Comparison of laser micro-inscribed pillars on flat versus tilted substrates

When laser micromachining is performed at a non-zero angle of incidence, it is possible to fabricate microstructures with tilted sidewalls that could not be achieved using a perpendicular beam. In this report, we examine the influence of the angle of incidence (0–70°), feature size and spacing (10–150 µm), lens focal length (25 and 100 mm), and spot radius (15–50 µm) on the minimum achievable feature width (Whalf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{{{\text{half}}}}$$\end{document}) of periodic arrays of square micropillars. A new empirical model is developed that is applicable to both flat and tilted substrates, and accurately predicts the pillar height as a function of the feature width. By comparing structures inscribed at different tilt angles, we found that when the angle of incidence is increased Whalf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{{{\text{half}}}}$$\end{document} also rises, such that it becomes more difficult to fabricate thin microstructures on angled substrates. Based on this knowledge, we establish a correlation that predicts Whalf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{{{\text{half}}}}$$\end{document} as a function of the angle of incidence and the beam’s spot radius. Therewith, we can predict the minimum feature width that can be achieved over a wide range of laser processing parameters, with an average error of only 4.5% within the bounds of this investigation. Finally, by testing lenses with different focal lengths, we found that a shorter focal length reduces Whalf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{{{\text{half}}}}$$\end{document} (improves the spatial resolution) when the angle of incidence is zero, but increases Whalf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{{{\text{half}}}}$$\end{document} when the angle of incidence is high. We attribute this discrepancy to the angular spread of the beam cone, which can interact significantly with adjacent pillars, leading to degradation of the intended microstructure and a reduction in pillar height.


Introduction
Laser micromachining has been of growing interest over the last two decades since it is a scalable, one-step fabrication process that does not require a clean room, and is applicable to most substrate materials. The majority of published literature on the fabrication of microstructures by fs laser micromachining concentrates on studies in which the laser beam strikes the surface at a perpendicular angle. In contrast, relatively few reports have performed angled laser micromachining using a tilted sample (or beam) [1][2][3][4][5][6][7][8][9]. This specialized technique allows for the fabrication of microstructures with tilted sidewalls that could not be achieved with a perpendicular beam, and is important for applications such as thin film sensor production [1], optics [2], microcutting tool fabrication [3,4], microfluidics and droplet manipulation [5,10], and methane capture devices [11].
The relevant equations for angled laser micromachining can be developed based on those for normal incidence angle. Figure 1a illustrates the 2D and cross-sectional intensity profiles of a Gaussian beam. It is standard in the field to define the beam's radius ( R ) based on the 1∕e 2 standard [8,12], such that the intensity at the perimeter (black dashed line) is equal to 13.5% of the maximum intensity. Accordingly, 95% of the radiation is contained within the defined beam radius, and 5% of the radiation extends into the 'beam tails'. For laser micromachining, the beam is passed through focusing optics, typically a convex lens. Figure 1b presents an illustration of the beam's shape profile after the focusing lens. The bottom of the image represents the focal plane, at which the beam reaches its smallest achievable spot radius, w 0 . The value of w 0 can be calculated using characteristics of the light and lens: where is the beam's wavelength, f is the focal length of the lens, R is the radius of the unfocused beam, and M 2 is the beam quality factor, which describes the variation of the beam's intensity profile from that of an ideal Gaussian beam. The position of the focal plane is taken as z = 0 on the z-axis shown in Fig. 1b. The theoretical spot radius at a non-zero z-position, w z , is given by Also shown in Fig. 1b is the beam's angular spread, which describes the angle at which the beam cone shrinks as it nears the focal plane. Longer-f lenses have the advantage of a lower and hence a greater depth of focus (DOF) but have the detriment of a wider w 0 . This leads to a wider minimum beam diameter, which limits the minimum achievable size of inscribed lines and structures.
When laser micromachining is performed on a flat surface, the beam spot is circular, with an area of A spot = w 2 z . However, when the laser beam has an incidence angle with respect to the plane of the surface, the spot radius is affected by the projection of the beam onto the angled plane of the surface. Hence, as illustrated in Fig. 1c, the spot radius is stretched in the downhill direction, such that the angled beam spot radius is given by w z, = w z ∕cos( ) [1]. Then, since the beam is not stretched in the y-axis direction, the spot takes the shape of an ellipse, with a minor radius of w z , a major radius of w z, , and an area of The single pulse peak fluence must also be adjusted to consider the influence of the incidence angle . For micromachining on flat surfaces, the fluence is given by F z = 2E∕ w 2 z . However, when the spot area of the tilted surface is considered, the fluence of angled micromachining is instead given by To date, very few articles have been published on the topic of angled laser micromachining. Emmony et al. [6] measured the periodicity of laser induced periodic surface structures (LIPSS) on a germanium mirror and observed a correlation between the ripple distance and . Nolte et al. [7] observed that the reflectivity of the laser beam on the tilted side walls of percussion drilled holes is influenced by the beam's linear polarization direction, leading to elliptical holes. Wang et al. [2] used an incidence angle of = 45 • to fabricate pyramid and cone-shaped optical devices on sapphire. Overmeyer et al. [1] developed the equations for the beam's spot size and fluence on tilted substrates (Eq. 4 and 5) and validated their theory through the successful fabrication of thin film sensors on 3D substrates. Ling et al. [8] (4) A spot, = w 2 z ∕ cos ( ). profile. The black dashed lines represent the 1∕e 2 spot radius. The graph below plots the relative intensity, with the beam tails shaded pink. b Illustration of a focused laser beam including the angular spread ( ), spot radius at focus ( w 0 ), and spot radius at an unfocused z-position ( w z ). c Illustration of the laser beam's spot radius on a flat ( w z ) versus an angled ( w z, ) substrate studied the formation mechanism of laser induced ellipsoidal cones during repeated raster scans of a steel surface, and deduced that the ellipsoidal shape is due to the different reflectivities of s-and p-polarized light incident on different sections of the cones' surfaces. Hajri et al. [3] combined Overmeyer's equations with a model published by Neuenschwander et al. [13] to establish a new model for the rate of material ablation during angled laser micromachining. Tuo et al. [5] fabricated angled micropillars on a steel surface, leading to directional wettability properties of water drops on the surface, as well as inherent drag reduction. Börner et al. [4] micromachined dielectric, cylindrical rods to fabricate end mill cutting tools with a diameter of 1 mm. Kravanja et al. [9] used = 45 • to engrave tilted grooves on a magnetoactive elastomer substrate, leading to tunable wettability between a sticky and sliding state for water droplets.
What is common among all of the publications listed above is that each was performed using a narrow range of laser processing parameters. Hence, a parametric study of how different processing parameters influence the resulting microstructures during angled laser micromachining is greatly needed. In this report, we examine the influence of the angle of incidence ( 0 < < 70 • ), feature size and spacing (10-150 µm), lens focal length (25 and 100 mm), and spot radius ( 15 < w z < 50 µm) on the final size and fidelity of inscribed micropillars. The goal of our analysis is to identify which of these parameters require special consideration for the case of angled micromachining. Our experimental results surround the production of periodic arrays of square micropillars, with a special emphasis on the fabrication of tall micropillars with well-defined valleys between them, which is difficult to achieve at high .
Femtosecond laser micromachining was performed using a Carbide CB3, 40 Watt pulsed laser (Light Conversion) with a central wavelength of = 1030 nm. The laser was operated with a repetition rate of 7.4 kHz and a pulse duration of approximately 250 fs. The beam was focused using either a 25 or 100 mm focal length lens, depending on the experiment. The spot radius was adjusted by positioning the sample with respect to the lens at specific z-positions corresponding to w z values of 15, 25.8, 35, or 50 µm (see Eq. 2). All experiments were performed using w z = 25.8 µm, except for the 'Influence of Spot Size' discussion section. To ensure comparability between all of our fabricated patches, a consistent peak fluence value of F z = 1.27 J/cm 2 was maintained throughout experiments. Since both the spot radius and the incidence angle affect the fluence value, we varied the pulse energy from E = 6.4 to 70.6 µJ, depending on the experiment. Due to this variation in E , the material removal rate per overscan pass also fluctuated for different patches. Thus, the number of overscan passes was also varied to maintain a consistent patch depth, based on the measured rate of material removal for different laser processing parameters. A full list of patches analyzed in this report, along with all of their laser processing parameters is available in Supporting Table 1. All patches were machined using p-polarized light, with respect to the plane of incidence of the beam on the tilted sample. P-polarized light was chosen because, according to Fresnel's equations, the reflectivity of p-polarized light is lower than that of s-polarized light. Accordingly, p-polarized light produces more ablation of the substrate material and hence a higher oval efficiency of the micromachining process [7,8].
The irradiation source was complimented by a 5D sample positioning system purchased from Newport Corp. (Irvine, CA, United States), consisting of two XMS50-S linear translation stages (x-and y-axes), one MVN80 vertical linear stage actuated by an LTA-HL motorized actuator (z-axis), and two URS50BCC rotational stages. The stages were controlled using a Newport XPS universal high-performance motion/driver controller, which was instructed using PVT (position-velocity-time) command codes produced using custom Python programs. Figure 2a presents an illustration of the raster scanning pattern that was used to fabricate square pillar microstructures on the substrate. It is important to note that the 2D plane of the scanning pattern is projected onto the x-y plane of the 3D experimental space (as indicated in Fig. 2b), and not the substrate plane. Adjacent lines are spaced 5 µm apart, while simultaneously adjusting the z-axis position to maintain the same focus position between the uphill and downhill portions of the patch. The beam shutter is opened and closed over specific sections to leave the substrate material to develop square pillars. As indicated on the illustration by dashed boxes, based on the number of adjacent lines that include beam shuttering, we can define the intended width ( W i ) of the pillars. We use the term 'intended' because on the final sample, the actual widths of the pillars will differ slightly from the width defined by the raster scanning pattern due to the tapering angle of the pillars' walls. In all experiments, the intended spacing ( S i ) is equal to W i , such that the period of the pattern is equal to 2 ⋅ W i . The intended height ( H i ) of the pillars in each patch is equal to the product of the single layer material removal rate and the number of overscan passes of the raster scan, such that H i ranged from 142 to 149 µm, as shown in Supporting Table 1. Figure 2B presents a 3D illustration of an angled micropillar, along with our naming scheme that defines the pillar's dimensions along different cross-sections of the surface. The height and width of the pillar along the blue crosssectional plane, which is parallel to the x-z plane, are defined as H and W , respectively. The subscript was chosen to designate that these measurements were taken on the cross-sectional plane that slopes down the face of the tilted surface. The height and width of the pillar along the orange cross-sectional plane, which is parallel to the y-z plane, are defined simply as H and W (respectively), since on that cross-sectional plane the tilt of the surface is not relevant.

Analysis
Confocal microscopy was performed using an Olympus LEXT OLS5000 laser scanning confocal microscope. 3D visualizations of this data were generated using the Olympus OLS5000 Analysis Application. To reliably measure the width, height, and spacing of microstructures, crosssections of the confocal data were exported from the Olympus OLS5000 Analysis Application into MATLAB. As shown by the black arrows in Fig. 2b, the feature heights were measured from the tallest point of a given pillar to the lowest point of an adjacent valley. Every measurement presented in this report is the average of at least 3 individual measurements.
Scanning electron microscopy (SEM) was performed using an FEI Inspect F50 microscope. = 0 • (normal angle of incidence), the micropillars stand perpendicular to the substrate plane (see Fig. 2b), whose orientation is illustrated by the white dashed lines in each 3-D image. Then, when is increased, the micropillars gain a corresponding tilt angle with respect to the substrate plane. This effect is further visualized by Fig. 3b, which shows scanning electron micrographs of the same pillar patches (higher-resolution SEM images are available in the supporting information, along with higher magnification images of LIPSS that formed on the pillars' side walls). Note that these images were acquired by placing the samples flat on the SEM's stage, so that the tilt of the pillars can be clearly visualized with respect to the substrate plane. Accordingly, at = 0 • the pillars appear as a simple square array, whereas at higher , the pillars lean progressively more towards the right side, appearing wider and overlapping each other. This topography can be better understood through Fig. 3c, which shows the topography of cross-sections of the = 0 and 60 • pillars. For the = 60 • patch, the pillars are tilted towards the right side, with the peaks of the pillars hanging over pores beneath.

Model for microstructure height evolution
Before considering the influence of different tilt angles on laser micromachined structures, we will first develop a model that will be useful for quantifying the influence of factors such as the incidence angle, spot size, and focal length. Figure 4 presents confocal microscope heatmaps of micropillars fabricated on a flat substrate ( = 0 • ) with different intended widths ( W i ). Since the same laser processing parameters and number of overscans were used for each patch, the total ablated depth in the substrate material was the same for all three cases. However, based on the shift in color between the three heatmap images, it is clear that pillar height increases for wider pillars (larger W i ).
This trend can be explained by considering the raster scan trajectories that were used to fabricate each patch (refer to Fig. 2a). Below each heatmap in Fig. 4 is a graph showing a series of overlapping Gaussian curves (black lines) representing the intensity profiles of adjacent lines in the scanning pattern. Some lines of the scan are shuttered to produce pillar structures, which leads to crests and troughs in the accumulated intensity profile (red line), which is calculated by summing all of the adjacent Gaussian distributions [14]. Hence, regions with a high accumulated intensity produce  Confocal microscope heatmaps of the topography of micropillars with different intended widths. The colored bar beneath the heatmaps correlates the topographical height in the images with different colors. Also below each heatmap is a graph of the theoretical accumulated intensity profile that was used to fabricate the patch valleys, whereas regions of low accumulated intensity produce pillars. For W i = 20 µm, the tails of the individual intensity profiles in the valley regions overlap significantly with the pillar regions, which causes the pillars' height to become truncated by the tails of the adjacent scan lines. In contrast, for W i = 70 µm, the spacing between pillars is wide enough that the accumulated intensity profile of the neighboring valley regions does not significantly encroach on the pillar sections, which allows for the development of tall, well-resolved pillars.
This relationship is demonstrated on a broader scale by Fig. 5a, which plots the relationship between the pillar height ( H ) and W i for pillars of different sizes. Starting from the left side of the graph, the value of H begins at 0 μm for small values of W i , which matches the observations of previous researchers who found that when pillars are fabricated too close together, no distinct microstructures form at all [15,16]. Then, H rises quickly in the range of 30 < W i < 70 μm and finally reaches an asymptote of around 170 μm when W i is large. This asymptote corresponds to the maximum possible depth of the patch, based on the rate of material removal and the number of overscans. This trend is further visualized in Fig. 5b, which shows 3-D confocal images corresponding to four representative data points. Based on the shift towards warmer colours in the heatmaps, it is clear that the pillars become taller when W i increases. It is also visible in the rightmost image that, when W i is very large, the pillars become truly square shaped since the beam tails no longer overlap with the centers of the pillars.
Combined, all of these factors produce the characteristic trend indicated by the solid line in Fig. 5a, which is described by a mathematical relationship called the logistic function: As indicated in Fig. 5a, U is the asymptote value, and W half represents the location on the x-axis where H reaches half of its asymptote value ( H = U∕2 ). The parameter k defines the slope of the curve. Equation 6 predicts that when W i < W half , the pillar height will be less than half of its asymptote value ( H < U∕2 ), whereas when W i > W half , H will be greater than U∕2 . Thus, for the sake of our ensuing discussion, we will define pillars with H > U∕2 as being well resolved, and pillars with H < U∕2 as being poorly resolved. From a practical standpoint, W half represents the minimum feature width that can be machined while retaining the option to produce tall, well-resolved microstructures, and will be used as a basis of comparison throughout this discussion.

Influence of angle of incidence
To assess the influence of incidence angle on the shape and quality of inscribed microstructures, patches of micropillars with intended widths ranging from W i = 10 to 150 μm were fabricated at = 0 , 45, 60, and 70 • . Figure 6 presents the pillar height evolution for a variety of different incidence . Fig. 5 a Pillar height versus the intended width for = 0 • . The trendline is given by Eq. 6. b 3-D confocal heatmaps of four representative data points. The shift to warmer colours indicates that H increases with W i angles ( = 0 , 45, 60, and 70 • ). In this graph the y-axis represents the dimensionless pillar height, H * = H ∕U , such that a value of H * = 1 represents a measurement where H is equal to the asymptote value fit using Eq. 6. This format facilitates our comparison of the four data sets, and the dimensional version of this plot ( H versus ) can be found in the supporting information. Also, note that the images presented in Fig. 3 correspond to the data points in Fig. 6 at W i = 40 µm, and can be used to better visualize the patches represented by the data. In Fig. 6, it is clear that Eq. 6 accurately models the pillar height evolution for all four tested values of . Comparing all four curve fits, it is apparent that while the slopes of the lines are quite similar (k remains essentially constant), the curves shift to the right as increases. This relationship is graphed in the figure inset, which demonstrates the dependence of W half on and shows that W half remains similar for = 0 and 45 • , but increases monotonically for progressively larger incidence angles. To model this trend, we have developed a simple empirical model that predicts W half as a function of : In this equation, W half,flat is equal to W half for = 0 • , and q is a fitting factor. Using MATLAB's nonlinearmodel function, we found that the best fit is obtained for values of W half,flat = 38.3 µm and q = 0.27 . Equation 7 is represented by the black line in the inset of Fig. 6, and provides a good fit for the data ( R 2 = 0.93 ). However, it should be noted that Eq. 7 is not derived based on theory, and was developed based on only four data points. Accordingly, this model is applicable to the scope of this investigation, but should be further validated by future research before being accepted for a broader application. Now, given that W half defines the smallest pillar width that will produce well-resolved pillars, and W half increases for higher , it follows that microstructures are generally less well-resolved when fabricated at higher . This trend is demonstrated in Fig. 7a, which plots the change in H * when is increased, for several discrete values of W i . For relatively thin pillars ( W i = 40 µm), H * drops substantially over the range of angles tested, from 0.54 to 0.09. Figure 7b plots the cross-sectional height of a series of W i = 40 µm pillars fabricated at different , in which the drop in H * with can be clearly observed.
In contrast to the downward trend observed in H * for narrow micropillars, very wide micropillars are less affected by changes in , as shown by the pink and black lines in Fig. 7a (7) W half = W half,flat 1 + q ⋅ (1∕ cos ( ) − 1) .

Fig. 6
Dimensionless pillar height of microstructures fabricated at different incidence angles versus the intended pillar width. The trendlines were fit using Eq. 6, and the inset plots the values of W half from Eq. 6 against the incidence angle. The trendline within the inset plot is given by Eq. 7 representing W i = 100 and 150 µm. This occurs because for those cases, W i remains much larger than W half even at high . Hence, as demonstrated by Fig. 4, the pillars are wide enough that the beam tails do not overlap significantly with the peaks of the pillars. As a result, H * remains close to its asymptote value for all incidence angles tested. Together, these observations demonstrate that for high angles of incidence, wider feature sizes must be designed to produce wellresolved microstructures on the finished workpiece, which represents a significant constraint in the design process of tilted microstructures.
The laser beam's angle of incidence also influences the wall angle of the inscribed micropillars. The wall angle is important because, when the laser beam removes material from the substrate during micromachining, the inscribed trenches do not have perpendicular walls. For example, when a single line is inscribed on the surface, a V-shaped groove is typically formed due to the Gaussian distribution of the beam's energy. Further, when multiple scans of the surface are performed, a characteristic wall angle forms due to a combination of beam scattering, reflectance, and fluence reduction occurring on the angled wall that was already created by previous scans [17]. Figure 8a plots the wall angles of both the uphill ( wall,u ) and downhill ( wall,d ) slopes of the micropillars (as illustrated in Fig. 8b)  The positive correlation between wall,d and is easily explained. Before micromachining begins, the downhill wall of the pillar is already tilted at an angle of . Then, successive raster scans repeatedly remove substrate material, which further steepens the downhill pillar wall, leading to larger values of wall,d for patches fabricated at higher . In contrast, before micromachining begins, the pristine substrate material that will be removed to form the uphill wall of the pillar is tilted in the direction opposing wall,u , as can be seen by comparing the dashed line in Fig. 8b with the bracket illustrating wall,u . Accordingly, it is surprising that wall,u and are also positively correlated. To explain this trend, we theorize that wall,u might increase at larger due to light reflecting off adjacent, uphill pillars, as shown in Fig. 8b. In this diagram, some light is reflected from the downhill side of the rightmost pillar, eventually ablating the uphill side of its neighbour. This mechanism explains the positive correlation between wall,u and , and could be validated in future experiments by comparing angled pillar patches on materials with different reflectivities.

Influence of spot radius
To determine the influence of the beam's spot size on angled microstructures, a series of experiments were performed at an incidence angle of = 45 • and various spot sizes of w z = 15 , 25.8, 35, and 50 µm. The resulting values of H * versus W i are plotted in Fig. 9 along with curve fits obtained using Eq. 6, which match the trend of the data. 3-D confocal Fig. 8 a Wall angle versus angle of incidence, for the uphill and downhill slopes of micropillars ( W i = 70 µm). Note that some error bars are too small to be visible behind the data points. b Illustration of the laser beam interacting with the tilted substrate during micromachining Fig. 9 Comparison of inscribed micropillars using four different beam spot radii. The 3-D confocal heatmaps below the graph represent the associated inscribed surfaces at W i = 40 µm (from left to right, w z = 15, 25.8, 35 , and 50 µm) heatmaps of pillars fabricated with W i = 40 µm at each w z are also presented below the graph for visual comparison. The inset graph plots the values of W half versus the different spot sizes tested. The black line in the inset graph represents the following equation: in which c 45 is a fitting factor, applicable to microstructures fabricated at = 45 • . Using a value of c 45 = 1.59 , Eq. 8 provides a good fit for the data in the Fig. 9 inset ( R 2 = 0.99 ). This demonstrates that there is a direct, linear relationship between the spot size of the beam and the minimum microstructure size that can be fabricated in a well-resolved fashion.
Furthermore, Eq. 8 was established using a series of patches fabricated at = 45 • , but it is reasonable to assume that the linear correlation between W half and w z would also hold for other incidence angles. Hence, for normal angle of incidence, we assume that W half,flat = c 0 w z , where c 0 is applicable to the case of = 0 • . Substituting this correlation into Eq. 7, we define a new, more comprehensive prediction of W half as a function of both w z and : Using all of the data presented above for different spot sizes and incidence angles, the best fit is obtained using q = 0.29 and c 0 = 1.45 . Table 1 provides a comparison between the values predicted by Eq. 9 and the measured values, and demonstrates that Eq. 9 is highly successful, with an average error of only 4.5%. Therefore, when fabricating tilted microstructures, the value of W half can be accurately predicted as a function of the beam's spot size and the incidence angle.
Our measured value of c 0 = 1.45 suggests that wellresolved microstructures cannot be produced with a width less than 1.45 times the beam radius based on the fluence use in our experiments. However, it is important to note the value of c 0 is highly dependent on application-specific parameters, specifically the relationship between the substrate's threshold fluence and the applied fluence, F z . For example, Noh et al. [12] fabricated an array of conical spikes on steel with a width of approximately W i = 5 µm using a beam with R = 17.5 µm, which suggests a value of c 0 ≈ 0.29 . However, this was accomplished using a much lower fluence value ( F z = 0.026 J/cm 2 ) than our experiments ( F z = 1.27 J/ cm 2 ), which is known to decrease the ablated line width [12,16]. Based on this comparison, we conclude that while the trend defined by Eq. 8 is widely applicable, the value of c 0 is highly process dependent.

Influence of lens focal length
All of the microstructures compared above were machined using a lens with a focal length of f = 100 mm. To determine how the focal length influences angled micromachining, we used an f = 25 mm lens to replicate our patches for incidence angles of = 0 and 70 • using the same laser processing parameters. Figure 10a compares the H values for a series of patches with different W i for both lenses at = 0 • , and Fig. 10b makes the same comparison for = 70 • . In both graphs, the asymptote pillar height reached by the 25 mm lens is significantly lower than the 100 mm lens, by around 30% in each case, indicating that the material removal rate was greater for the longer-f lens. The value of W half is also affected by the focal length: in Fig. 10a, W half is smaller for f = 25 mm, whereas in Fig. 10b, W half is larger for f = 25 mm. This discrepancy can be visualized through the 3-D confocal microscope heatmaps shown below the graphs. At = 0 • and W i = 40 µm, H is greater for the f = 25 mm lens because its W half is very small at that incidence angle. This is discernable in the 3-D heatmaps, in which the peaks of the f = 25 mm pillars reach a darker red colour (and hence a greater height), as indicated by the coloured bar. In contrast, at = 70 • , H is lower for the f = 25 mm lens because its W half is very large at that incidence angle. Due to the sloped surface's influence on the heatmap, this is not immediately discernable through the colour of the pillars. However, through visual inspection, it is clear that the pillars fabricated using the f = 100 mm lens are taller and more well-resolved than those made using the f = 25 mm lens. Therefore, the shorter-f lens produced taller pillars at = 0 • , but produced shorter pillars at = 70 • . This trend can be quantified by considering Eq. 7, in which the value of q quantifies the increase in W half (and hence the minimum feature size that can be well-resolved) when increases. Using the W half measurements shown in both graphs in Fig. 10, the 25 mm lens exhibits a value of q = 0.65 , which is much larger than the value measured for f = 100 mm ( q = 0.27 ). This difference suggests that shorter f lenses are more prone to undesirable reductions in microstructure resolution when machining at high . The dependance of q on f can be explained by considering the geometry of the tilted micropillars, and their interaction with the laser beam. As per Eq. 3, when a focused laser beam approaches a substrate, it takes the shape of a cone with an angular spread of . This is illustrated in the schematics presented in Fig. 11a, which show the beam cone approaching both a flat and a tilted surface. For the 100 mm lens (gray cone), the beam cone is extremely narrow, with = 1.4 • . However, for the 25 mm lens (red cone), the angular spread is much larger, resulting in a wider cone ( = 5.7 • ). This is important because the wall angle ( ∅ wall,d , as indicated in Fig. 11a) of the micropillars is different on the flat and tilted substrates. Taking patches with W i = 70 µm and f = 100 mm as a benchmark, the pillars fabricated at = 0 • have � wall,d = 71.2 • , whereas the pillars fabricated at = 70 • have a much higher wall angle of � wall,d = 79.4 • when measuring the downhill side of the pillar. Accordingly, the beam cone resulting from the 25 mm lens passes extremely close to the row of micropillars immediately uphill from its point of focus, with a separation between the beam cone and substrate of only 4.9 • (from Although the beam cone illustrated in Fig. 11a does not overlap with the substrate as drawn, some of the laser beam's energy will still interact with the uphill pillar due to the distribution of the beam's intensity profile. As shown in Fig. 1a, 5% of the beam's energy is contained in the beam tails, which extend beyond the boundaries of the beam spot radius. This energy is not accounted for in Eq. 3, which assumes that the beam forms a perfectly conical shape. Therefore, some of the laser beam's energy extends beyond the beam cone's angular spread (as defined by Eq. 3) and can ablate substrate material uphill from the beam's intended location, reducing the height of adjacent micropillars. Moreover, this effect is more prominent for shorter-f lenses due to their wider beam cones, which explains why W half changed more for the 25 mm lens than the 100 mm lens in Fig. 10 when was increased from 0 to 70°. This effect is demonstrated by Fig. 11b, which plots the cross-sectional height profile

Conclusions
To identify what laser processing parameters require special consideration for the case of angled micromachining, we fabricated periodic arrays of square micropillars using angles of incidence ranging from 0 < < 70 • , spot radiuses ranging from 15 < w z < 50 µm, two different lenses ( f = 25 and 100 mm), and a wide range of intended pillars widths 10 < W i < 150 µm.
By measuring the height of pillars fabricated with different widths, we developed a new model (Eq. 6) that predicts H as a function of W i . Equation 6 also allowed us to define a new variable, W half , which effectively characterizes the minimum achievable feature width, and hence can be used to define the spatial resolution limitations of the laser etching process on flat or tilted substrates. By comparing micropillars fabricated at different , we discovered that W half increases monotonically with , such that at higher angles of incidence, the minimum achievable feature width also increases. In addition, we found that the wall angle of the micropillars is positively correlated with . Notably, the downhill wall angle increased from wall,d = 71.2 to 79.4 • when was increased from 0 to 70 • . Accordingly, microstructures fabricated at high have more perpendicular sidewalls.
Next, by testing different spot radii, we discovered that there is a direct relationship between w z and W half . Accordingly, when w z increases, the minimum achievable width of the inscribed microstructures also increases proportionally. Based on this knowledge, a new model is introduced that predicts W half as a function of and w z . This model is useful to determine the minimum feature width that can be achieved over a wide range of processing parameters, and is shown to have an average error of only 4.5% within the bounds of this investigation.
Finally, we tested a lens with a shorter focal length ( f = 25 mm) than the one that was used for the main body of experiments (100 mm). Interestingly, decreasing f reduced W half at = 0 • , but increased W half at = 70 • . We attribute this discrepancy to the angular spread of the beam cone, which is wider for shorter-f lenses. At low , the wall angle is also relatively low, so the beam does not interact significantly with adjacent pillars. However, at high , the wall angle is very high, so the beam can ablate substrate material uphill from its present location, reducing the height of adjacent micropillars, leading to degradation of the intended topography.
For future work, we recommend that similar studies are performed using angled laser micromachining on a wider range of substrate materials, especially since we theorized that the positive correlation between wall,u and could be explained by the reflection of the laser beam on our metallic substrate. Also, similar studies should be performed using different microstructure topographies, such as grooves or laser induced periodic surface structures. Finally, we suggest that the influence of beam polarization, including linear s-and p-polarized light, along with circular and elliptical polarization states, should be investigated on tilted substrates.