Numerical Simulation and Experimental Investigation of Single-Point Picosecond Laser Ablation inside K9 Glass

Abstract

K9 glass is a classical transparent material widely used in high-power optical systems due to its high-temperature resistance. However, the precision machining of K9 glass is difficult. The laser processing method, characterized by being non-contact, having a small heat-affected zone, and having high processing precision, is commonly employed for processing intricate structures. In this study, the vector diffraction model is employed to simulate the internal electric field inside the material when focused by objective lenses with varying numerical apertures. Furthermore, the temperature field is simulated. The simulation considered the nonlinear absorption of the material, the stretching of the focal dot due to spherical aberration, and the energy loss of the laser during the focusing process. The experiment indicated that the ablated area consists of numerous small, ablated dots and that multiple ablated areas emerged under an NA of 0.6. This study can provide valuable references for the research of the interaction between lasers and glass materials.

1. Introduction

K9 (BK7) glass belongs to the family of optical borosilicate glasses and is an inorganic transparent material with main components including SiO₂, B₂O₃, Na₂O, and others [1]. K9 glass is distinguished by its exceptional transmittance of the visible and near-infrared spectra, remarkable chemical stability, low content of internal impurities and bubbles, and a high softening point of 719 °C [2]. These advantageous characteristics render it a prevalent optical glass extensively utilized in various optical systems, particularly in high-power laser systems [3]. Traditionally, K9 glass has been processed using milling methods [4]. However, milling is not only inefficient and challenging for processing intricate structures but it often leaves significant damage and sub-damage structures on the glass surface, requiring further removal through coarse and fine grinding, leading to high wastage rates [5]. Using a fixed abrasive polishing pad (FAP) [6] is a method for the ultra-precision machining of K9 glass. This technique processes K9 glass using grinding and polishing pads with varying particle sizes and types. The resultant products exhibit high precision, low damage, and superior surface quality. However, this method suffers from low processing efficiency and high costs, making it unsuitable for mass production.

The laser processing method utilizes short-pulse lasers to focus on the material processing area. The laser energy is strongly absorbed by the material in the focused area, generating a large number of free electrons and forming a plasma. The coupling between free electrons and the crystal lattice causes the lattice to heat up. When the density of free electrons exceeds the threshold for establishing a plasma, the material undergoes irreversible ablation damage, forming an ablated area and achieving the processing goal [7]. Short-pulse laser processing offers several advantages, including non-contact operation, a minimal heat-affected zone, and high precision. Laser processing can significantly enhance both the quality and efficiency of machining, improve material utilization, and reduce waste. As a result, it stands out among the various machining methods. In 2009, Stephen Lee et al. [8] used a 355 nm wavelength, 10 ps pulse width laser pulse to process the glass surface, with a laser focal dot diameter of only 14 μm, successfully creating micropores with a diameter of 100 μm on the glass surface.

Laser processing research is the study of material modification processes and the properties of the ablated material area. Currently, it is widely believed that the modification process begins with the rapid absorption of laser energy through a nonlinear excitation mechanism. The energy is then dissipated into the lattice, leading to changes inside the glass [9]. Consequently, extensive research is dedicated to explaining this complex nonlinear absorption mechanism. Shang et al. [10] suggest that the absorption of laser energy is primarily utilized in the material ionization process, with avalanche ionization predominating when materials are irradiated with long-pulsed lasers and multiphoton ionization predominating with short-pulsed lasers. Miyamoto et al. [11] established a thermal conduction model, simulating the nonlinear absorption of ultra-short laser pulses in borosilicate glass modification. The absorption rate increases with the increase in laser energy and laser pulse rate, reaching up to 90%. This is significant for the subsequent study of the temperature field formed inside the glass by ultra-short lasers. Furthermore, Miyamoto et al. [12] developed a simulation model of ultra-short pulse laser modification inside borosilicate glass that can determine the intensity distribution of laser energy absorption, nonlinear absorption rate, and temperature distribution inside the glass under different pulse repetition rates and pulse energies. The results indicate that the ablated area consists of an interior structure resembling droplets and an exterior structure resembling ellipses, corresponding to regions of intense laser absorption and thermally affected melting areas, respectively. Sun [13], through numerical analysis, investigated the roles of thermal ionization and electronic damage in borosilicate glass modification under conditions of high-repetition-rate picosecond laser pulses. It was found that the droplet-shaped interior structure corresponds to the damage regions caused by high-density free electrons. Their work elucidates the laser damage mechanism and nonlinear absorption mechanism of K9 glass, revealing the significant relationship between laser absorption and free electron ionization. However, these models simplify the complex transmission process of lasers inside material by reducing it to a thermal source with specific energy densities, which is typically represented as either surface heat inputs applied to the material boundary or a volumetric heat source embedded inside the material. Additionally, the influence of spherical aberration on laser focusing is significant, often causing stretching of the laser focal area. In this study, a picosecond pulse laser with a wavelength of 532 nm was utilized to perform single-point ablation inside K9 glass. The vector diffraction theory was employed to calculate the internal electric field of the material and subsequently derive the temperature field inside the material. This model comprehensively considers the electromagnetic transmission process of the laser inside the material, the nonlinear absorption effects of the material, the stretching effect of spherical aberration on the focused dot, and the energy loss during the focusing process. The relationship between the morphology dimensions of the ablated area and the laser pulse energy, as well as the depth of laser focusing, was systematically analyzed. Additionally, experimental investigations were conducted on K9 glass, and the results were fitted and analyzed in comparison with the numerical model.

2. Numerical Simulation

2.1. Vector Diffraction Model for a Focused Laser

The laser used in this study is a picosecond laser with a wavelength of 532 nm, and the laser is considered to follow a Gaussian distribution in both the time and spatial domains [14]. The laser pulse width is 10 ps, and the material lattice is rapidly heated on the picosecond time scale, with thermal diffusion processes much slower than this, resulting in minimal thermal impact on non-focused areas [15].

When calculating the electric field inside the material, the vector diffraction integral proposed by Richards and Wolf [16] is mainly relied on, and the theoretical model proposed by Török [17] is also employed. The schematic diagram of laser transmission in air and inside the material after being focused by the objective lens is shown in Figure 1. The refractive index of air is denoted as n₁, while the refractive index of the material is denoted as n₂. After the laser is refracted at the material surface, it is focused at the Gaussian focus inside the material. The original focal position of the laser with respect to the material surface is at a distance of d_nom, and the Gaussian focus is at a distance of d from the material. The Cartesian coordinate system is established with the Gaussian focus as the origin. ${\widehat{\mathit{s}}}_1$, ${\widehat{\mathit{s}}}_2$ are unit vectors aligned with the direction of the incident light and the refracted light, respectively, and ${\mathit{r}}_p={\widehat{\mathit{e}}}_{x}x+{\widehat{\mathit{e}}}_{y}y+{\widehat{\mathit{e}}}_{z}z$ is the vector pointing from the origin o to the field point P.

According to the vector diffraction theory, the electric field of the laser on the outer side of the material surface is represented as:

$${\mathit{E}}_1\left(x,y,-d\right)=-{\displaystyle \frac{i{k}_1}{2\pi }}{\iint }_{\Omega }{\displaystyle \frac{\mathit{a}\left(s_{1x},s_{1y}\right)}{s_{1z}}}\mathit{exp}\left[i{k}_1\left(s_{1x}x+s_{1y}y-s_{1z}{d}_{nom}\right)\right]d{s}_{1x}d{s}_{1y}$$

(1)

where $k_1={\displaystyle \frac{2\pi n_1}{\lambda }}$ is the wave number of the laser in air, Ω is the total solid angle formed by all geometric rays outside the pupil, $\mathit{a}\left(s_{1x},s_{1y}\right)$ is the complex amplitude of the electric field inside the pupil, and Equation (1) is the sum of plane waves. The electric field inside the material is still assumed to be in the form of a sum of plane waves, and the transformation from incident rays to refracted rays follows Snell’s law of refraction. Therefore, the electric field inside the material surface is represented as:

$$\begin{gathered} {\mathit{E}}_2\left(x,y,-d\right)=-{\displaystyle \frac{i{k}_1}{2\pi }}{\iint }_{\Omega }{\displaystyle \frac{\mathit{T}·\mathit{a}\left(s_{1x},s_{1y}\right)}{s_{1z}}}\mathit{exp}\left[i{k}_1\left(s_{1x}x+s_{1y}y \\ -s_{1z}{d}_{nom}\right)\right]d{s}_{1x}d{s}_{1y} \end{gathered}$$

(2)

where the operator $\mathit{T}$ represents the transformation from the incident wave to the refracted wave, and it is a function of the angle of incidence and the refractive indices $n_1$ and $n_2$. The electric field at point P for the laser can be presented as:

$${\mathit{E}}_2\left({\mathit{r}}_p\right)=-{\displaystyle \frac{i{k}_2}{2\pi }}{\iint }_{\Omega }\mathit{F}\left({\widehat{\mathit{s}}}_2\right)\mathit{exp}(i{k}_2{\widehat{\mathit{s}}}_2·{\mathit{r}}_p)\times {\mathit{J}}_0(s_{1x},s_{1y};s_{2x},s_{2y})d{s}_{1x}d{s}_{1y}$$

(3)

where the term ${\mathit{J}}_0\left(s_{1x},s_{1y};s_{2x},s_{2y}\right)={\left({\displaystyle \frac{k_1}{k_2}}\right)}^2$ is the Jacobian matrix introduced through coordinate transformation. By using Equation (2) as the boundary condition for Equation (3), $\mathit{F}\left({\widehat{\mathit{s}}}_2\right)$ can be expressed as:

$$\mathit{F}\left({\widehat{\mathit{s}}}_2\right)=\left({\displaystyle \frac{k_2}{k_1}}\right){\displaystyle \frac{\mathit{T}·\mathit{a}\left(s_{1x},s_{1y}\right)}{s_{1z}}}\mathit{exp}\left(i\Psi \right)$$

(4)

where the term $\Psi =d_{nom}\left(k_2{s}_{2z}-k_1{s}_{1z}\right)$. Substitute Equation (4) into Equation (3), expand the exponential terms in Equation (3), and utilize the vector of Snell’s law $k_1{s}_{1x}=k_2{s}_{2x}$, $k_1{s}_{1y}=k_2{s}_{2y}$. Arrange to derive:

$${\mathit{E}}_2\left(x,y,z\right)=\begin{array}{l}-{\displaystyle \frac{i}{2\pi k_1}}{\iint }_{\Omega }{\displaystyle \frac{\mathit{T}·\mathit{a}\left(s_{1x},s_{1y}\right)}{cos{\theta }_1}}\mathit{exp}\left(i\Psi \right)\\ \times \mathit{exp}\left(i{k}_{2z}z\right)\mathit{exp}\left[-i\left(k_{1x}x+k_{1y}y\right)\right]d{k}_{1x}d{k}_{1y}\end{array}$$

(5)

Li et al. [18] rewrote Equation (5) in a more concise form of Fourier transform.

$${\mathit{E}}_2\left(x,y,z\right)=-{\displaystyle \frac{i}{2\pi k_1}}F\left\{{\displaystyle \frac{\mathit{T}·\mathit{a}\left(s_{1x},s_{1y}\right)}{cos{\theta }_1}}\mathit{exp}\left(i\Psi \right)\mathit{exp}\left(i{k}_{2z}z\right)\right\}$$

(6)

The term $\mathit{T}·\mathit{a}\left(s_{1x},s_{1y}\right)$, where $\mathit{T}$ is the refraction operator, is determined by the Fresnel law, and $\mathit{a}\left(s_{1x},s_{1y}\right)$ is the complex amplitude of the electric field inside the pupil. The incident laser is all assumed as linearly polarized light in the x-direction. In this case, $\mathit{T}·\mathit{a}\left(s_{1x},s_{1y}\right)$ can be written as:

$$\mathit{T}·\mathit{a}\left(s_{1x},s_{1y}\right)=f\sqrt{cos{\theta }_1}{E}_1\left[\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}\right]$$

(7)

where f is the focal length of the objective lens, $E_1$ is the complex amplitude of the incident laser electric field, and the values in the matrix ${\left[\begin{array}{ccc}{a}_1& a_2& a_3\end{array}\right]}^T$ are:

$$\begin{gathered} a_1={\tau }_p{cos}^2\phi cos{\theta }_2+{\tau }_s{sin}^2\phi \\ {a}_1={\tau }_p{cos}^2\phi cos{\theta }_2+{\tau }_s{sin}^2\phi \\ {a}_3={\tau }_{p}cos\phi sin{\theta }_2 \end{gathered}$$

(8)

Thus, the internal electric field model inside the material has been constructed. In the optical frequency range, the magnetization mechanism of the medium is nearly frozen and the relative permeability ${\mu }_r\approx 1$. Therefore, the laser intensity can be obtained using the relationship $I\left(x,y,z\right)={\displaystyle \frac{n_2{\epsilon }_{0}c}{2}}{\left|{\mathit{E}}_2(x,y,z-d)\right|}^2$ (the coordinate origin is chosen at the surface of the material).

2.2. Temperature Field Model

K9 glass absorbs laser energy and rapidly heats up, melting, vaporizing, and even undergoing chemical reactions. Therefore, the critical temperature for glass modification is set to its softening point (719 °C) [19]. According to Fourier’s heat conduction law [20], the three-dimensional heat transfer equation for temperature T is:

$$k{\nabla }^{2}T+w=\rho c{\displaystyle \frac{\partial T}{\partial t}}$$

(9)

where $k$ represents the thermal conductivity of the glass, $w$ is the volumetric heat power density, which refers to the heat power absorbed per unit volume of the glass, $\rho$ denotes the density of K9 glass, and c is the specific heat capacity of the glass.

During laser processing, in addition to convective heat transfer between the glass surface and the surrounding environment, radiative heat transfer also needs to be considered. Thus, the boundary conditions include:

$$-k{\displaystyle \frac{\partial T}{\partial \overrightarrow{n}}}{|}_{\Omega }=h\left(T_0-T\right)+\sigma \epsilon \left(T_0^4-T^4\right)$$

(10)

where $T_0$ represents the environment temperature, $k$ is the thermal conductivity, $\sigma$ is the Stefan–Boltzmann constant, $\epsilon$ is the emissivity of the glass surface, $h$ is the convective heat transfer coefficient, and $\overrightarrow{n}$ is the unit vector in the direction normal to the glass surface.

Before laser heating, it is assumed that the glass temperature is equal to the environment temperature. Therefore, the initial condition is:

$$T{|}_{t=0}=T_0$$

(11)

After the laser is incident on the glass, the laser intensity will continuously increase during the focusing process. Nevertheless, due to the absorption of the glass material, the total intensity of the laser will decrease. The absorption coefficient of the glass is denoted as $\alpha$, and the focal depth of the laser is denoted as $d$. According to Lambert–Beer’s law [21], when ${\displaystyle \frac{1}{\alpha }}\gg d$, it can be considered that the loss caused by material absorption inside the focal depth range is very low. Therefore, the laser heat source can be approximated as:

$$w\left(x,y,z\right)={\displaystyle \frac{dI}{dz}}=\left(1-R\right)\alpha I\left(x,y,z\right)\exp \left(-\alpha z\right)$$

(12)

where $R$ represents the Fresnel reflectance at the glass surface. It is worth noting that the absorption coefficient $\alpha$ is temperature dependent. This is because under the action of a short-pulse laser, free electrons can be heavily ionized through processes such as multiphoton or avalanche ionization, leading to nonlinear effects in the absorption of laser energy. As a result, the absorption coefficient $\alpha$ is related to the laser energy and repetition rate [11].

3. Materials and Methods

The glass material used in the experiment is K9 (BK7) glass (produced by SCHOTT) with dimensions of 10 mm × 10 mm × 2 mm, and the relevant characteristic parameters are shown in

Table 1. Table of the characteristic parameters for K9 glass.

Characteristic Parameters	Parameter Values
Refractive index (n2)	1.5168
Density ($\rho$)	2.51 g/cm3
Thermal conductivity coefficient ($k$)	1.114 W/m·°C
Specific heat capacity (c)	858 J/kg·°C
Softening temperature	719 °C
Absorption coefficient (20 °C) ($\alpha$)	0.1632/m

. A 532 nm picosecond laser (sourced from COHERENT) with a pulse width of 10 ps was employed as the laser source. Two different focusing objectives were utilized: 20× M PLAN APO NIR, with a numerical aperture (NA) of 0.4 and a focal length (F) of 4 mm, and 50× M PLAN APO NIR, with NA = 0.6 and F = 4 mm.

The schematic diagram of the experimental setup is illustrated in Figure 2. The laser, after being reflected by a mirror, enters the objective lens and is then focused into the K9 glass after passing through the lens. The K9 glass sample is placed on an electronically controlled displacement platform, which is moved using a computer-controlled platform to perform laser scanning. The laser focal position is adjusted by controlling the height of the objective lens. The laser repetition rate f_k = 4 kHz, and the laser scanning speed v = 100 mm/s.

After the processing, the morphology and cross-sectional damage of the ablated area are evaluated using an optical microscope (OM, German ZEISS Axio Scope A1).

In the experiment, the focus is on investigating the influence of laser single-pulse energy Q and focal depth d on the morphology and cross-sectional dimensions of the ablated area under different numerical apertures. The experimental parameters are distributed at different levels, as shown in

Table 2. Table of experimental parameter settings and their level distribution.

Experimental Parameter	1	2	3	4	5
Objective NA	0.4	0.6	\	\	\
Single pulse energy of laser Q/μJ	5	15	25	35	\
Laser focal position d/μm	350	550	750	950	1150

, using orthogonal experimental methods. According to geometric optics [22], under the paraxial condition, the relationship between focal depth d and the nominal laser focal position $d_{nom}$ is as follows:

$$d=n_2{d}_{nom}$$

(13)

Based on this, the laser focal position d can be calculated, and the calculated values are recorded in Table 2.

4. Results and Discussion

This section is divided by subheadings. It should provide a concise and precise description of the experimental results, their interpretation, as well as the experimental conclusions that can be drawn.

4.1. Temperature Distribution

In this section, the temperature distribution in the laser focal area was simulated using the temperature field model. The relationship between the cross-sectional diameter Ф of the ablation area and the single-pulse energy Q and focal depth d was discussed, and the differences in the ablated area under two different numerical apertures were analyzed.

When the laser enters the interior of the K9 glass, some of it is reflected, while most of it transmits into the material. By using a vector diffraction model, the electric field distribution inside the material can be calculated, which allows for obtaining the optical intensity distribution inside the material used as the volumetric heat source inside the material. The temperature field simulation is carried out using the finite-difference time-domain method. To ensure simulation accuracy, the simulation domain is divided into sufficiently small subdomains with a subdivision accuracy of 0.1 μm and a time step of 1 ps. The power–time distribution of the laser pulse is shown in Figure 3. Since the pulse width of the laser pulse τ_p = 10 ps, the moment tm when the peak power is reached is set to 10 ps, and the observation time is set to 20 ps to ensure complete absorption of the laser energy.

The temperature field distribution simulation diagrams inside the K9 glass were obtained under the conditions of an objective numerical aperture of 0.4 and a focal depth d of 350 μm as the single-pulse energy Q increased from 5 μJ to 25 μJ, as shown in Figure 4a–c. In the diagrams, the purple curves represent the isotherms at 719 °C. It can be observed that due to diffraction effects, apart from the Gaussian focus, some diffraction beams exist along the laser propagation direction and on both sides. However, as the intensity of these diffraction beams is low, the temperature in these areas is insufficient to form an ablation area and can therefore be disregarded. With the increase in single-pulse energy, the laser energy density continuously increases, leading to an increase in the cross-sectional diameter (Ф) of the ablated area, which is consistent with expectations. It is noteworthy that due to the nonlinear absorption of the material, even at a single-pulse energy of Q = 5 μJ, the temperature at the focus exceeds the softening point of material, thus forming a very small ablation area.

Figure 5a–c present the temperature field distribution simulation diagrams inside the K9 glass under the conditions of a numerical aperture of 0.4 and a single-pulse energy of Q = 15 μJ as the focal depth d increases from 350 μm to 750 μm. With the increase in focal depth, the elongation of the laser dot length and the increase in transmission losses lead to a reduction in laser energy density. However, as long as the laser energy density is sufficiently high, an ablation area can still be formed. Therefore, with the increase in focal depth, the length of the ablation area also increases.

The temperature field under a larger numerical aperture differs significantly from that under a lower numerical aperture. Figure 6a–c show the simulated temperature distribution in K9 glass under the condition of an objective numerical aperture of 0.6 and a focal depth of d = 350 μm as the single-pulse energy Q increases from 5 μJ to 25 μJ. It is evident that in the focal area, in addition to a dot at the Gaussian focus, there is a series of dots with gradually decreasing light intensity along the direction of laser propagation. Under the condition of higher single-pulse energy, the light intensity of these dots is high enough to form several ablation areas along the direction of laser propagation. The increase in single-pulse laser energy will enhance the laser energy density, resulting in the expansion of the ablation area.

Figure 7a–c present the simulated temperature field distribution in K9 glass under the conditions of an objective numerical aperture of 0.6 and a single-pulse energy Q of 15 μJ as the focal depth d increases from 350 μm to 750 μm. When the focal depth is low, the adjacent ablation areas are almost connected together, as shown in Figure 7a. As the focal depth increases, there are cases where the ablation areas become separated. The largest ablation area is designated as the first ablation area, and the adjacent ablation area is designated as the second ablated area. The distance between the surfaces of the two ablation areas is denoted as Δ_i, as shown in Figure 7, and different Δ_i values under different focal depths d are recorded for experimental analysis.

4.2. Experimental Analysis

In the experiment, a single-pulse laser was used to scan and ablate the K9 glass. Because K9 glass exhibited maximum transmittance at 532 nm, this wavelength is used to completed experiments. Most of the laser energy was nonlinearly absorbed in the focal area, generating a large number of free electrons due to the coupling effect between the free electrons and the lattice, leading to lattice heating. Once the temperature reached the damage threshold, irreversible changes occurred, forming the ablated area. The laser repetition rate used in the experiment was f_k = 4 kHz, and the scanning speed was v = 100 mm/s. The morphology and dimensions of the ablated area’s longitudinal section were observed and recorded using an optical microscope.

The longitudinal section of the ablation area is shown in Figure 8a–d. The objective numerical aperture was 0.4, and the focal depth was 356.73 μm. The single-pulse energy gradually increased from 5 μJ. The laser propagated from the top to the bottom of the longitudinal section. The black area in the figure represents the area where the material was completely ablated, making it opaque to visible light. A melted zone formed around the ablated area, appearing as a transparent area with a clear boundary with the surrounding material. This was caused by the diffusion of the absorbed energy, raising the temperature above the melting point of the material. Due to the nonlinear absorption of the laser, even at a single-pulse energy of Q = 5 μJ, a small continuous ablated area was formed in the focal area. Observing the longitudinal sections of the ablated area at Q = 15 μJ and 25 μJ, it was found that the ablated area formed at the Gaussian focus consisted of several closely spaced ablated dots. Additionally, a series of discrete ablated dots formed along the direction of laser propagation outside the Gaussian focus. This phenomenon was speculated to be caused by the nonlinear refractive index of material and laser self-focusing. Figure 8e was used to illustrate this phenomenon. As the laser focused, the refractive index of the material changed due to the nonlinear interaction with the laser. The refractive index increased with higher light intensity, resulting in spherical distortion of the phase front of the laser and the formation of multiple foci along the optical path after refocusing. The critical laser power threshold for self-focusing was expressed as $P_{cr}={\displaystyle \frac{3.77{\lambda }^2}{8\pi n_2{n}_a}}$ [23], where λ is the laser wavelength, n₂ is the refractive index of K9 glass, and n_a is the nonlinear refractive index coefficient of K9 glass. The calculated value of P_cr was 1.2 MW. In the experiment, when the single-pulse energy was Q = 5 μJ with 10 ps pulse width, the peak power was 0.5 MW, which was lower than the critical power threshold. Thus, under the condition of Q = 5 μJ, a continuous ablated area was formed at the Gaussian focus. Under the conditions of Q = 15 μJ and 25 μJ, several closely spaced ablated dots were formed at the Gaussian focus and a series of discrete ablated dots were formed outside the Gaussian focus. When the single-pulse energy increased to 35 μJ, the morphology of the ablated area changed again, with only one very large ablated dot formed at the Gaussian focus and a series of discrete ablated dots formed outside the Gaussian focus. This was because the power density at the Gaussian focus was too high, leading to the formation of an ablated area due to the self-focusing process, resulting in a very large ablated dot at the Gaussian focus.

It is worth noting that in Figure 8d, multiple cracks can be clearly seen on the outer side of the melted zone. The formation of these cracks is attributed to the stress difference generated during the cooling process in the focal area. Near the focal point of the laser, the material is rapidly heated. Upon the completion of the laser, the focal area rapidly cools and solidifies in a slightly larger volume state than before. During the solidification process, the outer region of the focal area solidifies first and cannot shrink, essentially maintaining a tensile state. Therefore, residual tensile stress is presumed to exist inside the ablated area, while residual compressive stress exists externally. This stress difference leads to the generation and propagation of cracks inside the ablated area.

To observe the specific morphology of ablated dots, the morphology characteristics of the ablated dots were characterized using a scanning electron microscope (SEM). As shown in Figure 9a, with an objective numerical aperture of 0.4, a single-pulse energy Q of 25 μJ, and a focal depth of 1167.9 μm, the morphology of the ablated area longitudinal section obtained under an optical microscope is depicted. Figure 9b shows the morphology obtained via SEM of the area inside the red box in Figure 9a. From the SEM images, it can be observed that the ablated area generally appears as an irregular elongated ellipsoidal shape with internal voids. Multiple ellipsoidal voids are stacked together to form a larger ablated area.

Elemental mapping analysis (EDS) was performed on this area, with photos of the test area and distribution maps of carbon, oxygen, and silicon shown, respectively, in Figure 10a–d. It is evident from the images that the oxygen content in the ablated area is significantly reduced, while the carbon and silicon elements show minimal change. Due to the material’s nonlinear absorption of the laser, the focused area experiences intense heating. When the temperature exceeds the material’s boiling point, vaporization occurs. At excessively high temperatures, silicon dioxide material in the focused area ionizes to form plasma [24], generating O₂, CO₂, and mixtures of Si and SiO₂ [25,26]. Therefore, voids are formed in the ablated area, accompanied by a significant decrease in oxygen content.

The longitudinal section morphology of the ablated area is shown in Figure 11, with an objective numerical aperture of 0.6 and a single-pulse energy of Q = 5 μJ. The focal depth at each point is marked in the figure after measurement. It can be observed that there are two ablated areas at each focal area, separated by a certain distance. Due to the laser’s self-focusing effect, both the first and second ablated areas above and below contain a series of ablated dots. As the laser propagates, it is continuously absorbed until it cannot cause self-focus anymore, resulting in a continuous ablated area at the end of the second ablated area. From the experimental results, it can be observed that there are certain differences in the morphology of the ablation areas when NA = 0.4 and NA = 0.6. Compared to the ablation areas at NA = 0.4, those at NA = 0.6 exhibit a longer axial length but smaller lateral dimensions. Additionally, no cracks were observed near the ablation area at NA = 0.6. Both sets of ablation areas show a series of ablation spots within their interiors.

The distance between the upper vertex of the two ablated areas is denoted as Δ_act, and the Δ_act values at different focal depths are recorded and plotted in Figure 12. It can be seen that the distance between the upper surfaces of the two ablated areas is consistent with the numerical simulation model. This distance increases with the increase in focal depth, and the lengths of the two ablated areas also increase. As the laser focal position increases, the ablated area also enlarges, indicating that the laser energy is dispersed over a larger space. When the laser energy is relatively high, the energy density in the focal area remains above the material damage threshold, resulting in an increase in the length of the ablated area. However, when the focal depth reaches a point where the energy density is lower than the material damage threshold, no further damage occurs, and the ablated area may even contract.

5. Conclusions

This study utilizes a vector diffraction model to calculate the electric field inside K9 glass and simulates the temperature field inside K9 glass through the finite-difference time-domain method. It comprehensively considers the nonlinear absorption of the material, the stretching effect of spherical aberration on the focal area, and the energy loss of the laser during the focusing process. Reasonable simulation results are obtained. The conclusions drawn from the experiments are summarized as follows:

(1)

During the laser focusing process, due to the difference in refractive index between the material and air, a series of diffraction beams will form behind the Gaussian focus. Compared with the Gaussian dot, these diffraction beams have low energy under low numerical apertures and cannot ablate the material. However, under high numerical apertures, these diffraction beams have relatively high energy, causing damage to the material. These diffraction beams weaken the optical intensity at the Gaussian focus and form multiple ablated areas.

(2)

In the experiment, it was found that multiple ablated dots were formed at the Gaussian focus, and a series of ablated dots were formed along the laser propagation direction outside the Gaussian focus, this phenomenon was speculated to be caused by the nonlinear refractive index of material and laser self-focusing. When the laser peak power exceeds the self-focusing critical power threshold, multiple ablated dots will form at the Gaussian focus, and when the laser peak power continues to increase and the energy density at the defocusing point exceeds the material damage threshold, adjacent ablated dots will merge, forming a larger ablated dot at the Gaussian focus.

(3)

As the focal depth increases, the laser focal area will expand. Under high laser energy, this expansion will cause the ablated area to expand first and then contract with increasing focal depth. However, under low laser energy, this expansion will cause the laser energy density to be lower than the material damage threshold, resulting in the contraction of the ablated area.