direction were tested. This was of interest since in glasses the occurrence and orientation of nanocracks – which influence the etching rate significantly – depend on the polarization of the inscribing laser beam [20].

In Fig. 2(a) the dimensions of the samples, as well as the nomenclature for the following images, are depicted. The beam was focused 360 µm below the surface of the sample with three different lenses with focal lengths of 3.1 mm, 4.51 mm, and 8 mm and numerical apertures of 0.68, 0.55, and 0.5, respectively. The corresponding Rayleigh lengths in the crystal of 1.7 µm, 2.0 µm and 3.1 µm set a lower limit for the focal dimensions in the crystal. As described in [21,22] the focus disperses towards the output face, due to spherical aberrations resulting in so called focal displacements of 40 µm, 52 µm, and 97 µm, respectively. However, the resulting track heights ht in transversal writing scheme are always much shorter (see Fig. 2(b)). To obtain more regular diameters, we also inscribed several tracks next to each other into the material, yielding larger quadratic or rectangular cross- sections of the modified area. By moving the focus upwards in z-direction (longitudinal writing scheme), we could generate single track modifications with circular cross- sections in the xy-plane (see Fig. 2(c)), too.

 figure: Fig. 2.

Fig. 2. (a) Sketch of a sample. The black lines indicate the transversally (1) and longitudinally (2) written tracks. (b) Cross sections of tracks of fs-modified material inscribed transversally (in x-direction) and (c) longitudinally (in z-direction, upwards) to the laser beam before etching.

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We investigated the dependency of the track height ht on the pulse energy and writing velocity in the parameter range between the threshold for the material modification (> 0.1 µJ) and the threshold for the onset of cracks along the track (≤ 1.8 µJ). As expected the track height ht increases with the deposited energy. It is always considerably smaller than the height of the focal displacements calculated above and no substantial differences are found for ht when the inscribing light is polarized parallel or perpendicular to the writing direction. For all examined parameters we did not find any significant change of the inscribed track width wt and it was always found to be (2 ± 1) µm.

2.2 Selective etching: method and parameters

Since the laser focus is distorted at the edges of the sample when parts of the beam do not enter the crystal, the inscribed tracks usually do not end at the edges of the samples in the transversal writing scheme. Therefore, the residual unmodified regions were removed by lapping and polishing prior to the etching process. In longitudinal writing scheme, the tracks do end at the surface. For etching, we used 48 % hydrofluoric acid, 85 % ortho-phosphoric acid, 96 % sulfuric acid, or a 1:1 mixture of sulfuric and phosphoric acid. Different etching temperatures between room temperature and 150 °C were tested. The temperature was measured with a mercury glass thermometer at the bottom of the acid container. With hydrofluoric acid (48 %) no etching effect was obtained. Samples with tracks inscribed with identical writing parameters were etched under different conditions. To determine the selective etching rates of the fs-pulse-modified material, the samples were taken out of the etching agent at different times and the etched depth was measured with a light microscope. For etching temperatures below 135 °C, we could not observe any etching of the unmodified material at the outer surfaces of the YAG sample.

3. Results and discussion

3.1 Examination of the etching process

To investigate the selective etching process in single tracks, we varied the inscription parameters and the etching conditions. The etched depth was measured at different etching times to obtain the etching rate.

3.1.1 Dependency on etching parameters

The etched depth de vs. the etching time t exhibits a square-root dependency for all etching temperatures, see Fig. 3(a), which shows square-root fits of de for three examples of etching temperatures. This dependency can be described by the well-known Brownian diffusion equation [23]

$${{d}_{e}}{ = D} \cdot {{t}^{{1/2}}}.$$

 figure: Fig. 3.

Fig. 3. (a) Etching depth vs. etching time for different etching temperatures and square root fits, (b) fitting parameter D vs. temperature Te and viscosity η and linear fit. The single tracks in the [111] oriented samples were inscribed with 0.45 µJ pulse energy, a σ-polarized beam, 100 µm/s translation velocity and etched in phosphoric acid.

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In this equation the fitting parameter D shows the same dependency on the temperature Te of the etching agent like the diffusion coefficient [24]:

$${D} \sim {({{T}_e}/\eta )^{1/2}},{\ }{\textrm{with}}{\ }\eta \sim {exp}({{E/R}{{T}_{e}}} ),$$
where η is the viscosity, E the activation energy, and R is the universal gas constant [25]. Figure 3(b) shows the expected dependency of the fitting parameter D on the viscosity of phosphoric acid [26] and the etching temperature.

These results indicate that the selective etching process of fs-laser written structures in YAG is mainly based on diffusion, similar as had been observed for fused silica [27]. For etching temperatures between 75 °C and 135 °C, D increases from 54 µm/h1/2 to 96 µm/h1/2. It should be noted that actual channels can be etched twice as fast if the etching starts from both sides into the middle of the channel. In different samples inscribed with the same parameters, the fitting parameter D is reproducible within a range of less than ±10 % root mean square deviation. In most cases tow etched samples were investigated and compared after the use of identical laser inscription parameter sets. We attribute this uncertainty mainly to the high nonlinear sensitivity to instabilities and/or different alignment conditions of the structuring laser beam as well as uncertainties of the temperature measurement of the etching agent.

According to Eq. (2) etching agents with a lower viscosity should increase the etching velocity. Thus, we tested also sulfuric acid [28] and a 1:1 mixture of sulfuric and phosphoric acid, as well as hydrofluoric acid (48%), since these exhibit lower viscosities than phosphoric acid (see Fig. 4).

 figure: Fig. 4.

Fig. 4. Etching depth vs. etching time for different etching agents. The orientation of the sample was [111]. Inscription data and fitting parameters D for square-root time dependencies are given aside the figure.

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The etching progress at 110 °C could be enhanced by more than a factor of two from D = 74.8 µm/h1/2 for concentrated phosphoric acid to D = 162.8 µm/h1/2 for the mixture of concentrated sulfuric and phosphoric acid, yielding four times shorter etching times.

3.1.2 Dependency on fs-laser structuring parameters

The dependency of the selective etching process on the structuring parameters can be quantified by the fitting parameter D. To this end, samples structured with different parameters were etched under identical conditions in phosphoric acid.

No significant dependency of D on the investigated orientation, polarization and focal volume, i.e. the chosen focal length, was observed. Unlike for fused silica [20], no nanocracks were observed for σ- and π-polarized light. Different pulse energies (Fig. 5(a)) and writing velocities (Fig. 5(b)) cause variations of the fitting parameter D of below 10 %, which is much lower than the influence of the etching temperature in the investigated parameter range.

 figure: Fig. 5.

Fig. 5. Fitting parameter D in dependency of pulse energy (a) and writing velocities (b) for different polarizations of the inscribing beam and etching temperatures in phosphoric acid. The samples were oriented in [111] direction. The dashed lines are guides to the eye.

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The fastest etching occurs at pulse energies around 0.36 µJ, and cracks observed at higher inscription pulse energies seem to decelerate the etching progress. Figure 6 shows that the etching progress is also decreased at higher writing velocities, which arises from the lower spatial overlap of successive pulses and could potentially be circumvented by higher repetition rates of the inscribing laser.

 figure: Fig. 6.

Fig. 6. Fitting parameter D in dependency of pulse energy for different writing velocities. The tracks were written in σ-polarization with a lens of 4.51 mm focal length in an [111] oriented crystal and etched in 85 °C phosphoric acid.

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3.1.3 Selectivity of the etching process

The selectivity is the ratio between the etching rate of the modified and unmodified material. The taper of the microchannels can be used to quantify the selectivity [7]. A reduced selectivity will lead to more unmodified material to be etched away from the inner walls in the front part of the channel, increasing the taper. Since the cross sections of our tracks are not circular, we measured the cross section in x- as well as in y-direction (cf. Fig. 2). We defined the selectivity S as twice the ratio of the channel depth de in x-direction to the change of its track width Δwch = wch1wch2 in y-direction over this length:

$${S\ = 2{\times }}{{d}_{e}}{/\Delta}{{w}_{\textrm {ch}}}.$$
Due to the high aspect ratio, it is difficult to determine the width of a buried channel with a precision better than 1 µm by microscopy. Consequentially the uncertainty of S increases with rising selectivity and a corresponding decrease of Δwch. A reasonable selectivity measurement is possible if the channel width changes more than 4 µm over a length shorter than ≈ 1.5 mm, i.e. for selectivities lower than ≈ 750. Using the estimated measurement precision of 1 µm for the width, we calculated selectivities and corresponding measurement errors for different channels. These results are shown in Fig. 7. A typical selectivity for etching temperatures of 110 °C and below is between 300 and 680, decreasing toward higher temperatures. The selectivity does not vary significantly between the two different etching agents.

 figure: Fig. 7.

Fig. 7. Selectivity in dependency of the etching temperature for different samples etched with phosphoric acid or a 1:1 mixture of phosphoric and sulfuric acid at temperatures between 80 °C and 135 °C.

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3.2 Selective etching results of single- and multi-track structures

Channels etched from simple single tracks, written in a transversal writing scheme show the expected elongated cross section (as described in section 2.1) with dimensions wt × ht in a range of 10 µm - 160 µm × 2 µm - 75 µm, depending on the writing parameters and the temperature of the etching acid. In contrast, multiple tracks written next to each other in 1 µm distance yield etched channels of nearly rectangular cross- sections. Channels of single tracks written in longitudinal writing scheme have round cross sections with the smallest diameter of only 5 µm (see Fig. 8).

 figure: Fig. 8.

Fig. 8. Quasi-three-dimensional microscopic images taken with a Keyence Vhx-6000 microscope. Microchannels resulting from etching of (1) single tracks written in x-direction (4.51 mm focal length, π-polarization, 100 µm/s, 0.27 µJ) (2) multiple tracks written with 1 µm horizontal distance (same writing parameters like (1)) (3) single tracks with round cross sections written in longitudinal writing mode in z-direction (4.51 mm focal length, π-polarization, 0.5 µJ, from left to right: 100 µm/s, 50 µm/s, 25 µm/s). The scale holds for both images. The artifacts arise from an imperfect surface polishing prior to the etching, which was affected during the etching process.

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Figure 8 shows a quasi-three-dimensional picture of such selectively etched microchannels (phosphoric acid, Te = 85 °C) recorded with a digital light microscope (Vhx-6000, Keyence) with the focus scanned through the sample. The sample was gold-coated to enhance the visibility of the end facet.

Additionally, channels with lengths up to 8.9 mm and a nearly quadratic cross section of 15 µm × 15 µm, corresponding to an aspect ratio of 593, were achieved. These are the longest microchannels in YAG up to now. The 8.9 mm long channel was fabricated by selective etching of a modified structure containing 15 single tracks inscribed with a y-distance of 1 µm to each other, yielding a rectangular cross -section of the modified area of 15 µm × 15 µm. In a proof-of-principle experiment, the sample was etched in phosphoric acid at 85 °C for 9 months. The etching process was highly selective (S = 1300 ± 470) and there is no evident etching of the unmodified material. It should be noted that with the accelerated etching process achieved in the 1:1 mixture of sulfuric and phosphoric acid the required etching time would be decreased by nearly an order of magnitude to about 1 month. Even after this long etching time, the polished YAG surfaces are nearly unaffected and exhibit only few small scratches. The longest microchannels obtained from single tracks (4.51 mm focal length, π-polarization, 100 µm/s writing velocity, 0.9µJ pulse energy) had an elliptical cross- section of 35 µm × 5 µm at a length of 4.3 mm, with no measurable taper. This channel was etched from both sides until the acid could flow through completely. The diameter of 5 µm is the smallest channel diameter we could reach at this channel length with the applied focusing parameters, however, we did not investigate the longitudinal writing scheme in such detail, yet.

­To evaluate the diameters of the etched single-track microchannels over the length of the channels, the side facets of crystal samples were polished and photographed under a light microscope in xz- and xy-plane (orientations as stated in Fig. 2).

Figure 9 shows side views of single tracks inscribed with 0.45 µJ pulse energy, a translation velocity of 100 µm/s and the inscribing beam σ-polarized to the writing direction after selective etching in phosphoric acid for 35 days at 100 °C, 85 °C, and 75 °C. For a temperature of 135 °C an image of the sample after etching for only 78 h is shown since the etching velocity is much higher at this temperature and the channel was etched through completely after a much shorter time than for the lower temperatures. It can be seen that the diameter of the channels increases with etching temperature and the channels exhibit a taper from the left, where the etching started, to the right, as described in section 3.1.3. The black pattern at the left end of each channel results from an image error: The reduced selectivity with increasing temperature affects the transparency of the YAG edge surface, leading to increasing shadowing effects.

 figure: Fig. 9.

Fig. 9. Side views of selectively etched single tracks, etched at different temperatures in phosphoric acid, structured with identical parameters in [111] oriented YAG crystals. Inset (a) shows the magnified front part inside the red circle.

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After long etching times, we could observe another – real – effect at the entrance of the channels. In Fig. 9 it only occurs for the channel- etched at 100 °C (see inset (a)), as for the other temperatures the etching time was too short for this effect to occur. The front view of samples inscribed under similar conditions (0.45 µJ, 100 µm/s, 4.51 mm focal length, π-polarization) but etched for 50 days is shown in Fig. 10. One can recognize faceted polygonal shapes of the entrance of each waveguide, which differ clearly from the shape of the material modification (cf. Fig. 2 (b)) and also from the shape of the channels (see Fig. 9 and channel 1 in Fig. 8). This points toward a limitation of the selectivity after very long etching times. The shapes can possibly be attributed to the symmetry of the cubic YAG structure in [111] orientation, however, more investigations are required to understand this phenomenon.

 figure: Fig. 10.

Fig. 10. Channel cross sections at the channels front facet in xz-plane. Channel e) was inscribed in a longitudinal writing scheme, all other channels were inscribed in the transversal scheme. Writing conditions were 0.45 µJ, 100 µm/s, 4.51 mm focal length, π-polarization, the samples were etched in phosphoric acid for 50 d.

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4. Conclusion and outlook

We presented a systematic parameter study on the selective etching process of fs-laser structured YAG crystals. We used a 1-kHz, 150-fs laser system to inscribe tracks inside the crystal volume under variation of pulse energy, translation velocity as well as polarization of the laser beam, the orientation of the sample, and arrangement of the tracks (single/multi). The tracks were etched in different etching agents at different temperatures. We found that the progress of the etching depth depends on the square root of time, just like the well-known Brownian motion. Also, the temperature dependency follows Brownian diffusion laws, both suggesting a mainly diffusion-driven process. The equation allows predicting the required etching time for a given structure with a deviation of less than ± 10%.

By using a 1:1 mixture of phosphoric and sulfuric acid as etching agent instead of pure phosphoric acid we reduced the etching time by a factor of 4 compared to our results with only phosphoric acid. Selective etching of multiple fs-laser inscribed tracks enabled the fabrication of micro-channels with up to 8.9 mm length at a diameter of 15 µm, which is more than a factor of 2 longer than the longest previously reported microchannels in YAG [9]. The longest single-track etched channel was 4.3 mm at an elliptic cross section of 35 µm × 5.3 µm. The taper was very low since both channels were etched until the acid could flow through completely. By measuring the taper of channels etched from one side only, we found a decreasing trend of the selectivity towards higher etching temperatures. For samples etched at temperatures below 110 °C we determined very high selectivity values in excess of 500, and for the lowest etching temperatures as well as for channels with through-flow of the acid the resulting channels were basically parallel. Due to the very high selectivity it should be possible to inscribe even narrower channels by utilizing different focusing conditions, e.g. in the longitudinal writing scheme. However, the etching process is not yet well investigated in this parameter range.

For future applications, it seems crucial to predictably control the selective etching process. Especially the fabrication of long and complex structures is limited by the time it takes to inscribe and etch the structures. The inscription velocity of our system is restricted by the fixed repetition rate of 1 kHz to below 2 mm per second, but pulsed laser systems with MHz repetition rates and similar pulse parameters are readily available to circumvent this bottleneck. Besides the beneficial influence on the inscription velocity, also an increased diffusion constant and thus etching velocity seems feasible by using MHz-repetition rate lasers for the inscription as the stronger pulse-to-pulse overlap supposedly enhances the laser-structuring effect. Further systematic investigations are required, but very interesting first results on MHz-inscribed sub-µm-channels were reported very recently [9]. It should, however, be noted, that aberration effects limited the inscription depth of these structures to less than 0.2 mm below the surface. In our system, the inscription depth of up to 2 mm is limited by the available pulse energy, with aberration effects increasing the track height for deeper channels.

At our highest observed diffusion constants of 162.8 µm/h1/2, selective etching of a 4 mm long channel currently takes about 1 week at a temperature of 110 °C. This is – as far as we know – the highest etching velocity reported for selective etching of YAG up to now. While this time appears long as compared to the required inscription time, it should be noted that it is on the same order like the growth process of YAG crystals and in a mass production process, large numbers of samples could be etched in the same etching bath. Due to the reasonably low etching temperatures, the process is easy to control and cost-efficient, too. While similar structures of shorter length have been previously reported in sapphire [8], the etching progress in our experiments in YAG is much faster and does not require dangerous HF as an etching agent like in the case of sapphire. Moreover, in contrast to sapphire, YAG can be easily doped with rare earth ions like Yb3+ and Nd3+, allowing for the simultaneous integration of microstructures and versatile active optical functionalities like temperature sensors [29] or fs-laser inscribed lasers [5,30] on the same platform.

Thus, we are convinced that our results pave the way toward the well-defined fabrication of long and complex microstructures from crystalline materials. Such structures are of importance e.g. for high -pressure microjet nozzles, where crystals can bear higher pressures than currently used glasses or plastics. Further applications e.g. in the field of crystalline photonic metamaterials are conceivable but require even smaller channels.

Funding

Deutsche Forschungsgemeinschaft (DFG - EXC 2056) (project ID 390715994).

Acknowledgments

This work is supported by the Cluster of Excellence ‘Advanced Imaging of Matter’ of the Deutsche Forschungsgemeinschaft (DFG) - EXC 2056 - project ID 390715994e.

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21. 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]  

22. C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. 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(1), 013517 (2005). [CrossRef]  

23. M. von Smoluchowski, “Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen,” Ann. Phys. 326(14), 756–780 (1906). [CrossRef]  

24. 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]  

25. C. V. Raman, “A theory of the viscosity of liquids,” Nature 111(2790), 532–533 (1923). [CrossRef]  

26. Potash Corp “Purified Phosphoric Acid,” Tech. Inf. Bull. (2012).

27. F. Venturini, W. Navarrini, G. Resnati, P. Metrangolo, R. M. 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]  

28. Landolt-Börnstein, “Part 5a, Viscosity and Diffusion,” in Properties of Matter in its Aggregated States, 2, 6th ed. (Springer-Verlag, 1969).

29. H. Berthou, “Optical-fiber temperature sensor based on upconversion-excited fluorescence,” Opt. Lett. 15(19), 1100–1102 (1990). [CrossRef]  

30. C. Grivas, “Optically pumped planar waveguide lasers, Part I : Fundamentals and fabrication techniques,” Prog. Quantum Electron. 35(6), 159–239 (2011). [CrossRef]  

References

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  • |

  1. R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
    [Crossref]
  2. F. Chen and J. R. V. de Aldana, “Optical waveguides in crystalline dielectric materials produced by femtosecond- laser micromachining,” Laser Photonics Rev. 8(2), 251–275 (2014).
    [Crossref]
  3. D. Choudhury, J. R. Macdonald, and A. K. Kar, “Ultrafast laser inscription: perspectives on future integrated applications,” Laser Photonics Rev. 8(6), 827–846 (2014).
    [Crossref]
  4. K. Sugioka and Y. Cheng, “Femtosecond laser three-dimensional micro- and nanofabrication,” Appl. Phys. Rev. 1(4), 041303 (2014).
    [Crossref]
  5. T. Calmano and S. Müller, “Crystalline Waveguide Lasers in the Visible and Near-Infrared Spectral Range,” IEEE J. Sel. Top. Quantum Electron. 21(1), 401–413 (2015).
    [Crossref]
  6. 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(Part 2), L1146–L1148 (1999).
    [Crossref]
  7. S. Juodkazis, K. Nishimura, 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]
  8. D. Wortmann, J. Gottmann, N. Brandt, and H. Horn-Solle, “Micro- and nanostructures inside sapphire by fs-laser irradiation and selective etching,” Opt. Express 16(3), 1517–1522 (2008).
    [Crossref]
  9. 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]
  10. J. Siebenmorgen and K. Petermann, “Femtosecond laser written stress-induced Nd: Y3Al5O12 (Nd: YAG) channel waveguide laser,” Appl. Phys. B: Lasers Opt. 97(2), 251–255 (2009).
    [Crossref]
  11. D. Choudhury, A. Ródenas, L. Paterson, F. Díaz, 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]
  12. 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]
  13. C. A. Ross, D. G. MacLachlan, D. Choudhury, and R. R. Thomson, “Optimisation of ultrafast laser assisted etching in fused silica,” Opt. Express 26(19), 24343 (2018).
    [Crossref]
  14. N. Jornod, V. J. Wittwer, C. Kränkel, D. Waldburger, U. Keller, T. Südmeyer, and T. Calmano, “High-power amplification of a femtosecond vertical external-cavity surface-emitting laser in an Yb:YAG waveguide,” Opt. Express 25(14), 16527–1605 (2017).
    [Crossref]
  15. K. Hasse, T. Calmano, B. Deppe, C. Liebald, and C. Kränkel, “Efficient Yb3+:CaGdAlO4 bulk and femtosecond-laser-written waveguide lasers,” Opt. Lett. 40(15), 3552 (2015).
    [Crossref]
  16. A. G. Okhrimchuk and A. V. Shestakov, “Depressed cladding, buried waveguide laser formed in a YAG: Nd3+ crystal by femtosecond laser writing,” Opt. Lett. 30(17), 2248–2250 (2005).
    [Crossref]
  17. Y. Tan, F. Chen, J. R. Vázquez De Aldana, G. A. Torchia, A. Benayas, and D. Jaque, “Continuous wave laser generation at 1064 nm in femtosecond laser inscribed Nd: YVO4 channel waveguides,” Appl. Phys. Lett. 97(3), 031119 (2010).
    [Crossref]
  18. M. Trebbin, K. Krüger, D. Deponte, S. V. Roth, H. N. Chapman, and S. Förster, “Microfluidic liquid jet system with compatibility for atmospheric and high-vacuum conditions,” Lab Chip 14(10), 1733–1745 (2014).
    [Crossref]
  19. D. Choudhury, W. T. Ramsay, R. Kiss, N. A. Willoughby, L. Paterson, and A. K. Kar, “A 3D mammalian cell separator biochip,” Lab Chip 12(5), 948–953 (2012).
    [Crossref]
  20. C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “Polarization-selective etching in femtosecond laser-assisted microfluidic channel fabrication in fused silica,” Opt. Lett. 30(14), 1867 (2005).
    [Crossref]
  21. 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]
  22. C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. 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(1), 013517 (2005).
    [Crossref]
  23. M. von Smoluchowski, “Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen,” Ann. Phys. 326(14), 756–780 (1906).
    [Crossref]
  24. 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]
  25. C. V. Raman, “A theory of the viscosity of liquids,” Nature 111(2790), 532–533 (1923).
    [Crossref]
  26. Potash Corp “Purified Phosphoric Acid,” Tech. Inf. Bull. (2012).
  27. F. Venturini, W. Navarrini, G. Resnati, P. Metrangolo, R. M. 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]
  28. Landolt-Börnstein, “Part 5a, Viscosity and Diffusion,” in Properties of Matter in its Aggregated States, 2, 6th ed. (Springer-Verlag, 1969).
  29. H. Berthou, “Optical-fiber temperature sensor based on upconversion-excited fluorescence,” Opt. Lett. 15(19), 1100–1102 (1990).
    [Crossref]
  30. C. Grivas, “Optically pumped planar waveguide lasers, Part I : Fundamentals and fabrication techniques,” Prog. Quantum Electron. 35(6), 159–239 (2011).
    [Crossref]

2019 (1)

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]

2018 (1)

2017 (1)

2015 (2)

K. Hasse, T. Calmano, B. Deppe, C. Liebald, and C. Kränkel, “Efficient Yb3+:CaGdAlO4 bulk and femtosecond-laser-written waveguide lasers,” Opt. Lett. 40(15), 3552 (2015).
[Crossref]

T. Calmano and S. Müller, “Crystalline Waveguide Lasers in the Visible and Near-Infrared Spectral Range,” IEEE J. Sel. Top. Quantum Electron. 21(1), 401–413 (2015).
[Crossref]

2014 (4)

F. Chen and J. R. V. de Aldana, “Optical waveguides in crystalline dielectric materials produced by femtosecond- laser micromachining,” Laser Photonics Rev. 8(2), 251–275 (2014).
[Crossref]

D. Choudhury, J. R. Macdonald, and A. K. Kar, “Ultrafast laser inscription: perspectives on future integrated applications,” Laser Photonics Rev. 8(6), 827–846 (2014).
[Crossref]

K. Sugioka and Y. Cheng, “Femtosecond laser three-dimensional micro- and nanofabrication,” Appl. Phys. Rev. 1(4), 041303 (2014).
[Crossref]

M. Trebbin, K. Krüger, D. Deponte, S. V. Roth, H. N. Chapman, and S. Förster, “Microfluidic liquid jet system with compatibility for atmospheric and high-vacuum conditions,” Lab Chip 14(10), 1733–1745 (2014).
[Crossref]

2013 (1)

D. Choudhury, A. Ródenas, L. Paterson, F. Díaz, 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]

2012 (1)

D. Choudhury, W. T. Ramsay, R. Kiss, N. A. Willoughby, L. Paterson, and A. K. Kar, “A 3D mammalian cell separator biochip,” Lab Chip 12(5), 948–953 (2012).
[Crossref]

2011 (1)

C. Grivas, “Optically pumped planar waveguide lasers, Part I : Fundamentals and fabrication techniques,” Prog. Quantum Electron. 35(6), 159–239 (2011).
[Crossref]

2010 (2)

F. Venturini, W. Navarrini, G. Resnati, P. Metrangolo, R. M. 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]

Y. Tan, F. Chen, J. R. Vázquez De Aldana, G. A. Torchia, A. Benayas, and D. Jaque, “Continuous wave laser generation at 1064 nm in femtosecond laser inscribed Nd: YVO4 channel waveguides,” Appl. Phys. Lett. 97(3), 031119 (2010).
[Crossref]

2009 (2)

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]

J. Siebenmorgen and K. Petermann, “Femtosecond laser written stress-induced Nd: Y3Al5O12 (Nd: YAG) channel waveguide laser,” Appl. Phys. B: Lasers Opt. 97(2), 251–255 (2009).
[Crossref]

2008 (2)

2006 (1)

S. Juodkazis, K. Nishimura, 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]

2005 (4)

A. G. Okhrimchuk and A. V. Shestakov, “Depressed cladding, buried waveguide laser formed in a YAG: Nd3+ crystal by femtosecond laser writing,” Opt. Lett. 30(17), 2248–2250 (2005).
[Crossref]

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “Polarization-selective etching in femtosecond laser-assisted microfluidic channel fabrication in fused silica,” Opt. Lett. 30(14), 1867 (2005).
[Crossref]

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]

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. 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(1), 013517 (2005).
[Crossref]

1999 (1)

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(Part 2), L1146–L1148 (1999).
[Crossref]

1990 (1)

1923 (1)

C. V. Raman, “A theory of the viscosity of liquids,” Nature 111(2790), 532–533 (1923).
[Crossref]

1906 (1)

M. von Smoluchowski, “Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen,” Ann. Phys. 326(14), 756–780 (1906).
[Crossref]

1905 (1)

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]

Benayas, A.

Y. Tan, F. Chen, J. R. Vázquez De Aldana, G. A. Torchia, A. Benayas, and D. Jaque, “Continuous wave laser generation at 1064 nm in femtosecond laser inscribed Nd: YVO4 channel waveguides,” Appl. Phys. Lett. 97(3), 031119 (2010).
[Crossref]

Berthou, H.

Bhardwaj, V. R.

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “Polarization-selective etching in femtosecond laser-assisted microfluidic channel fabrication in fused silica,” Opt. Lett. 30(14), 1867 (2005).
[Crossref]

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. 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(1), 013517 (2005).
[Crossref]

Brandt, N.

Calmano, T.

Cerullo, G.

F. Venturini, W. Navarrini, G. Resnati, P. Metrangolo, R. M. 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]

Chapman, H. N.

M. Trebbin, K. Krüger, D. Deponte, S. V. Roth, H. N. Chapman, and S. Förster, “Microfluidic liquid jet system with compatibility for atmospheric and high-vacuum conditions,” Lab Chip 14(10), 1733–1745 (2014).
[Crossref]

Chen, F.

F. Chen and J. R. V. de Aldana, “Optical waveguides in crystalline dielectric materials produced by femtosecond- laser micromachining,” Laser Photonics Rev. 8(2), 251–275 (2014).
[Crossref]

Y. Tan, F. Chen, J. R. Vázquez De Aldana, G. A. Torchia, A. Benayas, and D. Jaque, “Continuous wave laser generation at 1064 nm in femtosecond laser inscribed Nd: YVO4 channel waveguides,” Appl. Phys. Lett. 97(3), 031119 (2010).
[Crossref]

Cheng, Y.

K. Sugioka and Y. Cheng, “Femtosecond laser three-dimensional micro- and nanofabrication,” Appl. Phys. Rev. 1(4), 041303 (2014).
[Crossref]

Choudhury, D.

C. A. Ross, D. G. MacLachlan, D. Choudhury, and R. R. Thomson, “Optimisation of ultrafast laser assisted etching in fused silica,” Opt. Express 26(19), 24343 (2018).
[Crossref]

D. Choudhury, J. R. Macdonald, and A. K. Kar, “Ultrafast laser inscription: perspectives on future integrated applications,” Laser Photonics Rev. 8(6), 827–846 (2014).
[Crossref]

D. Choudhury, A. Ródenas, L. Paterson, F. Díaz, 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]

D. Choudhury, W. T. Ramsay, R. Kiss, N. A. Willoughby, L. Paterson, and A. K. Kar, “A 3D mammalian cell separator biochip,” Lab Chip 12(5), 948–953 (2012).
[Crossref]

Corkum, P. B.

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. 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(1), 013517 (2005).
[Crossref]

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “Polarization-selective etching in femtosecond laser-assisted microfluidic channel fabrication in fused silica,” Opt. Lett. 30(14), 1867 (2005).
[Crossref]

Corrielli, G.

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]

de Aldana, J. R. V.

F. Chen and J. R. V. de Aldana, “Optical waveguides in crystalline dielectric materials produced by femtosecond- laser micromachining,” Laser Photonics Rev. 8(2), 251–275 (2014).
[Crossref]

Deponte, D.

M. Trebbin, K. Krüger, D. Deponte, S. V. Roth, H. N. Chapman, and S. Förster, “Microfluidic liquid jet system with compatibility for atmospheric and high-vacuum conditions,” Lab Chip 14(10), 1733–1745 (2014).
[Crossref]

Deppe, B.

Díaz, F.

D. Choudhury, A. Ródenas, L. Paterson, F. Díaz, 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]

Ebisui, T.

S. Juodkazis, K. Nishimura, 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]

Einstein, A.

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]

Förster, S.

M. Trebbin, K. Krüger, D. Deponte, S. V. Roth, H. N. Chapman, and S. Förster, “Microfluidic liquid jet system with compatibility for atmospheric and high-vacuum conditions,” Lab Chip 14(10), 1733–1745 (2014).
[Crossref]

Gattass, R. R.

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
[Crossref]

Gong, Q.

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]

Gottmann, J.

Grivas, C.

C. Grivas, “Optically pumped planar waveguide lasers, Part I : Fundamentals and fabrication techniques,” Prog. Quantum Electron. 35(6), 159–239 (2011).
[Crossref]

Gu, M.

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]

Hashimoto, S.

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]

Hasse, K.

Hirao, K.

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(Part 2), L1146–L1148 (1999).
[Crossref]

Hnatovsky, C.

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. 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(1), 013517 (2005).
[Crossref]

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “Polarization-selective etching in femtosecond laser-assisted microfluidic channel fabrication in fused silica,” Opt. Lett. 30(14), 1867 (2005).
[Crossref]

Horn-Solle, H.

Jaque, D.

D. Choudhury, A. Ródenas, L. Paterson, F. Díaz, 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]

Y. Tan, F. Chen, J. R. Vázquez De Aldana, G. A. Torchia, A. Benayas, and D. Jaque, “Continuous wave laser generation at 1064 nm in femtosecond laser inscribed Nd: YVO4 channel waveguides,” Appl. Phys. Lett. 97(3), 031119 (2010).
[Crossref]

Jiang, H.

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]

John, S.

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]

Jornod, N.

Juodkazis, S.

S. Juodkazis, K. Nishimura, 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]

Kar, A. K.

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]

D. Choudhury, J. R. Macdonald, and A. K. Kar, “Ultrafast laser inscription: perspectives on future integrated applications,” Laser Photonics Rev. 8(6), 827–846 (2014).
[Crossref]

D. Choudhury, A. Ródenas, L. Paterson, F. Díaz, 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]

D. Choudhury, W. T. Ramsay, R. Kiss, N. A. Willoughby, L. Paterson, and A. K. Kar, “A 3D mammalian cell separator biochip,” Lab Chip 12(5), 948–953 (2012).
[Crossref]

Keller, U.

Kiss, R.

D. Choudhury, W. T. Ramsay, R. Kiss, N. A. Willoughby, L. Paterson, and A. K. Kar, “A 3D mammalian cell separator biochip,” Lab Chip 12(5), 948–953 (2012).
[Crossref]

Kiyama, S.

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]

Kondo, Y.

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(Part 2), L1146–L1148 (1999).
[Crossref]

Kränkel, C.

Krüger, K.

M. Trebbin, K. Krüger, D. Deponte, S. V. Roth, H. N. Chapman, and S. Förster, “Microfluidic liquid jet system with compatibility for atmospheric and high-vacuum conditions,” Lab Chip 14(10), 1733–1745 (2014).
[Crossref]

Liebald, C.

Liu, Y.

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]

Macdonald, J. R.

D. Choudhury, J. R. Macdonald, and A. K. Kar, “Ultrafast laser inscription: perspectives on future integrated applications,” Laser Photonics Rev. 8(6), 827–846 (2014).
[Crossref]

MacLachlan, D. G.

Matsuo, S.

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]

S. Juodkazis, K. Nishimura, 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]

Mazur, E.

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
[Crossref]

Metrangolo, P.

F. Venturini, W. Navarrini, G. Resnati, P. Metrangolo, R. M. 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]

Misawa, H.

S. Juodkazis, K. Nishimura, 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]

Mitsuyu, T.

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(Part 2), L1146–L1148 (1999).
[Crossref]

Morihira, Y.

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]

Müller, S.

T. Calmano and S. Müller, “Crystalline Waveguide Lasers in the Visible and Near-Infrared Spectral Range,” IEEE J. Sel. Top. Quantum Electron. 21(1), 401–413 (2015).
[Crossref]

Navarrini, W.

F. Venturini, W. Navarrini, G. Resnati, P. Metrangolo, R. M. 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]

Nishimura, K.

S. Juodkazis, K. Nishimura, 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]

Okada, T.

S. Juodkazis, K. Nishimura, 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]

Okhrimchuk, A. G.

Osellame, R.

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]

F. Venturini, W. Navarrini, G. Resnati, P. Metrangolo, R. M. 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]

Paiè, P.

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]

Paterson, L.

D. Choudhury, A. Ródenas, L. Paterson, F. Díaz, 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]

D. Choudhury, W. T. Ramsay, R. Kiss, N. A. Willoughby, L. Paterson, and A. K. Kar, “A 3D mammalian cell separator biochip,” Lab Chip 12(5), 948–953 (2012).
[Crossref]

Petermann, K.

J. Siebenmorgen and K. Petermann, “Femtosecond laser written stress-induced Nd: Y3Al5O12 (Nd: YAG) channel waveguide laser,” Appl. Phys. B: Lasers Opt. 97(2), 251–255 (2009).
[Crossref]

Qiu, J.

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(Part 2), L1146–L1148 (1999).
[Crossref]

Raman, C. V.

C. V. Raman, “A theory of the viscosity of liquids,” Nature 111(2790), 532–533 (1923).
[Crossref]

Ramsay, W. T.

D. Choudhury, W. T. Ramsay, R. Kiss, N. A. Willoughby, L. Paterson, and A. K. Kar, “A 3D mammalian cell separator biochip,” Lab Chip 12(5), 948–953 (2012).
[Crossref]

Rayner, D. M.

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “Polarization-selective etching in femtosecond laser-assisted microfluidic channel fabrication in fused silica,” Opt. Lett. 30(14), 1867 (2005).
[Crossref]

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. 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(1), 013517 (2005).
[Crossref]

Resnati, G.

F. Venturini, W. Navarrini, G. Resnati, P. Metrangolo, R. M. 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]

Ródenas, A.

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]

D. Choudhury, A. Ródenas, L. Paterson, F. Díaz, 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]

Ross, C. A.

Roth, S. V.

M. Trebbin, K. Krüger, D. Deponte, S. V. Roth, H. N. Chapman, and S. Förster, “Microfluidic liquid jet system with compatibility for atmospheric and high-vacuum conditions,” Lab Chip 14(10), 1733–1745 (2014).
[Crossref]

Shestakov, A. V.

Siebenmorgen, J.

J. Siebenmorgen and K. Petermann, “Femtosecond laser written stress-induced Nd: Y3Al5O12 (Nd: YAG) channel waveguide laser,” Appl. Phys. B: Lasers Opt. 97(2), 251–255 (2009).
[Crossref]

Simova, E.

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “Polarization-selective etching in femtosecond laser-assisted microfluidic channel fabrication in fused silica,” Opt. Lett. 30(14), 1867 (2005).
[Crossref]

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. 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(1), 013517 (2005).
[Crossref]

Südmeyer, T.

Sugioka, K.

K. Sugioka and Y. Cheng, “Femtosecond laser three-dimensional micro- and nanofabrication,” Appl. Phys. Rev. 1(4), 041303 (2014).
[Crossref]

Sun, Q.

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]

Tan, Y.

Y. Tan, F. Chen, J. R. Vázquez De Aldana, G. A. Torchia, A. Benayas, and D. Jaque, “Continuous wave laser generation at 1064 nm in femtosecond laser inscribed Nd: YVO4 channel waveguides,” Appl. Phys. Lett. 97(3), 031119 (2010).
[Crossref]

Taylor, R. S.

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. 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(1), 013517 (2005).
[Crossref]

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “Polarization-selective etching in femtosecond laser-assisted microfluidic channel fabrication in fused silica,” Opt. Lett. 30(14), 1867 (2005).
[Crossref]

Thomson, R. R.

Torchia, G. A.

Y. Tan, F. Chen, J. R. Vázquez De Aldana, G. A. Torchia, A. Benayas, and D. Jaque, “Continuous wave laser generation at 1064 nm in femtosecond laser inscribed Nd: YVO4 channel waveguides,” Appl. Phys. Lett. 97(3), 031119 (2010).
[Crossref]

Trebbin, M.

M. Trebbin, K. Krüger, D. Deponte, S. V. Roth, H. N. Chapman, and S. Förster, “Microfluidic liquid jet system with compatibility for atmospheric and high-vacuum conditions,” Lab Chip 14(10), 1733–1745 (2014).
[Crossref]

Vazquez, R. M.

F. Venturini, W. Navarrini, G. Resnati, P. Metrangolo, R. M. 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]

Vázquez De Aldana, J. R.

Y. Tan, F. Chen, J. R. Vázquez De Aldana, G. A. Torchia, A. Benayas, and D. Jaque, “Continuous wave laser generation at 1064 nm in femtosecond laser inscribed Nd: YVO4 channel waveguides,” Appl. Phys. Lett. 97(3), 031119 (2010).
[Crossref]

Venturini, F.

F. Venturini, W. Navarrini, G. Resnati, P. Metrangolo, R. M. 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]

von Smoluchowski, M.

M. von Smoluchowski, “Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen,” Ann. Phys. 326(14), 756–780 (1906).
[Crossref]

Waki, R.

S. Juodkazis, K. Nishimura, 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]

Waldburger, D.

Willoughby, N. A.

D. Choudhury, W. T. Ramsay, R. Kiss, N. A. Willoughby, L. Paterson, and A. K. Kar, “A 3D mammalian cell separator biochip,” Lab Chip 12(5), 948–953 (2012).
[Crossref]

Wittwer, V. J.

Wortmann, D.

Yang, H.

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]

Yoko, T.

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(Part 2), L1146–L1148 (1999).
[Crossref]

Zhou, Y.

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]

Adv. Mater. (1)

S. Juodkazis, K. Nishimura, 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]

Ann. Phys. (2)

M. von Smoluchowski, “Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen,” Ann. Phys. 326(14), 756–780 (1906).
[Crossref]

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]

Appl. Phys. B: Lasers Opt. (1)

J. Siebenmorgen and K. Petermann, “Femtosecond laser written stress-induced Nd: Y3Al5O12 (Nd: YAG) channel waveguide laser,” Appl. Phys. B: Lasers Opt. 97(2), 251–255 (2009).
[Crossref]

Appl. Phys. Lett. (2)

D. Choudhury, A. Ródenas, L. Paterson, F. Díaz, 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]

Y. Tan, F. Chen, J. R. Vázquez De Aldana, G. A. Torchia, A. Benayas, and D. Jaque, “Continuous wave laser generation at 1064 nm in femtosecond laser inscribed Nd: YVO4 channel waveguides,” Appl. Phys. Lett. 97(3), 031119 (2010).
[Crossref]

Appl. Phys. Rev. (1)

K. Sugioka and Y. Cheng, “Femtosecond laser three-dimensional micro- and nanofabrication,” Appl. Phys. Rev. 1(4), 041303 (2014).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

T. Calmano and S. Müller, “Crystalline Waveguide Lasers in the Visible and Near-Infrared Spectral Range,” IEEE J. Sel. Top. Quantum Electron. 21(1), 401–413 (2015).
[Crossref]

J. Appl. Phys. (1)

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. 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(1), 013517 (2005).
[Crossref]

J. Opt. A: Pure Appl. Opt. (1)

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]

J. Phys. Chem. C (2)

F. Venturini, W. Navarrini, G. Resnati, P. Metrangolo, R. M. 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]

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]

Jpn. J. Appl. Phys. (1)

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(Part 2), L1146–L1148 (1999).
[Crossref]

Lab Chip (2)

M. Trebbin, K. Krüger, D. Deponte, S. V. Roth, H. N. Chapman, and S. Förster, “Microfluidic liquid jet system with compatibility for atmospheric and high-vacuum conditions,” Lab Chip 14(10), 1733–1745 (2014).
[Crossref]

D. Choudhury, W. T. Ramsay, R. Kiss, N. A. Willoughby, L. Paterson, and A. K. Kar, “A 3D mammalian cell separator biochip,” Lab Chip 12(5), 948–953 (2012).
[Crossref]

Laser Photonics Rev. (2)

F. Chen and J. R. V. de Aldana, “Optical waveguides in crystalline dielectric materials produced by femtosecond- laser micromachining,” Laser Photonics Rev. 8(2), 251–275 (2014).
[Crossref]

D. Choudhury, J. R. Macdonald, and A. K. Kar, “Ultrafast laser inscription: perspectives on future integrated applications,” Laser Photonics Rev. 8(6), 827–846 (2014).
[Crossref]

Nat. Photonics (2)

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
[Crossref]

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]

Nature (1)

C. V. Raman, “A theory of the viscosity of liquids,” Nature 111(2790), 532–533 (1923).
[Crossref]

Opt. Express (3)

Opt. Lett. (4)

Prog. Quantum Electron. (1)

C. Grivas, “Optically pumped planar waveguide lasers, Part I : Fundamentals and fabrication techniques,” Prog. Quantum Electron. 35(6), 159–239 (2011).
[Crossref]

Other (2)

Potash Corp “Purified Phosphoric Acid,” Tech. Inf. Bull. (2012).

Landolt-Börnstein, “Part 5a, Viscosity and Diffusion,” in Properties of Matter in its Aggregated States, 2, 6th ed. (Springer-Verlag, 1969).

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Figures (10)

Fig. 1.
Fig. 1. Sketch of the fs-laser structuring setup. The red arrows indicate the E-field vectors of the laser beams used for structuring with respect to the orientation of the inscribed tracks.
Fig. 2.
Fig. 2. (a) Sketch of a sample. The black lines indicate the transversally (1) and longitudinally (2) written tracks. (b) Cross sections of tracks of fs-modified material inscribed transversally (in x-direction) and (c) longitudinally (in z-direction, upwards) to the laser beam before etching.
Fig. 3.
Fig. 3. (a) Etching depth vs. etching time for different etching temperatures and square root fits, (b) fitting parameter D vs. temperature Te and viscosity η and linear fit. The single tracks in the [111] oriented samples were inscribed with 0.45 µJ pulse energy, a σ-polarized beam, 100 µm/s translation velocity and etched in phosphoric acid.
Fig. 4.
Fig. 4. Etching depth vs. etching time for different etching agents. The orientation of the sample was [111]. Inscription data and fitting parameters D for square-root time dependencies are given aside the figure.
Fig. 5.
Fig. 5. Fitting parameter D in dependency of pulse energy (a) and writing velocities (b) for different polarizations of the inscribing beam and etching temperatures in phosphoric acid. The samples were oriented in [111] direction. The dashed lines are guides to the eye.
Fig. 6.
Fig. 6. Fitting parameter D in dependency of pulse energy for different writing velocities. The tracks were written in σ-polarization with a lens of 4.51 mm focal length in an [111] oriented crystal and etched in 85 °C phosphoric acid.
Fig. 7.
Fig. 7. Selectivity in dependency of the etching temperature for different samples etched with phosphoric acid or a 1:1 mixture of phosphoric and sulfuric acid at temperatures between 80 °C and 135 °C.
Fig. 8.
Fig. 8. Quasi-three-dimensional microscopic images taken with a Keyence Vhx-6000 microscope. Microchannels resulting from etching of (1) single tracks written in x-direction (4.51 mm focal length, π-polarization, 100 µm/s, 0.27 µJ) (2) multiple tracks written with 1 µm horizontal distance (same writing parameters like (1)) (3) single tracks with round cross sections written in longitudinal writing mode in z-direction (4.51 mm focal length, π-polarization, 0.5 µJ, from left to right: 100 µm/s, 50 µm/s, 25 µm/s). The scale holds for both images. The artifacts arise from an imperfect surface polishing prior to the etching, which was affected during the etching process.
Fig. 9.
Fig. 9. Side views of selectively etched single tracks, etched at different temperatures in phosphoric acid, structured with identical parameters in [111] oriented YAG crystals. Inset (a) shows the magnified front part inside the red circle.
Fig. 10.
Fig. 10. Channel cross sections at the channels front facet in xz-plane. Channel e) was inscribed in a longitudinal writing scheme, all other channels were inscribed in the transversal scheme. Writing conditions were 0.45 µJ, 100 µm/s, 4.51 mm focal length, π-polarization, the samples were etched in phosphoric acid for 50 d.

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

d e = D t 1 / 2 .
D ( T e / η ) 1 / 2 ,   with   η e x p ( E / R T e ) ,
S = 2 × d e / Δ w ch .

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