1. Introduction

Direct femtosecond laser inscribing of transparent glasses offers a facile means [1] for guiding chemical etching [2] to open structures [3] with flexible three-dimensional (3D) geometry. Such femtosecond irradiation followed by chemical etching (FLICE) enables integrated micro/nano fluidic, optofluidic and microelectromechanical systems (MEMS) [4–7] to form seamlessly without the 2D limitation of photolithographic processing [5]. In fused silica, the overlapping exposures lead to a self-organized assembly of periodic nanograting structures [8] oriented orthogonally to the laser polarization. Hnatovsky et al. noted a highly selective chemical etching response only along the nanograting planes [9], permitting a wide birth of 3D micro-structuring applications in fused silica and other transparent materials. FLICE permits fabrication of optical components such as micro-mirrors [10] and micro-lenses [11] or mechanical components such as actuators [6,12], flexures [13] and micro-channels [14–16]. Such multi-component elements can be further integrated into microsystems such as a lab-on-a-chip [5], an optofluidic micro-rheometer [16], a lab-in-fibre [17], a microscope-on-chip [11,18], and a high-harmonic optical generator [15].

The FLICE technique typically entails scanning of a focussed Gaussian beam to form single isolated or multiple overlapping nanograting tracks. For the case of transverse scanning, a single scan track results in micro-holes of elliptical-like profile, as dictated by the focal beam waist (w₀) and depth of focus (DOF). At the threshold exposure, the nanograting track can be reduced to a single nano-grating plane and provide an extraordinary small nano-hole with cross-sectional dimensions of ∼40 nm by ∼1.5 µm [19].

In order to etch other channel shapes or to process larger volumes, parallel modification tracks are typically scanned into an assembly that permits opening of the desirable volume shape [12,15,17]. Alternatively, beam shaping presents a more direct approach to control the cross-sectional profile of a single modification track. A slit aperture [20] and cylindrical lens [21] have been applied in transverse scanning to balance the DOF and w₀ ratio and provide circularly symmetric holes. Spatiotemporal beam shaping was also applied to generate a channel with a circular cross-section [22] and also benefit from an improved spatial resolution [23]. A variety of alternate beam shapes have been applied more broadly to 3D femtosecond processing of transparent materials [24–28]. For example, spatial light modulators (SLMs) have been used to correct for spherical aberration of the surface (hereafter surface aberration) when inscribing deeply focussed optical waveguides [24,25], to split beams into multi-foci for parallel processing [26], to stretch beams axially to form high aspect ratio nanochannels [27], and to accelerating beams for shaped micromachining [28]. The potential for moulding the nanograting structure to follow a Bessel beam shape has been demonstrated with an axicon [29], but without an examination of the potential FLICE responses over the shaped modification volume. There remain significant gaps in knowledge around the type of morphology expected from the laser interactions and their potential for FLICE processing when the beams have become so heavily deviated from the traditional Gaussian beam shape. The combined influence of nonlinear optical propagation and the deep focussing distortion due to surface aberration further remains unexplored. With the advent of high averaged powered (i.e., kW) femtosecond lasers today, the possible tailoring of the nanograting volume to follow non-traditional beam shapes is desirable for scaling up the speed and volume in FLICE processing.

In this paper, the manipulation of the laser interaction volume and resulting shape of nanograting structure for FLICE processing was explored in fused silica by modifying a Gaussian laser beam with diverging (positive) and converging (negative) conical phase fronts. The further influence of surface aberration on deep focusing to 1500 μm depth was also examined. On focusing, the conical phase fronts formed a vortex beam with zero angular momentum. However, the high intensity interactions of the convex and concave phase fronts were dominated by Gaussian-Bessel beam shapes forming before or after the focal plane, respectively. On transverse scanning, the beam shaping resulted in tuning of modification tracks from circular to highly asymmetric cross-sections of up to 1:13 aspect ratio which would not be possible to produce in a single scan with traditional Gaussian beam focusing. The beam shape strongly influenced the chemical etching rate, the resolution, and the morphology of the resulting micro-channels, opening new avenues for both geometric shape and chemical process control. Beam shaping thus opens new ways to harness higher laser pulse energy for cross-sectional shaping and for scaling up of processing volume in 3D FLICE nanomachining.

2. Experiment setup and beam shaping

An ytterbium-doped fiber laser (Amplitude; Satsuma) of 250 fs pulse duration and 1030 nm wavelength provided Gaussian beam quality of M_x² = 1.14 and M_y² = 1.01. The laser beam was frequency doubled and expanded to w₀ ${\cong} $ 2 mm beam waist (1/e² radius), underfilling the SLM (Hamamatsu; X10468-04) aperture. The phased modulated beam was relayed through a 4f optical beam system as shown in Fig. 1(a) and reconstructed with 2.43× demagnification onto to the back focal plane of a fabrication lens as described previously [30]. The fabrication lens (aspherical 0.55 NA, Newport; 5722-A-H) minimized spherical aberration for focusing in air, but was employed (Fig. 1(a)) to complete a final Fourier transform of the SLM image plane to a focal plane positioned inside of a fused silica substrate (Nikon; NIFS-S). To minimize surface aberration, the fabrication lens was underfilled (Fig. 1(a)) to a spot size of 0.9 mm (1/e² radius), creating an effective focusing NA of 0.20. A diffraction limited beam waist of ∼1 µm (1/e² radius) and a DOF of 11 µm (full width at half maximum) was calculated for inside of the fused silica. The laser polarization was linear and aligned perpendicular with respect to the writing direction.

 

Fig. 1. Simplified schematic (a) of the SLM beam shaping and focusing arrangement, providing conical phase front for femtosecond laser processing of fused silica. Simulated longitudinal intensity profiles of Gaussian-Bessel beams forming in the focal volume of fused silica for complementary conical phase front angles of θ = –1.03 (b), –0.26 (c), 0 (d), +0.26 (e) +1.03 (f) mrad. The simulations included surface aberration for 600 µm paraxial focusing depth (white dashed line). Scale bar applies to all beam profiles. The green arrows show the direction of laser propagation.

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Laser modification tracks were formed by transverse scanning at 0.4 mm/s speed and 500 kHz repetition rate, using parameters optimized previously for FLICE with the same laser configuration [31]. In this way, nano-gratings were assembled with an effective exposure of at least 2500 overlapping pulses. Laser pulse energy was varied from 52 nJ to 421 nJ. The tracks were written at depths of 100, 600, and 1500 µm to intercept the side facets, and then immersed in 5% aqueous hydrofluoric acid (HF) for 30 minutes to etch along the tracks. Laser modification tracks were formed with opposite scanning directions into and out of the facet, but had negligible influence on the morphology of laser tracks and micro-channels, or on the chemical etching rates. The influence of conical phase front angle on the shape of modification tracks prior to and after chemical etching were evaluated in terms of resolution, morphology, aspect ratio, and etching rates with an optical microscope (Olympus; BX51).

Conical phase fronts with complimentary angles, θ, of –2.5 < θ < + 2.5 mrad were phase wrapped (0 - 2$\pi $) and imposed onto the SLM. The conical angles were magnified by 2.43-fold at the SLM image plane (Fig. 1(a)) by the fixed 4f relaying optics. These angles resulted in vortex beams of 0^th order forming with radii up to 40 µm by the 4.5 mm focal length lens. The focusing was further influenced by increasing surface aberration with deeper focusing. For comparison, unmodified beams (θ = 0) were also applied with and without pre-compensation of aberration, by imposing the aberration correction (AC) phase from [24,32] onto the SLM.

Fourier propagation was employed to simulate the longitudinal intensity profile expected to form inside of the fused silica under the influences of various degrees of surface aberration and conical phase front. The surface aberration phase front expected for each focusing depth [24] was translated to the SLM image plane (Fig. 1(a)) as described in [24], positioned 4.5 mm before the aspherical lens. The aberration phase front was corrected for the magnification in the 4f beam delivery and combined with the conical phase front at the SLM plane. The resulting light field was Fourier transformed and propagated under the thin lens approximation to provide the longitudinal intensity profile in the focal volume of the fused silica substrate.

3. Results and discussion

3.1 Simulated beam shape

The anticipated beam shapes under the combined influence of conical phase front and surface aberration are first examined by comparing focusing at 100 and 600 µm depths (Fig. 1(b)-(f)). The longitudinal cross-section of intensity profile for an unmodified Gaussian beam (θ = 0) at 600 µm depth shows (Fig. 1(d)) a relatively tight focus of ∼1 µm (1/e² radius) has been retained near the paraxial focus (dashed line) but elongated by the surface aberration from 11 µm to 17 µm DOF for the respective shallow (100 µm, not shown) to deep (600 µm) focusing conditions. The DOF further stretches to 24 µm at 1500 µm focusing depth (not shown), indicating a moderately strong surface aberration effect arising under the present 0.20 NA focusing. Returning to 600 µm focusing depth, Fig. 1(e) shows a conical phase front angle of θ = +0.26 mrad to have compressed the DOF to 12 μm, nearly recovering to the aberration-free beam shape (i.e., 11 µm DOF). Positive conical phase front thus serves as a form of aberration correction. Further increase of the angle to θ = +1.03 mrad reimposes the influence of conical phase front, stretching the DOF by 7-fold to 87 µm while also shifting the position of peak intensity downward by 94 µm below the paraxial focus (600 µm dashed line in Fig. 1(f)). In contrast, conical phase fronts with negative angles work in concert with surface aberration to provide large elongation effects, for example, stretching the DOF to 32 µm and 113 µm for θ = –0.26 and –1.03 mrad conical angles, respectively. The peak intensity positions are also shifted upward by 19 and 94 µm, respectively, due to the concave wavefront shape.

The Fourier simulation provided a guidance on the potential beam elongation available with varying laser pulse energy and conical phase front angle as presented in Fig. 2 for the case of 600 μm focusing depth. An anticipated height, ${{\rm{C}}_{\rm{h}}}$, for the etched microchannel was calculated (false color) as a function of conical phase front angle (-2.5 < θ < +2.5 mrad) and pulse energy (52 < ${\rm{E}}$ < 421 nJ) by scaling the exposure to an observed threshold energy, ${{\rm{E}}_{{\rm{th}}}}$ = 40 nJ, for modification assessed for the reference case of non-aberrated and unmodified focusing (i.e., z = 0 µm depth, θ = 0 mrad). A threshold axial energy density for modification was defined as $ \frac{\mathrm{E}_{\mathrm{th}}}{0.5 \times \mathrm{DOF}_{\mathrm{th}}}=\frac{40 \mathrm{~nJ}}{6.3 \,\mathrm{\mu} \mathrm{m}} $. Fourier simulation was employed to calculate the vertical range of exposure, $ \mathrm{C}_\mathrm{h}(\mathrm{E}, \mathrm{\theta}) $, exceeding this threshold density according to a simple DOF scaling given by $\frac{{{{\rm{E}}_{{\rm{th}}}}}}{{{\rm{DO}}{{\rm{F}}_{{\rm{th}}}}}} = \frac{{\rm{E}}}{{{{\rm{C}}_\textrm{h}({{\rm{E}},{\mathrm{\theta }}} )}}}$. In Fig. 2, a boundary for the onset of etching begins at a minimum exposure threshold of ∼40 nJ found at θ = +0.26 mrad (off scale in Fig. 2) and projects from this vertex into a cone accommodating an increasing magnitude of conical phase front angle with rising energy. The cone is centered near θ = +0.26 mrad (dotted white line) that marks the minimization of surface aberration for 600 μm focusing depth. Under this tightest focussing case, the channel heights are predicted to be shortest. With increasing or decreasing angle, the channel heights rise before an abrupt fall at a threshold exposure for maximum beam stretching. The white dashed line marks a boundary exposure reaching to three-times the threshold, $\frac{{3 \times {{\rm{E}}_{{\rm{th}}}}}}{{{\rm{DO}}{{\rm{F}}_{{\rm{th}}}}}}$ . Inside this region, the modification tracks are highly over-exposed, and distortions from Kerr focusing, plasma defocusing, and intensity clamping are anticipated. Outside of this zone, the one anticipates a beam elongation stretching from ${{\rm{C}}_\textrm{h}}$ = ∼6 µm at low pulse energy exposure of 50 nJ to ∼62 μm high channels for θ = ${\pm} $1 mrad conical beams at 400 nJ exposure. The trend lines in Fig. 2 remain similar for focusing at other depths, except for a shift of the symmetry axis (θ = + 0.26 mrad) according to the degree of surface aberration compensation required.

 

Fig. 2. Calculated microchannel height shown in false colour as a function of conical phase front angle (-2.5 < θ < + 2.5 mrad) and pulse energy (52 nJ to 421 nJ) for focussing depth of 600 µm. The data were calculated by Fourier simulation and based on a threshold pulse energy of ${{\rm{E}}_{{\rm{th}}}}$ = 40 nJ forming a channel height of 6.3 µm for shallow focussing and unmodified beam $({\mathrm{\theta }} = 0 \,{\rm{mrad}}$). The white dashed line identifies the laser exposure corresponding to 3$\; \times {{\rm{E}}_{{\rm{th}}}}$. The minimum DOF of 12 µm was generated at θ = +0.26 mrad (white dotted line).

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3.2 Morphology of laser tracks

A comprehensive set of laser modification tracks was evaluated over widely varying conical phase front angle and increasing degrees of surface aberration as focal depth was increased from 100 to 1500 µm. A representative example illustrating the influence of conical phase front at 600 µm focusing depth is presented in Fig. 3, showing facet and top-sided views of the track morphology before and after modification by chemical etching. The facet morphology observed in Fig. 3(a) prior to chemical etching are presented for 157 and 305 nJ pulse energies, which defined an optimal exposure range found for forming tracks having slow and precise to moderately fast etching rates at shallow focal depth (i.e., ∼100 µm). Positive and negative conical phase fronts compressed and stretched the focal interaction volume (Fig. 3(a)), respectively. As previously studied [33], the white and grey zones (Fig. 3(a)) were inferred to be regions of positive and negative refractive index change [25], respectively. One notes an asymmetric response in the contrast and relative sizes of the grey and white zones according to the application of negative and positive conical angles, respectively. As a reference for the unmodified beam (θ = 0), similar vertical track heights of 12 and 14 μm (Fig. 3(a)) were observed in the facet images for the 157 and 305 nJ exposures.

 

Fig. 3. Optical microscopy images (a) showing the end facet morphology of laser scanning tracks in fused silica formed with 157 nJ (top) and 305 nJ (bottom) pulse energy for varying conical phase front angles ranging from $- $1.03 mrad to $+ $1.03 mrad, left to right. The horizontal axis marks the vertical position of modification by the unmodified beam (θ = 0 mrad). Optical microscopy images (backlighting) of facet morphology (top) and side view (bottom) of the modification tracks in (a) following 30 minutes of chemical etching (5% aqueous HF), revealing opened channels (dark zones) for laser exposure of 157 nJ (b) and 305 nJ (c) pulse energy. Laser tracks were inscribed at 500 kHz repetition rate, 0.4 mm/s scan speed, and 0.20 NA focusing optic. The direction of laser propagation was downward in the facet views and into the page for the side views.

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The influence of conical phase front on the track morphology is seen (Fig. 3(c)) in stretching the vertical track height by 86% and 29% to 26 and 18 μm, respectively, for angles of θ = – 0.26 mrad and θ = +0.26 mrad, respectively, when using 305 nJ exposure. These stretching compares favorably with stretching values of 50% and 37% with respect to simulated DOF values from Fig. 1(c) and (e), for the same respective conical angles, θ = –0.26 mrad and θ = + 0.26 mrad. The Fourier propagation simulations (Fig. 1) thus provide a guide in predicting the track heights arising from the combined effects of surface aberration and conical phase front angle. However, the larger DOF required an increasing pulse energy to reach the modification threshold intensity over the increased length. The countering influences of positive and negative conical angles with surface aberration as predicted in Fig. 1(b)-(f) was observed and confirmed in the morphology images (Fig. 3(a)). This manifested in the asymmetric angle limits of θ = –0.51 mrad and θ = + 0.68 mrad for the maximum stretching angles available to reach threshold modification as shown for the 157 nJ pulse energy example in Fig. 3(a). These boundaries expand to θ = –1.03 and θ = +1.03 mrad for the higher pulse energy case of 305 nJ (Fig. 3(a)).

The tuning of conical phase front angle has thus enabled an adjustable modification height together with a variable ratio of positive (white) and negative (grey) refractive index changes (Fig. 3(a)). The underlying assembly of nanograting volumes is unveiled next by examining the morphological zones opened by chemical etching. The highly contrasting black zones of the facet images for low 157 nJ (Fig. 3(b), top) and high 305 nJ (Fig. 3(c), top) energy exposure reveal the cross-sectional profile of micro-channels that have opened into the bulk glass after 30 minutes of etching time. The side view images, positioned below the facet views in the bottom images of Fig. 3(b) and (c) show a widely varying etching depth and hole width that appear strongly influenced by the conical phase front angle. However, the facet views (Fig. 3(a)) show the etching volume to be localized to the grey zones that are seen to dominate the morphology in the pre-etching facet images (Fig. 3(a)), where nanograting structures are inferred [7,34] to have formed to guide chemical etching. The relative size and shape of the etched ‘black’ zones (facet views in Fig. 3(b) and (c)) are seen to elongate or compress by the varying conical phase angle, that approximately follow the beam shaping trends in Fig. 1(b)-(f). Conical phase front shaping thus opens a way to manipulate the cross-sectional symmetry of the etched holes.

The chemically etched tracks reached maximum depths of ∼110 and ∼188 µm (Fig. 3(b) and (c), bottom) for 157 nJ and 305 nJ pulse energy, respectively, which appear enhanced by the positive phase front angles in the range of θ = $ + $0.08 to $+ $0.26 mrad over the unmodified beam case (θ = 0). In Fourier propagation analysis, aberration compensation was found optimal at $+ $0.26 mrad (Fig. 1(e)), which aligns well with the deepest observed etching tracks (Fig. 3(c) side view) and shortest observed channel heights (Fig. 3(c) facet view). In contrast, negative phase front angle inhibited the FLICE process, indicating an asymmetric benefit from the positive conical phase fronts that may arise from compensation of surface aberration.

Alternatively, negative conical phase fronts enabled a novel tailoring of micro-channel shapes, providing high aspect ratio channel cross-sections as high as 1:7 and 1:13, with respective 157 nJ and 305 nJ laser exposures and respective θ = –0.51 and –1.03 mrad angles. This processing zone is delineated (red dotted lines) in the facet view images of Fig. 3(b) and (c) and presents a new way to create smooth, narrow open fissures with a single laser scan of a negative conical-phase-shaped beam. The deepest etching tracks here (i.e., –1.03 mrad in Fig. 3(c)) were noted to form a smooth and symmetric perimeter wall (facet view) while also retaining a uniform cross-sectional profile into the depth (side view). In contrast, a channel roughness of ∼1 µm was evident along the channel depth (Fig. 3(c), side view) for positive angles near $+ $0.26 mrad that partially compensated for the surface aberration. The wall roughness may be attributed to an over exposure condition observed from θ = –0.17 to +0.51 mrad for exposure of 305 nJ pulse energy. This exposure range falls into the hatched overexposure zone as predicted in Fig. 2. However, one may employ a larger conical phase front angle to avoid or mitigate the overexposure, for example, for conical angles of θ > +1.03 and θ < –1.03 mrad. In Fig. 3(c) (facet views), these larger angles provided microchannel heights of 24 and 28 µm, respectively, which are only 2-fold smaller than the predicted channel height of ∼ 50 µm in Fig. 2 for 305 nJ and θ = ${\pm} $1.03 mrad exposure conditions. Hence, the principles of combining conical angle and pulse energy (Fig. 2) to mitigate over-exposure and facilitate control of channel cross-section (Fig. 3) have been verified.

3.3 Tuning microchannel etching

A global assessment of conical phase front and surface aberration effects on chemical etching of laser tracks are presented in terms of etching rate and lateral resolution by the false colour graphics in Fig. 4. The graphics show the combined influence of laser pulse energy, conical phase front beam shaping and surface aberration on the FLICE processing. The pulse energy was increased with steps of about 20%. The growing influence of surface aberration appears in the progression of shifting processing zones in Fig. 4(a), (b), and (c) as focussing depth was increased over values of 100, 600, and 1500 µm, respectively. The colour scale presents the average etching rate (µm/min) as observed over a 30 min etching time. Average etching rates of ∼2.5 µm/min are noted at 52 nJ pulse energy that just exceeds the modification threshold exposure. The etch rates are seen to fall from here before rising again with increasing exposure to maximum rates that scaled upward from ∼6.7 to 7.5 µm/min for a focusing depth increasing from 100 µm to 1500 µm, respectively. These highest rates were observed at pulse energy exposures in a narrow range of 341 to 347 nJ. The etching rates were notably influenced by the conical phase front angle, providing maximum values with angles shifting from 0 to +0.17 and +0.43 mrad as the focussing depth was increased from 100 μm to 1500 μm.

 

Fig. 4. Observed etching rates (µm/min), averaged over 30 min, presented in false colour as a function of conical phase front angle (-2.5 < θ < +2.5 mrad) and pulse energy (52 nJ to 421 nJ) for depths of 100 (a), 600 (b), and 1500 µm (c), respectively. The contour lines identify the lateral width of channels (2 to 5 µm). The short-dashed white lines (a, b, and c) mark an iso-intensity exposure calculated for near-threshold modification that provided the highest aspect ratio channel cross-section. The white dotted lines mark the conical angle minimizing the effect of surface aberration. Optimal conical phase front angles (d) for forming micro-channels with highest resolution (green squares) and fastest etching rate (red circles), plotted as a function of depth, and including examples of channels with high aspect ratio cross sections (blue diamonds). The three inset images are representative examples of varying cross-sectional etching profile. The red line is a guide to the eye. The green bars mark a flexible processing window for generating higher resolution micro-channels with <3 µm width and supporting etching rates of at least 50% of the peak value. The black dashed line indicates the conical phase front angles calculated for minimizing surface aberration.

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While high etch rates are desirable in forming micro-holes, the highest rates in Fig. 4 were associated with rougher wall morphology, higher cross-sectional asymmetry, and larger cross-sectional area. To reflect such factors, the etch rate data were superimposed with contour lines marking the lateral width (2-5 µm) of the etched tracks. The graphics thus capture the most influential ranges for conical phase front angle ($- $2.5 < θ < $+ $2.5 mrad) and pulse energy (52 to 421 nJ) for controlling etching rates and the structural resolution. The outer boundaries of the etching zones can be predicted by calculating an iso-exposure condition marked by the white dashed curve on Fig. 4(a), (b), and (c). Here, an above threshold pulse energy exposure of 52 nJ was selected for the case of shallow and non-aberrated focusing and scaled linearly by the stretching of the beam focus by both of aberration and conical phase front relative to the minimum DOF of ∼11 µm. The iso-exposure indicates a boundary where sufficient laser exposure was delivered to enable a modification for guiding chemical etching.

Examination of the shallow focussing data (100 µm) in Fig. 4(a) notes a low-energy onset (∼52 nJ) for enabling chemical etching that centres symmetrically in a narrow range of conical phase front angle near to θ = 0 mrad. This unmodified beam condition provided modest etching rate of ∼2.5 µm/min, opening holes with high resolution (∼2 μm width) and the most symmetric cross-section. Without beam modification (θ = 0 mrad), etch rates could be increased to 6.4 μm/min with a higher pulse energy of 305 nJ, but at the cost of wider track size (∼ 5.1  μm) and rougher wall surface. These observations align well with the literature available on HF etching under similar exposure and shallow focussing conditions [35,36].

With little beam distortion from surface aberration, Fig. 4(a) shows the conical beam shaping to only diminish the etching rate symmetrically, for example at 341 nJ exposure, falling from a peak value of 6.7 µm/min at θ = ∼0 mrad to values of ∼3 µm/min at the symmetric conical phase front angles of θ ${\cong}\,{\pm} $0.43 mrad. These conical angles gave rise to a lower peak intensity exposure due to the beam elongation, resulting in a narrowing of the track from ∼5.7 µm to ∼ 3 µm while also improving the surface morphology. These benefits are commensurate with an increasingly asymmetric cross-section transformation of the etch channel that approximately followed the conical beam simulation. Because there is little surface aberration, the iso-exposure curve (white dashed line) marks a symmetric boundary encompassing the processing window about θ = 0  mrad. However, there is an absence of graphical data for negative conical angles θ $\le $ $- $0.5 mrad due to a one-sided beam elongation reaching the top surface to induce laser ablation (Fig. 4(a)). Conical beam shaping provides one key benefit for the case of shallow focussing (∼ 100 μm) by permitting higher laser exposure to open micro-holes with higher aspect ratio cross-section while avoiding surface roughening.

The influence of the conical phase front shaping became increasingly skewed by the increasing degree of surface aberration with deeper focusing depth. This can be seen in Fig. 4(b) for the case of 600 μm focussing depth by the vertical shift of the peak etching rate from θ = ∼0 mrad in Fig. 4(a) to up to ∼$+ $0.34 mrad in Fig. 4(b). An angle of $+ 0.26\;$ mrad defined the centre angle for a maximum etching rate of 6.3 µm/min for exposure of 380 nJ pulse. Fast etching rates were observed over a broad range of all-positive conical phase angles from $+ $0.17 to $+ $0.34 mrad (see also Fig. 3(b) and (c)). At near-threshold exposure of 52 nJ, the etching rates slowed to 1.9 µm/min, and centered into a narrow processing zone of conical phase front angles from $- $0.08 to $+ $0.08 mrad.

A bifurcation of the chemical etching into two distinct angular processing zones, previously noted in Fig. 3(b) and (c), manifests in Fig. 4(b) into a broad zone of fast etching rate (∼5 to 6.3 µm/min) appearing over positive conical phase front angles spanning from ∼0 to $+ $0.51 mrad and a slower etching zone (∼1 to 4.3 µm/min) appearing in a narrow range of negative conical phase front angle. The latter zone grows increasingly negative with rising pulse energy, trending diagonally (Fig. 4(b)) from 0 to $- $1.71 mrad phase angle for pulse energy increasing from 52 nJ to 421 nJ. This new processing zone is confined to narrow angle ranges of θ =$- $0.08${\pm} $0.08 mrad at 54 nJ pulse energy and θ = $- $0.51${\pm} $0.17 mrad at 182 nJ pulse energy.

The slow and fast etching zones mark significant difference in the surface quality, symmetry, and resolution of the etched holes. The slower etching zone provides moderately long holes (∼ 26 µm) with smooth surface morphology and narrow (2 to 3 µm) cross-sectional profile (i.e., Fig. 3(b) and (c), facet view) that reached a maximum aspect ratio of 1:13 for 305 nJ exposure. In contrast, the fast-etching zone provided micro-holes with a more symmetrical cross-section, as reported above in the ranges of θ = 0 to $+ $0.34 mrad for 157 nJ pulse energy (Fig. 3(b), facet view) and θ = 0 to $+ $0.51 mrad for 305 nJ pulse energy (Fig. 3(c), facet view). In Fig. 4(b), the most symmetric and narrowest (width < 3 µm) etching tracks were formed in the range of positive conical angles increasing monotonically from θ = 0.17 ${\pm} $ 0.17 mrad at 80 nJ to θ = 0.51 ${\pm} \;$0.17 mrad at 182 nJ. The width of the symmetric holes increased above 3 µm with higher exposure, and eventually formed asymmetric cross-section at the highest 421 nJ exposures tested.

The formation of symmetric cross-section was associated with a positive conical angle of +0.26 mrad (Fig. 1(e)) providing partial compensation of surface aberration for the present 600 µm depth. Higher positive angles (i.e., θ = $+ $0.51 to $+ $0.69 mrad for 157 nJ pulse exposure, Fig. 3(b)) of conical phase front thus delivered an increasing large beam elongation, resulting in high aspect ratio cross-sections reaching as high as 1:12 for θ = $+ $1.03 mrad (305 nJ, Fig. 3(c)). The benefits of narrow channel width (2.3 µm), high aspect ratio (1:12) and smooth surface morphology are similar with those reported above in the slow etching zone but obtained with slower etching rates of 1.8 µm/min (Fig. 3(c), θ= $+ $1.03 mrad for 305 nJ) than 2.3 µm/min (Fig. 3(c), θ= $- $1.03 mrad for 305 nJ), respectively.

The trends noted in Fig. 4(b) for 600 µm depth were further amplified at the deeper focusing depth of 1500 µm (Fig. 4(c)). At 1500 µm depth, the rapid etching zone has broadened moderately on angle and shifted to larger positive (diverging) conical angles with etching rates peaking at slightly higher rates of 7.46 µm/min near θ =$\; + $0.51 mrad (see Fig. 4(c)) in contrast with 6.3 µm/min at θ = $+ 0.26\;$mrad (Fig. 4(b)) at 600 μm depth. Narrow holes (< 3 µm) having the most symmetrical cross-section were observed at conical phase front angle of $+ $0.51 mrad that aligns with the best compensation of surface aberration available for this depth by a conical phase front. At this angle, peak etching rates of 7.46 µm/min were found to be 1.6× faster than values observed (∼ 4.5 µm/min) with the unmodified beam (θ = 0 mrad) for the same 347 nJ pulse energy. For the same pulse energy, the micro-hole cross-section could be tuned to high aspect-ratio by shifting away for the aberration compensation angle (θ =$\; + $0.51 mrad) to either high positive (θ <$+ $2.4 mrad) or moderate negative (θ >$- $1.03 mrad) angles. This angular processing zone narrowed with lowering of the pulse energy, providing narrower and lower aspect ratio holes with improving morphology. Outside of this trendline, one finds several new processing zones (Fig. 4(c)) that provided high resolution holes (< 2 µm) with high etch rates of ∼7 µm/min by using only modest pulse energies of 107 to 146 nJ for respective conical angles of θ = $ + $1.03 and $+ $1.37 mrad.

The iso-exposure contours for near-threshold modification (white dashed lines) in Fig. 4(b) and (c) are seen to shift upward and broaden with increasing focussing depth as larger positive values of conical phase front are required to compensate for the increasing surface aberration relative to the shallow focus case at 100 µm depth (Fig. 4(a)). The narrowest etch channels with smoothest surface morphology were observed to form near this threshold level of exposure. Hence, the iso-exposure contours are a guide for controlling the geometric shape and size of microchannels by tuning energy and conical angles to compensate for surface aberration and thus provide the highest quality channels without damage due to over-exposure.

For comparison purposes, laser modification tracks were generated at 600 and 1500 µm depth with beams pre-compensated for the surface aberration by the SLM. For 600 μm depth, the aberration correction could not improve over the etch rates reported (Fig. 4(b)) for the uncorrected (θ = 0 mrad) or optimized conical-shaped (i.e., θ ${\cong} $ 0.3 to 0.7 mrad) beams. A maximum etch rate of 4.6 µm/min was observed at 239 nJ pulse energy. For 1500 μm depth, the AC modified beams offered a moderate improvement in etch rate, peaking at 5 µm/min for 169 nJ relative to 1.4 μm/min (Fig. 4(c)) for the uncorrected beam applied at the same energy. This rate matches with the 5 µm/min rate observed at the same energy applied in shallow focusing (100 µm), demonstrating the expected aberration correction for low pulse energy interaction. However, AC had less influence at higher pulse energy, providing lower etching rates than observed under shallow focusing (100 μm) and also falling short of the peak etch rates noted for the conically shaped beams (i.e., 7.46 µm/min, 347 nJ, Fig. 4(c)). On the other hand, compensation of surface aberration improved on the cross-sectional symmetry of the micro-holes, but without benefits in improving etching rates or tuning cross-sectional shapes as demonstrated with the conical phase front.

The processing zones as plotted in Fig. 4(a), (b), and (c) conform well with the conical-shaped processing window as predicted in Fig. 2. Microchannels with the largest heights were generated when using the largest magnitude of conical angle for the largest pulse energies tested. However, an anomalous zone of slow etching rate was observed for a large range of negative conical phase front angles in the cases of 600 μm (Fig. 4(b)) and 1500 μm (Fig. 4(c)) focussing depth while Fig. 2 indicated that exposures in this zone were well above the modification threshold. Some form of nonlinear propagation and absorption effect may be disturbing the assembly of nanogratings for this combination of surface aberration and negative conical focussing effect. The maximum pulse energy exposure of 421 nJ here only moderately exceeded the 221 nJ threshold to drive Kerr lensing in fused silica. As previously studied [30], the focal elongation effects of conical phase front beams and surface aberration were found to inhibit the self-focusing and plasma defocusing distortions in single-pulse exposure. Hence, a further study is required to unravel the pulse accumulation effects under negative conical angles that appear to inhibit the chemical etching (Fig. 4(a), (b) and (c)).

A global summary of the benefits in conical phase front shaping for FLICE processing is presented in Fig. 4(d). On a measure of fastest etching rate, a monotonical increase in conical phase front angle from $+ $0.17 to $+ $0.51 (Fig. 4(d), red circles) provided peak values ranging from 6.67 to 7.46 μm/min for increasing depth from 100 to 1500 µm. However, the resulting micro-holes were wide (>3 µm) and showing rough morphology (Fig. 4(d), inset image on red arrow). Alternatively, a broader range of conical phase front angles (Fig. 4(d), green bars) represent tracks forming with high resolution (<3 µm width, inset image on green arrow) and moderate etching rates meeting at least 50% of the peak values. These processing zones follow approximately with the optimal conical phase front angle (Fig. 4(d), black dashed line) for compensating a large part of the surface aberration. Hence, the diverging conical phase front ( >0 mrad) enabled a full recovery and further enhancement of the etch rate of laser tracks formed at the deep depths (up to 1500 µm) of fused silica. In contrast, focal elongation with converging conical phase front (< 0 mrad) provided a novel geometry for etching open high aspect ratio (1:7 and 1:13) channels with modest etching rates (Fig. 4(d), blue diamonds, inset image on blue arrow).

The processing zones in Fig. 4(d) map key trend lines but exclude the influence of pulse energy as better depicted in Fig. 4(a)-(c). These more detailed graphs also identified other processing zones as previously discussed, for example, the fast (∼7 µm/min) and high resolution (< 2 µm) etching zone observed for low pulse energy (107 to 146 nJ) applied at larger conical phase front angles ($+ 1$.0 to +1.37 mrad) at 1500 μm focussing depth (Fig. 4(c)).

4. Conclusion

The combined influence of conical phase front beam shaping with surface aberration (Fig. 1), working together or against each other, was shown to provide broader latitude for controlling FLICE processing over shallow to deep focusing depth. Diverging conical phase fronts (positive angles) were tuned to partially compensate the surface aberration (Fig. 4(d)) and recover or enhance etching rates to provide high resolution and near symmetric etching channels for up to 1500 μm focusing depth (Fig. 4(a)-(c)). More generally, positive and negative phase angles offered a way to configure the relative size and shapes of the black and white modification zones (Fig. 3), offering more latitude in tailoring the geometric shape of the 3D microstructure from circular to high-aspect ratio cross-section (Fig. 2 and 3). The conical beam shaping promises wider latitude to generated variable 3D surface landscapes as presented in [37] for surface machining of silicon. Hence, the SLM conical phase front shaping demonstrated a diversity of FLICE control in fused silica over etch rate, track size, surface roughness, and cross-section geometry when used in tandem with variable pulse energy.