Evaluation of the Cross-Sectional Shape of μ–Grooves Produced in Stainless Steel 304 by Laser-Induced Etching Technique

The variation in cross-sectional profile of a microgroove fabricated with focused and diverging laser irradiation is demonstrated with ray tracing. To verify the result of ray tracing, stainless-steel 304 microgrooves were manufactured utilizing the conventional lens-based and optical fiber-based laser-induced etching techniques in phosphoric acid solution. Three distinctive groove geometries, i.e., a flat surface with no groove, an intermediate stage groove, and a fully developed V-groove, were rendered for numerical analysis. For focusing mode, the first and second reflections were caused by high laser intensity and the second reflected beam could lead to variation in the groove shape such as a U-shaped groove or a V-shaped groove in accordance with etchant concentration. On the contrary, a weak laser entirely distributed at the groove sidewall could not induce a chemical reaction, leading to a V-shaped groove. The effect of process variables such as laser power (intensity) and etchant concentration on the cross-sectional profiles of a groove are closely examined through the computed simulation results.


Introduction
A microgroove is a key configuration of numerous microelectromechanical systems based on microthermal and microfluidic devices such as a micro heat pipe [1][2][3], micro pump [4,5], micro mixer [6,7], micro reactor [8,9], and micro fuel cell [10]. Due to the relative easiness in fabrication and simplicity in geometry, microgrooves are increasingly adopted in the design of microdevices; accordingly, researches about effective micromachining techniques of grooves are also increasing. Since the performance of microdevices is almost exclusively decided by the capillary pressure force and sidewall roughness, the controllable shape of a microgroove is absolutely demanded in fabrication technology. Among the fabrication technologies of a microgroove, laser precision microfabrication has been frequently adopted for the manufacture of microgrooves on a variety of materials in the past few decades. The laser material processing in association with a direct writing technique enables to easily provide a rapid prototype of microstructures with no mask or additional processes. Laser-based micromachining of metals has often been achieved with high-intensity pulsed laser irradiation. However, a large heat-affected zone (HAZ) with redeposition of evaporated or resolidificated material and mechanical load on the processed area can have a negative effect on the functional properties of microstructures [11,12].
As a possible approach to realize material removal while avoiding such issues, laserinduced etching (LIE) may be employed. In LIE, the workpiece is immersed in a liquid etchant that helps the fabrication of microgrooves with almost no melting or debris due to the cooling effect. The micromachining of a workpiece in LIE is done through a thermochemical reaction between the etchant and the workpiece, while the laser beam irradiates the workpiece to locally raise the surface temperature. A relatively simple setup compared to a picosecond or sub-picosecond pulsed laser machining system is required for LIE at a substantially lower equipment or maintenance cost. Due to these advantages of LIE, Figure 1a shows the schematic diagram of a conventional lens-based LIE system. A diode-pumped solid-state continuous wave laser with a maximum power of 8 W and a center wavelength of 532 nm was used as the heat source. The laser beam was expanded five times and then tightly focused using a focusing lens (5×, focal length = 40 µm, working distance = 37.5 mm, numerical aperture = 0.14, depth of focus = 14 µm) on the surface of the workpiece after passing through a linear polarizer and quarter-wave plate. The laser beam was circularly polarized at the surface of the workpiece to avoid a possible polarization dependence of the etching process [18]. The direct writing of grooves was done by translating the workpiece using a high-resolution (1 µm/pulse) motorized XYZ stage. The workpiece was placed inside a Teflon ® chamber that was installed on the XYZ stage.
In conventional LIE, the laser light from a source travels through several optical elements such as a beam expander, a polarizing beam splitter, and quarter-wave plate before being focused by a focusing lens and irradiating the workpiece as shown in Figure 1a. A disadvantage of this configuration is that alignment of the laser beam through the optical elements is frequently required, which is tedious and potentially dangerous to a user. To overcome the shortcomings of conventional LIE, an optical fiber can be employed for laser beam delivery, as illustrated in Figure 1b. A pulsed Nd:YAG laser (center wavelength = 532 nm, maximum power = 60 W, pulse duration = 150 ns, repetition rate = 10 KHz) was used to manufacture microgrooves. The laser beam was launched into the optical fiber by a focusing lens (5×, focal length = 40 µm, numerical aperture = 0.14, depth of focus = 14 µm). Commercially available multimode fibers with core diameter of 50 µm and 105 µm (cladding diameter = 125 µm, numerical aperture = 0.22 for both fibers) were used as the light waveguide and machining tool. Note that, unlike a conventional LIE system in which the laser beam is transported to the workpiece through many optical components, the optical fiber delivers the laser beam directly to the surface of the workpiece with no intermediate optical elements, and no focusing lens is used to focus the laser beam on the workpiece. The fiber tip was housed inside a titanium jig with a jig mount and installed on a high-resolution (1 µm/pulse) motorized XYZ stage. The workpiece was placed inside a Teflon ® chamber through which the etchant is continuously circulated by a peristaltic pump. the laser beam is transported to the workpiece through many optical components, the optical fiber delivers the laser beam directly to the surface of the workpiece with no intermediate optical elements, and no focusing lens is used to focus the laser beam on the workpiece. The fiber tip was housed inside a titanium jig with a jig mount and installed on a high-resolution (1 μm/pulse) motorized XYZ stage. The workpiece was placed inside a Teflon ® chamber through which the etchant is continuously circulated by a peristaltic pump.
A stainless-steel 304 foil (Fe72/Cr18/Ni10) with a thickness of 500 μm was used as the workpiece. Additionally, phosphoric acid (H3PO4, 85%) diluted with distilled water was used as the etchant to etch the stainless-steel 304 foil. All etching processes were monitored in real time with a charge-coupled device (CCD) camera. Etching results were examined using a scanning electron microscope after an ultrasonic cleaning of the workpiece.
(a) (b)  Figure 2 shows an illustration of the geometry of the model for ray tracing. The material used for modeling was stainless steel 304. To investigate the variation in laser intensity over the groove surface for the abovementioned two different types of laser irradiation, a three-dimensional ray tracing was performed with cartesian coordinates.  A stainless-steel 304 foil (Fe72/Cr18/Ni10) with a thickness of 500 µm was used as the workpiece. Additionally, phosphoric acid (H 3 PO 4 , 85%) diluted with distilled water was used as the etchant to etch the stainless-steel 304 foil. All etching processes were monitored in real time with a charge-coupled device (CCD) camera. Etching results were examined using a scanning electron microscope after an ultrasonic cleaning of the workpiece. Figure 2 shows an illustration of the geometry of the model for ray tracing. The material used for modeling was stainless steel 304. To investigate the variation in laser intensity over the groove surface for the abovementioned two different types of laser irradiation, a three-dimensional ray tracing was performed with cartesian coordinates. light waveguide and machining tool. Note that, unlike a conventional LIE system in which the laser beam is transported to the workpiece through many optical components, the optical fiber delivers the laser beam directly to the surface of the workpiece with no intermediate optical elements, and no focusing lens is used to focus the laser beam on the workpiece. The fiber tip was housed inside a titanium jig with a jig mount and installed on a high-resolution (1 μm/pulse) motorized XYZ stage. The workpiece was placed inside a Teflon ® chamber through which the etchant is continuously circulated by a peristaltic pump.

Modeling for Ray Tracing
A stainless-steel 304 foil (Fe72/Cr18/Ni10) with a thickness of 500 μm was used as the workpiece. Additionally, phosphoric acid (H3PO4, 85%) diluted with distilled water was used as the etchant to etch the stainless-steel 304 foil. All etching processes were monitored in real time with a charge-coupled device (CCD) camera. Etching results were examined using a scanning electron microscope after an ultrasonic cleaning of the workpiece.
(a) (b)  Figure 2 shows an illustration of the geometry of the model for ray tracing. The material used for modeling was stainless steel 304. To investigate the variation in laser intensity over the groove surface for the abovementioned two different types of laser irradiation, a three-dimensional ray tracing was performed with cartesian coordinates.   The ray tracing calculation was conducted for three different groove geometries as presented in Table 1: (i) a plane surface with no groove, (ii) a developing groove in an intermediate stage with width and depth of 30 and 85 µm, respectively, for the focused irradiation condition and 100 and 85 µm for the diverging beam from an optical fiber, and (iii) a completed V-shaped groove structure with width and depth of 50 and 300 µm, respectively, for the focused irradiation and 100 and 300 µm for the diverging laser beam. Table 1. Three types of groove geometry for ray tracing calculation.

Groove Geometry
Irradiation Mode

Focused Beam Diverging Beam
Micromachines 2021, 12, x The ray tracing calculation was conducted for three different groov presented in Table 1: (i) a plane surface with no groove, (ii) a developin intermediate stage with width and depth of 30 and 85 μm, respectively, irradiation condition and 100 and 85 μm for the diverging beam from an o (iii) a completed V-shaped groove structure with width and depth of 50 spectively, for the focused irradiation and 100 and 300 μm for the divergi The distances between the optical source and the workpiece in the s 40 mm for the focused beam condition and 150 μm for the diverging beam an examination of laser intensity distribution on groove walls, optical rece ing the incident rays were installed at the top, sidewall, and bottom of the Ray tracing is a rendering technique based on the reflection and ref Therefore, to obtain the distribution of laser intensity over the irradiated flectivity of the material should be known. A reflectivity value of 0.7 (at 53 for the stainless steel 304 [19,20], and a specular reflection was assume reflected at the groove surface were allowed to transmit through the sur transmitted rays could be detected by the receivers of the model. The ray tracing calculation was conducted for three differe presented in Table 1: (i) a plane surface with no groove, (ii) a intermediate stage with width and depth of 30 and 85 μm, resp irradiation condition and 100 and 85 μm for the diverging beam (iii) a completed V-shaped groove structure with width and dep spectively, for the focused irradiation and 100 and 300 μm for th

Etch Threshold Temperature
The distances between the optical source and the workpiec 40 mm for the focused beam condition and 150 μm for the diver an examination of laser intensity distribution on groove walls, op ing the incident rays were installed at the top, sidewall, and botto Ray tracing is a rendering technique based on the reflectio Therefore, to obtain the distribution of laser intensity over the ir flectivity of the material should be known. A reflectivity value of for the stainless steel 304 [19,20], and a specular reflection wa reflected at the groove surface were allowed to transmit throug transmitted rays could be detected by the receivers of the model The ray tracing calculation was conducted for three di presented in Table 1: (i) a plane surface with no groove, (i intermediate stage with width and depth of 30 and 85 μm, irradiation condition and 100 and 85 μm for the diverging be (iii) a completed V-shaped groove structure with width and spectively, for the focused irradiation and 100 and 300 μm fo

Etch Threshold Temperature
The distances between the optical source and the work 40 mm for the focused beam condition and 150 μm for the d an examination of laser intensity distribution on groove wall ing the incident rays were installed at the top, sidewall, and Table 1. Three types of groove geometry for ray tracing calculation

Width 50
Depth 300 Ray tracing is a rendering technique based on the refle Therefore, to obtain the distribution of laser intensity over th flectivity of the material should be known. A reflectivity valu for the stainless steel 304 [19,20], and a specular reflection reflected at the groove surface were allowed to transmit th transmitted rays could be detected by the receivers of the m The distances between the optical source and the workpiece in the simulation were 40 mm for the focused beam condition and 150 µm for the diverging beam condition. For an examination of laser intensity distribution on groove walls, optical receivers for detecting the incident rays were installed at the top, sidewall, and bottom of the groove models.

Etch Threshold Temperature
Ray tracing is a rendering technique based on the reflection and refraction of light. Therefore, to obtain the distribution of laser intensity over the irradiated surfaces, the reflectivity of the material should be known. A reflectivity value of 0.7 (at 532 nm) was used for the stainless steel 304 [19,20], and a specular reflection was assumed. The rays not reflected at the groove surface were allowed to transmit through the surface so that the transmitted rays could be detected by the receivers of the model.

Etch Threshold Temperature
To investigate the surface temperature variation of the workpiece during laser irradiation, the workpiece temperature was calculated using the thermal model in our previous research [21]. Temperature fluctuations of the workpiece during laser irradiation are a major concern in LIE, and the quality of the machined area depends significantly upon melting and solidification of the workpiece.
Prior to the etching experiments, the etch threshold laser power was investigated via irradiation of the diverging continuous wave (CW) laser beam. Figure 3 shows the hemisphere-like microstructure produced with an experimentally verified threshold laser power of about 0.7 W. The diameter and depth of the etched structure were about 60 µm and 2 µm, respectively. For this threshold power, the computed maximum temperature rise at the workpiece surface was about 360 K for a stationary laser beam; for a 2 µm/s scan speed in the CW etching experiment, the maximum dwelling time of the laser beam at a location was 40 s, as presented in Figure 4. To investigate the surface temperature variation of the workpiece during laser irradiation, the workpiece temperature was calculated using the thermal model in our previous research [21]. Temperature fluctuations of the workpiece during laser irradiation are a major concern in LIE, and the quality of the machined area depends significantly upon melting and solidification of the workpiece.
Prior to the etching experiments, the etch threshold laser power was investigated via irradiation of the diverging continuous wave (CW) laser beam. Figure 3 shows the hemisphere-like microstructure produced with an experimentally verified threshold laser power of about 0.7 W. The diameter and depth of the etched structure were about 60 μm and 2 μm, respectively. For this threshold power, the computed maximum temperature rise at the workpiece surface was about 360 K for a stationary laser beam; for a 2 μm/s scan speed in the CW etching experiment, the maximum dwelling time of the laser beam at a location was 40 s, as presented in Figure 4.

Comparison of Microgrooves Fabricated by Two Different Laser Irradiation Modes
The etching reaction and, thus, fabrication results are highly dependent upon the etchant concentration. Since the velocity of etchant solution varies with concentration, the etchant flow over the workpiece and, more importantly, inside a microstructure under To investigate the surface temperature variation of the workpiece during laser irradiation, the workpiece temperature was calculated using the thermal model in our previous research [21]. Temperature fluctuations of the workpiece during laser irradiation are a major concern in LIE, and the quality of the machined area depends significantly upon melting and solidification of the workpiece.
Prior to the etching experiments, the etch threshold laser power was investigated via irradiation of the diverging continuous wave (CW) laser beam. Figure 3 shows the hemisphere-like microstructure produced with an experimentally verified threshold laser power of about 0.7 W. The diameter and depth of the etched structure were about 60 μm and 2 μm, respectively. For this threshold power, the computed maximum temperature rise at the workpiece surface was about 360 K for a stationary laser beam; for a 2 μm/s scan speed in the CW etching experiment, the maximum dwelling time of the laser beam at a location was 40 s, as presented in Figure 4.

Comparison of Microgrooves Fabricated by Two Different Laser Irradiation Modes
The etching reaction and, thus, fabrication results are highly dependent upon the etchant concentration. Since the velocity of etchant solution varies with concentration, the etchant flow over the workpiece and, more importantly, inside a microstructure under

Comparison of Microgrooves Fabricated by Two Different Laser Irradiation Modes
The etching reaction and, thus, fabrication results are highly dependent upon the etchant concentration. Since the velocity of etchant solution varies with concentration, the etchant flow over the workpiece and, more importantly, inside a microstructure under fabrication can differ significantly for low-and high-concentration cases. Figure 5 shows an example of the stainless-steel 304 microgroove manufactured at two different H 3 PO 4 (etchant) concentrations of 10% and 40% with a lens-based LIE system. The grooves fabricated at these concentrations showed a clear difference in the cross-sectional profile. Those fabricated at the low concentration in Figure 5a,b had a rather uniform width over depth with a round U-shaped bottom. On the other hand, the grooves for the high concentration in Figure 5c,d showed a triangular cross-section due to a gradual decease in groove width with increasing depth. Note that the surface pattern of the grooves for both 10% and 40% cases show no difference. The observed difference in cross-sectional profile was considered to be due to the viscosity changes between these two concentrations. The viscosity of 40% (≈3.8) H 3 PO 4 is about three times greater than that of the 10% (≈1.3) etchant [22,23], which may significantly the influence on the etchant flow inside a narrow groove. For the thermochemical reaction to be sustained, fresh etchant should be continuously supplied into the groove. At high concentrations, however, viscous resistance may hinder an easy refreshment of etchant into the groove, possibly leading to the loss of radicals or even bleaching of etchant. A decease in radicals would result in a natural decrease in groove width because, in this case, the etching reaction would be confined to only near the center of the Gaussian beam where the maximum intensity occurs.
Micromachines 2021, 12, x 6 of 14 fabrication can differ significantly for low-and high-concentration cases. Figure 5 shows an example of the stainless-steel 304 microgroove manufactured at two different H3PO4 (etchant) concentrations of 10% and 40% with a lens-based LIE system. The grooves fabricated at these concentrations showed a clear difference in the cross-sectional profile. Those fabricated at the low concentration in Figure 5a,b had a rather uniform width over depth with a round U-shaped bottom. On the other hand, the grooves for the high concentration in Figure 5c,d showed a triangular cross-section due to a gradual decease in groove width with increasing depth. Note that the surface pattern of the grooves for both 10% and 40% cases show no difference. The observed difference in cross-sectional profile was considered to be due to the viscosity changes between these two concentrations.  For the optical fiber-based LIE, grooves with only a triangular cross-section were fabricated at all concentrations. The reason for this result is that the laser beam emerging from a fiber tip diverges very rapidly, where the degree of divergence depends upon the numerical aperture of an optical fiber. Due to the divergence, the average irradiance decreases rapidly with the distance from the fiber tip. Accordingly, as the etching process is completed and, thus, the groove surface retreats, the etching reaction is active only in the middle of the beam at which the irradiance remains over a threshold value. In the optical fiber-based LIE case, the aspect ratio of the groove can be adjusted by varying etchant concentration and the number of scans over the same groove. Figure 6 shows a representative example of groove fabrication for which a highly uniform and clear surface pattern with no undesired thermal effects was observed. The cross-sectional profile of this groove had a uniform triangular shape and was reproducible. These results demonstrate that the proposed optical fiber-based LIE is an effective method to fabricate metallic microgrooves.
The width of microgrooves that can be fabricated with the optical fiber-based LIE depends on the diameter of optical fiber, and about minimum 50 µm at the surface can be achieved. Note that, for the conventional lens-based LIE, the groove width can be as small as 15 µm because, in this case, the laser beam spot can be focused to a much smaller size. The depth of grooves varies with laser power but it can also be adjusted by varying the number of scans. For a single scan, the groove depth reaches 100 to 200 µm depending on laser power; however, by repeatedly scanning over the same groove, the depth can be increased to 300-400 µm.
Micromachines 2021, 12, x 7 of 14 numerical aperture of an optical fiber. Due to the divergence, the average irradiance decreases rapidly with the distance from the fiber tip. Accordingly, as the etching process is completed and, thus, the groove surface retreats, the etching reaction is active only in the middle of the beam at which the irradiance remains over a threshold value. In the optical fiber-based LIE case, the aspect ratio of the groove can be adjusted by varying etchant concentration and the number of scans over the same groove. Figure 6 shows a representative example of groove fabrication for which a highly uniform and clear surface pattern with no undesired thermal effects was observed. The cross-sectional profile of this groove had a uniform triangular shape and was reproducible. These results demonstrate that the proposed optical fiber-based LIE is an effective method to fabricate metallic microgrooves. The width of microgrooves that can be fabricated with the optical fiber-based LIE depends on the diameter of optical fiber, and about minimum 50 μm at the surface can be achieved. Note that, for the conventional lens-based LIE, the groove width can be as small as 15 μm because, in this case, the laser beam spot can be focused to a much smaller size. The depth of grooves varies with laser power but it can also be adjusted by varying the number of scans. For a single scan, the groove depth reaches 100 to 200 μm depending on laser power; however, by repeatedly scanning over the same groove, the depth can be increased to 300-400 μm.

Optical Rendering
For experiments using the lens-based case, the radius of laser spot at the surface of workpiece was 7 μm as measured with a knife edge method for P = 2 W and f = 40 mm. On the contrary, the 1/e 2 -radius of the laser beam emitted for the optical fiber was about

Optical Rendering
For experiments using the lens-based case, the radius of laser spot at the surface of workpiece was 7 µm as measured with a knife edge method for P = 2 W and f = 40 mm. On the contrary, the 1/e 2 -radius of the laser beam emitted for the optical fiber was about 75 µm for P = 2 W and D gap = 150 µm. Since the laser intensity is inversely proportional to the laser spot area, the temperature of the irradiated surface for these two different modes of LIE may also have a large difference. Furthermore, since multiple reflections of the laser beam at the groove walls can affect the groove wall temperature, the cross-sectional shape of a groove may also be related to the intensity of the reflected beam. Figure 7 shows the computed intensity distributions at the nonetched surface of stainless steel 304 for the "focusing mode" and "diverging mode". The number of rays irradiating the workpiece was limited to 3,000,000 due to calculation time. For the calculation, an optical receiver was numerically placed beneath the surface to detect the incident ray, and the sizes of optical receivers for the focusing mode and diverging mode were 50 µm × 50 µm and 200 µm × 200 µm, respectively. The computed beam diameters agreed with the value of experiments. For the focusing mode, the maximum intensity reached about 3165 W/mm 2 , while that for the diverging mode was only about 1.7 W/mm 2 . For these laser intensities, the maximum surface temperatures calculated using the thermal model for the focusing and diverging modes were about 1490 K and 470 K, respectively, higher than the etch threshold temperature of 360 K. Therefore, etching of the flat workpiece was expected for both irradiation modes, subsequently developing into a groove. During this initial period, the groove shapes were expected to be similar, although their sizes could be different. 75 μm for P = 2 W and Dgap = 150 μm. Since the laser intensity is inversely proportional to the laser spot area, the temperature of the irradiated surface for these two different modes of LIE may also have a large difference. Furthermore, since multiple reflections of the laser beam at the groove walls can affect the groove wall temperature, the cross-sectional shape of a groove may also be related to the intensity of the reflected beam. Figure 7 shows the computed intensity distributions at the nonetched surface of stainless steel 304 for the "focusing mode" and "diverging mode". The number of rays irradiating the workpiece was limited to 3,000,000 due to calculation time. For the calculation, an optical receiver was numerically placed beneath the surface to detect the incident ray, and the sizes of optical receivers for the focusing mode and diverging mode were 50 μm × 50 μm and 200 μm × 200 μm, respectively. The computed beam diameters agreed with the value of experiments. For the focusing mode, the maximum intensity reached about 3165 W/mm 2 , while that for the diverging mode was only about 1.7 W/mm 2 . For these laser intensities, the maximum surface temperatures calculated using the thermal model for the focusing and diverging modes were about 1490 K and 470 K, respectively, higher than the etch threshold temperature of 360 K. Therefore, etching of the flat workpiece was expected for both irradiation modes, subsequently developing into a groove. During this initial period, the groove shapes were expected to be similar, although their sizes could be different.  However, once a groove is formed, the irradiation pattern over the groove walls and the associated development of groove shape are expected to be quite different between the two modes. Figure 8 shows the intensity distribution over the sidewalls of the intermediate stage groove (width = 30 µm and depth = 85 µm) for the focusing and diverging modes. For the focusing mode, absorption of the incident rays over the sidewall increased almost exponentially with groove depth with little absorption at the upper part of the sidewall (Figure 8a). The absence of rays near the top of the groove was due to the difference in propagation directions of rays and the angle of sidewalls. In other words, since the incident beam was confined within a narrow angle (less than about 16 • by the numerical aperture (NA) of the objective lens, NA = 0.14) that was smaller than the angle formed by the groove walls (about 20 • for the intermediate stage groove of consideration), most of the rays directly hit the lower portion of the groove wall, as shown in Figure 8b. Accordingly, the groove width at the surface did not change noticeably, whereas the bottom portion of the groove was continuously etched away, forming a deeper groove.
Micromachines 2021, 12, x 9 of 14 However, once a groove is formed, the irradiation pattern over the groove walls and the associated development of groove shape are expected to be quite different between the two modes. Figure 8 shows the intensity distribution over the sidewalls of the intermediate stage groove (width = 30 μm and depth = 85 μm) for the focusing and diverging modes. For the focusing mode, absorption of the incident rays over the sidewall increased almost exponentially with groove depth with little absorption at the upper part of the sidewall (Figure 8a). The absence of rays near the top of the groove was due to the difference in propagation directions of rays and the angle of sidewalls. In other words, since the incident beam was confined within a narrow angle (less than about 16° by the numerical aperture (NA) of the objective lens, NA = 0.14) that was smaller than the angle formed by the groove walls (about 20° for the intermediate stage groove of consideration), most of the rays directly hit the lower portion of the groove wall, as shown in Figure 8b. Accordingly, the groove width at the surface did not change noticeably, whereas the bottom portion of the groove was continuously etched away, forming a deeper groove. On the contrary, the diverging mode showed an entirely different tendency. Unlike the focusing mode, relatively low intensity with a maximum value of about 1.3 W/mm 2 was distributed almost over the entire sidewall of the intermediate stage groove (width = 100 μm and depth = 85 μm), as shown in Figure 8c. This was possibly due to a characteristic of the diverging beam delivered through the optical fiber. In other words, since the On the contrary, the diverging mode showed an entirely different tendency. Unlike the focusing mode, relatively low intensity with a maximum value of about 1.3 W/mm 2 was distributed almost over the entire sidewall of the intermediate stage groove (width = 100 µm and depth = 85 µm), as shown in Figure 8c. This was possibly due to a characteristic of the diverging beam delivered through the optical fiber. In other words, since the size of the incident beam was larger than the groove width at the surface, the rays could irradiate from the top to the bottom of the sidewall, as shown in Figure 8d.
The intensity distributions at the bottom of intermediate-stage grooves for the focusing and diverging modes are presented in Figure 9. As expected, in the case of focusing mode, Figure 9a, most of the incident rays directly propagated to the bottom of the groove, possibly inducing a strong etching reaction near the bottom portion and leading to the formation of a high-aspect-ratio groove.
Micromachines 2021, 12, x 10 of 14 size of the incident beam was larger than the groove width at the surface, the rays could irradiate from the top to the bottom of the sidewall, as shown in Figure 8d.
The intensity distributions at the bottom of intermediate-stage grooves for the focusing and diverging modes are presented in Figure 9. As expected, in the case of focusing mode, Figure 9a, most of the incident rays directly propagated to the bottom of the groove, possibly inducing a strong etching reaction near the bottom portion and leading to the formation of a high-aspect-ratio groove.
For the diverging mode, a very weak concentration of incident rays at the center of the groove bottom occurred. Moreover, the laser beam intensity at the bottom of the groove (see Figure 9c) was almost the same as that at the sidewall of Figure 8c. Therefore, it was expected that the sidewall and bottom could be etched away at a similar rate, maintaining the original triangular cross-sectional profile.  Figure 10a,b present the intensity distributions at the sidewall of a fully developed microgroove with the focusing mode. The modeled width and depth of the microgrooves were 50 μm and 300 μm, respectively, and the calculated groove angle was about 10°.
The intensity distribution over the sidewall of the fully developed groove in Figure 10a showed two distinctive peaks, namely, the "first reflection" and "second reflection" For the diverging mode, a very weak concentration of incident rays at the center of the groove bottom occurred. Moreover, the laser beam intensity at the bottom of the groove (see Figure 9c) was almost the same as that at the sidewall of Figure 8c. Therefore, it was expected that the sidewall and bottom could be etched away at a similar rate, maintaining the original triangular cross-sectional profile. Figure 10a,b present the intensity distributions at the sidewall of a fully developed microgroove with the focusing mode. The modeled width and depth of the microgrooves were 50 µm and 300 µm, respectively, and the calculated groove angle was about 10 • . groove, allowing the fabrication of a U-shaped groove for the low etchant concentration of 10% if a focused laser beam is used. For the case of the relatively higher etchant concentration of 40%, microbubbles and etch byproducts cannot easily escape from the groove due to the increased viscosity and, in this case, the laser beam may be scattered by the bubbles or insoluble byproducts, losing power before the second reflection; accordingly, etching of the sidewall near the bottom portion cannot occur, which ultimately results in the formation of a groove. For the diverging mode, the laser intensity over the sidewall (see Figure 10c) and at the bottom of the groove (see Figure 11b) for the fully developed groove appeared too low to induce thermochemical reaction. To verify this, a numerical calculation of the surface temperature variation was carried out with the maximum laser intensity on the sidewall (0.6 W/mm 2 ), and the results are presented in Figure 12. For this intensity, the computed The intensity distribution over the sidewall of the fully developed groove in Figure 10a showed two distinctive peaks, namely, the "first reflection" and "second reflection" peaks, both being sufficiently high to induce the etching reaction. Between these two reflection peaks, the more critical one in the determination of groove shape could be the second reflection peak. As demonstrated earlier in Section 3.2, thanks to the low viscosity at the relatively low etchant concentration of 10%, the removal of etch byproduct and supply of fresh etchant can occur relatively easily. Therefore, the thermochemical reaction by the first and second reflection peaks can be induced at the sidewall near the bottom of the groove, allowing the fabrication of a U-shaped groove for the low etchant concentration of 10% if a focused laser beam is used. For the case of the relatively higher etchant concentration of 40%, microbubbles and etch byproducts cannot easily escape from the groove due to the increased viscosity and, in this case, the laser beam may be scattered by the bubbles or insoluble byproducts, losing power before the second reflection; accordingly, etching of the sidewall near the bottom portion cannot occur, which ultimately results in the formation of a groove.
For the diverging mode, the laser intensity over the sidewall (see Figure 10c) and at the bottom of the groove (see Figure 11b) for the fully developed groove appeared too low to induce thermochemical reaction. To verify this, a numerical calculation of the surface temperature variation was carried out with the maximum laser intensity on the sidewall (0.6 W/mm 2 ), and the results are presented in Figure 12. For this intensity, the computed maximum temperature rise at the sidewall was about 349 K, which is far less than the etch threshold temperature of 360 K for a stationary laser beam. Therefore, etching of the sidewall and bottom of the groove could not occur any further for the fully developed groove. According to these results for ray tracing, it could be verified that the fabrication of both U-shaped and V-shaped grooves was feasible for the focusing mode by appropriately adjusting the etchant concentration, whereas only a V-groove could be formed in the diverging mode regardless of the etchant concentration.
Micromachines 2021, 12, x 12 of 14 maximum temperature rise at the sidewall was about 349 K, which is far less than the etch threshold temperature of 360 K for a stationary laser beam. Therefore, etching of the sidewall and bottom of the groove could not occur any further for the fully developed groove. According to these results for ray tracing, it could be verified that the fabrication of both U-shaped and V-shaped grooves was feasible for the focusing mode by appropriately adjusting the etchant concentration, whereas only a V-groove could be formed in the diverging mode regardless of the etchant concentration.

Conclusions
The influence of laser irradiation modes such as focused laser beam and diverging laser beam on the cross-sectional shape of a microgroove was investigated using ray tracing for three groove geometries, i.e., an initial flat surface with no groove, a developing groove of intermediate depth, and a fully developed V-groove.
Regardless of the laser irradiation modes, etching of the workpiece is expected in case of the flat surface with no groove due to the initial high laser intensity. However, once the groove is formed, an obvious difference appears between the focusing mode and the di- Micromachines 2021, 12, x 12 of maximum temperature rise at the sidewall was about 349 K, which is far less than the etc threshold temperature of 360 K for a stationary laser beam. Therefore, etching of the sid wall and bottom of the groove could not occur any further for the fully developed groov According to these results for ray tracing, it could be verified that the fabrication of bot U-shaped and V-shaped grooves was feasible for the focusing mode by appropriately ad justing the etchant concentration, whereas only a V-groove could be formed in the diverg ing mode regardless of the etchant concentration.

Conclusions
The influence of laser irradiation modes such as focused laser beam and divergin laser beam on the cross-sectional shape of a microgroove was investigated using ray tra ing for three groove geometries, i.e., an initial flat surface with no groove, a developin groove of intermediate depth, and a fully developed V-groove.
Regardless of the laser irradiation modes, etching of the workpiece is expected in cas of the flat surface with no groove due to the initial high laser intensity. However, once th groove is formed, an obvious difference appears between the focusing mode and the d

Conclusions
The influence of laser irradiation modes such as focused laser beam and diverging laser beam on the cross-sectional shape of a microgroove was investigated using ray tracing for three groove geometries, i.e., an initial flat surface with no groove, a developing groove of intermediate depth, and a fully developed V-groove.
Regardless of the laser irradiation modes, etching of the workpiece is expected in case of the flat surface with no groove due to the initial high laser intensity. However, once the groove is formed, an obvious difference appears between the focusing mode and the diverging mode and, thus, the probable cross-sectional profiles of a groove are noticeably distinguished. For focusing mode related to the typical lens-based LIE, the high first and second reflected laser intensities could sufficiently induce a thermochemical reaction. Above all, the second reflected laser intensity has a deeper relationship with the cross-sectional profile and, according to the etchant concentration, a U-shaped groove or a V-shaped groove can be produced. On the other hand, for diverging mode related to the optical fiber LIE, since the laser intensity distributed on the sidewall and bottom portion is too weak, no further etching occurs and the V-groove is maintained. According to this study, it was found that the ray tracing results enable prediction of the shape of the microgroove that can be manufactured through LIE.
It is believed that the LIE could be a practical micromachining technique for the manufacture of microthermal and microfluidic devices such as a micro heat pipe, micro heat spreader, and micro reactor, based on metallic microgrooves.