Melt pool size of optical glass irradiated by semiconductor laser

ABSTRACT Using a continuous laser to soften glass locally is a novel way of selective glass forming. Due to gaussian distribution of laser power density, a temperature gradient will be generated on glass treated with localized irradiation by laser, which brings about the coexistence of a glass state and a viscoelastic state. This situation creates a bulge on the surface of the softened area. Based on this phenomenon, we propose a measurement approach and a calculation model for the softening zone width of glass irradiated locally by laser and verify the calculation model through simulations and experiments. The maximum temperature deviation of the proposed temperature field prediction model is 4.7°C, while the minimum deviation is 0.53°C; making the deviation between the calculation model of the softening zone width and the experimental measurement result less than 0.319%.


Introduction
Laser processing of brittle materials, especially glass materials, has received increasing attention due to recently its advantages of non-contact heating, selected area heating, and simple processing with laser as a heat source [1]. Under laser irradiation, the glass exhibits a certain degree of fluidity and plasticity when it reaches temperatures above the softening point locally, making it easy to process and mold [2]. Various optical surfaces can be processed in the softened area using the rheoforming method, these providing a novel way of selective glass forming. The temperature field and the size of the softening zone are therefore essential parameters comprising primary physical characteristics of the laser-glass interaction, which have a direct impact on the whole physical process of subsequent glass processing and forming. Study of the temperature field and softening zone is of great importance for laser processing of glass.
Most previous research on laser-glass interactions applies the instantaneous effect of a short or ultrashort pulse laser to materials [3][4][5][6], such as the rapid forming and removal of glass [7][8][9]. Besides, continuous laser forming research mostly uses a CO 2 laser [10][11][12], which the glass can readily absorb, since the vibration band of SiO 2 is 9-11 μm. By contrast, semiconductor lasers produce different effects on glass due to the glass material's weak laser absorption. The semiconductor laser has a wide range of effects on glass, equivalent to those of a body heat source [13]. Heat conduction plays a vital role in glass cooling during strong absorption, while radiation and convection are the main factors during weak absorption [14]. Weak absorption easily widens and deepens the liquid pool on the glass, making it more beneficial to the subsequent processing and formation. Therefore, this study employs a semiconductor laser with a wavelength of 1060 nm as a heat source to achieve selective laser formation by making a stable softening zone in a local region of the optical mirror.
In this research, by comparing the results of the temperature field simulations and the multi-point temperature measurement experiments, we verified the accuracy of the glass temperature field prediction model under laser irradiation. We then compared the temperature field distribution data obtained from the prediction model and the width data for the bulge formed on the irradiated glass surface to demonstrate that the bulge width can measure the width of the softening zone. By analyzing data gained from many experiments and simulations, moreover, we produced a calculation model of the softening zone width using the data-fitting method. This model can accurately calculate the softening zone width of laser-irradiated optical glass.

The basic equation of transient heat conduction
As shown in Figure 1, a continuous semiconductor laser beam with a certain amount of defocus is applied to heat the soda-lime glass sample continuously by shooting the beam perpendicularly onto the surface. Assuming that the glass material is isotropic, the temperature field T inside the sample can be described by the heat conduction equation [15], shown in Equation (1). where ρ, c, and k represent the glass sample's density, specific heat, and heat transfer coefficient, respectively. Ñ is the gradient operator, t is the time, R is the reflectance of the material surface to the laser, α is the absorption coefficient, I 0 is the power density (light intensity) of the incident laser beam with x = 0, and Q denotes other heat sources.
where t = 0 is the moment when the laser begins to act, the initial temperature distribution is shown in Equation (2). In the process of laser irradiation of the glass, the glass has a uniform temperature distribution at the beginning. The following distribution function can be considered to equal the constant, i.e. the room temperature value:

Thermal boundary condition of the temperature field
The glass sample used in this study is cylindrical. Thermal energy on its upper and lower surfaces escapes to the outside through convection and radiation heat transfer. The boundary condition equation [16] expresses this phenomenon as follows: where k is the thermal conductivity of the glass, T is the temperature of the glass, T 0 is the initial temperature of the glass, n ! is the unit vector in the normal exterior direction of the glass surface, h is the coefficient of convective heat transfer, ε is the degree of blackness of the glass, and B is the Stefan-Boltzmann constant. Ceramic fiber composites are used to make a heatinsulating layer for the side surface of the cylindrical glass sample. This material has the advantages of low thermal conductivity and large specific heat capacity, which ensures the adiabatic state of the side surface. The following equation expresses the boundary condition of the heat-insulating layer:

Heat source model for the laser beam
The intensity distribution of laser beams is spatially nonlinear. Even if parameters such as the power and waist radius are the same, the different longitudinal section shapes of the beam will make a significant difference in the temperature field inside the glass, imposing challenging burdens on the forming effect [17].
This study aims to form a stable softening zone with a slight temperature gradient in a specific glass region by laser irradiation. However, the energy of an ordinary Gaussian beam is concentrated in the laser target spot. The temperature decreases sharply with increases in the distance from the center of the spot, resulting in a large temperature gradient within the range of the laser spot. It even exceeds the melting point of the glass in the local range, causing a glass ablation and limiting the formation of the softening zone.
The power density function of the flattened Gaussian [18] distributed laser and ordinary Gaussian distributed laser are shown in Figure 2. The colors in the figure range from blue to red, indicating an increase in laser intensity. Compared with ordinary Gaussian beams, the flattened Gaussian beam employed in this study brings a uniform power density to each position in the irradiation region. The slight temperature gradient inside the glass can improve the softening effect of laser irradiation and facilitate control of the softening area. The laser intensity at the center is expressed as follows: where I 0 is the power density of the laser, N F is the beam order N F � 0 ð Þ, and the spot radius r F is the coordinate value of r when the intensity decreases to the center value.

Laser-induced protrusions on glass surface
Unlike metallic materials, glass is colorless and transparent in both the glassy and viscous states, making it challenging to observe the softened areas formed during the experiment cleanly. However, after the laserirradiated glass sample was cooled, two spherical bulges appeared on the upper and lower surfaces of the glass. Figure 3 shows the bulges on the upper and lower surfaces of the glass sample under laser irradiation.
This phenomenon occurs because of the coexistence of viscoelastic and glass states in the local region of the glass under laser irradiation. The glass has a higher density in the glass state than in the viscoelastic state and is less prone to deformation. When an extrusion force is formed at the interface of the two states, the viscoelastic glass is deformed into a curved surface protruding from the surface [19][20][21][22]. With the end of laser irradiation, the optical glass cools to retain the curved shape, forming the shape shown in Figure 3. Glass material deformation in the softening region is determined by the glass viscosity and the instantaneous shear modulus. In the research on glass laser processing, the softening temperature of the glass material is usually taken as the target temperature. In this study, the region above the softening temperature point was considered the approximate forming region, which is the softening region of the glass.
The bulge width represents the viscoelastic boundary of the glass formed by laser irradiation, that is, the softening zone width of the irradiated glass. The unsoftened area with higher density forms an extrusion force along the boundary of the columnar softened area, which squeezes the softened glass and creates a surface bulge. The bulge develops a curved shape because of the surface tension. The softening zone width of the glass irradiated with different laser parameters can therefore be obtained by using a surface profilometer to measure the width of the curved surface formed on the glass.

Materials
The material chosen for the experiment was sodalime super-white glass with compositions of 72.05% SiO 2 , 13.78% Na 2 O, 0.22% K 2 O, 8.75% CaO, 3.99%   Table 1. The thermal conductivity and specific heat capacity change linearly with the temperature, and the variation curve [23] is shown in Figure 4.
Glass has different absorption intensities with lasers of different wavelengths, and Doppler optics display the experimental sample glass's absorption rates for lasers of different wavelengths. The variation in absorption coefficients of selected experimental samples at wavelengths from 800 to 1500 nm was measured using a Doppler spectrometer, as shown in Figure 5. The absorption coefficient of the soda-lime glass was 8 m −1 at a wavelength of 1064 nm when irradiated with the semiconductor laser used in this experiment.

Numerical simulation
The temperature field was simulated by COMSOL. In the simulation of the temperature field formed in the glass sample under laser irradiation, free triangular meshes were used to divide the experimental sample. Due to the high power density during semiconductor laser irradiation, a dense mesh grid was employed in this simulation to accurately calculate the temperature field of the laser irradiation-induced softening zone and the glass distribution on the glass surface ( Figure 6).
As soda-lime glass does not have a high absorption coefficient for semiconductor lasers, the glass absorbs lasers by depositing power layer by layer.
The Beer-Lambert law is therefore used to simulate the absorption phenomenon of the laser within a glass sample. In addition to the portion of the laser that is reflected and absorbed by the optical glass, another portion transmits, and thus leaves, the optical glass. In the simulation, the surface of the material bottom was therefore set as a zero flow boundary to achieve this phenomenon, and the ambient temperature was set as 20°C. In the simulation, the computed step of the transient calculation was set to 0.1s, and its starting and stopping times were set to 0s and 300s, respectively. During the analysis, the following assumptions were used: (1) the glass material was assumed to be homogeneous and isotropic; (2) the laser power density was assumed to be uniform; and (3) the emissivity ε of the glass material was assumed to be constant and not to vary with temperature and angle [24].

Experiment research
The light source used in this experiment was an LDM semiconductor laser from the Laser Line. The laser beam was emitted by the laser and transmitted through the optical fiber to the laser head, where it was focused on the processing surface by its internal optical system. The maximum output power of the laser was 3300W. The output laser was a flat-topped Gaussian circular beam with an adjustable spot radius and a laser wavelength of 1064 ± 20 nm and its output was continuous. The laser power was set to 230W, 250W and 300W, respectively, and the laser spot radius was set to 1.5 mm.
The dimensions of the glass sample selected for the experiment were consistent with those of the simulation. To obtain accurate temperatures at various points in the glass, a precision machining center was utilized to drill three blind holes, each with a diameter of 1 mm and a depth of 3 mm, into the glass sample. Thermocouples with a 1 mm diameter (type KPS-IN600-K-1 manufactured by KAIPUSEN) were then inserted into the holes for temperature measurements.
In order to ensure that the temperature measurement points were not within the softening area of the optical glass caused by the laser irradiation, the selected temperature measurement holes were set at 11 mm, 10 mm and 9 mm from the edge of the glass sample. The distribution of the temperature measurement holes is shown in Figure 7. Thermocouples were inserted vertically into the temperature measurement holes. As thermocouples are temperature sensitive at the top, the gaps were filled with silicone grease to ensure adequate contact between the thermocouple heads and the glass material. Temperature data was collected at 1s intervals during the measurement process, and the accuracy of the thermometer in the experiment was 0.1°C.
Preheating the glass before laser processing can reduce the maximum thermal stress generated inside the glass under laser irradiation. We simulated and analyzed the distribution of internal thermal stress in glass during laser forming under different preheating temperatures to obtain the optimal preheating temperature. In the simulation, the laser power was set to   250W, the spot radius to 1.5 mm and the irradiation time to 1 min. The thermal stress distribution of the soda-lime glass was calculated under the conditions of no preheating, preheating at 200°C, preheating at 300°C and preheating at 400°C, respectively. The thermal stress results at each point on the diameter of the upper glass surface were summarized, and a thermal stress variation curve was obtained. The dashed line in Figure 8 shows the fracture strength of the soda-lime glass. As is apparent from the graph, when preheats of 300°C and 400°C are applied, the thermal stresses generated in the glass under laser irradiation were lower than the fracture strength of the material and did not cause any material damage. Therefore, the glass material was preheated in a muffle furnace at 300°C before the laser irradiation experiment.

Numerical simulation results
When setting the laser power at 250W, the spot radius at 1.5 mm and the irradiation time at 5 minutes, the temperature field distribution on the glass surface was as shown in Figure 9(a). In the laser irradiation zone, the temperature is higher and the contour line is denser than in the rest of the glass, and the maximum temperature at the center rises to 779°C (relative to the ambient temperature). Due to the high transmittance of glass, the temperature of the underside of the glass also increases significantly. Figure 9(b) is a contour map of the temperature distribution in the longitudinal section of the glass under the same conditions. It can be seen that the highest temperature reached by the glass is not on the upper or lower surfaces, but on the inside of the glass, since the heat from the upper and lower surfaces is dissipated by convective and radiative heat transfer. Figure 10 shows the softening zones of a multisection and single section of the glass. The red area represents the liquid that has softened above the softening point, and the blue area represents the solid that has not softened. After irradiation for 5 minutes, the upper surface width of the softening zone is 1.95 mm, and the lower surface width is 1.92 mm, thus showing similar results. The softening zone runs through the glass at a depth if 5 mm to form a "parenthesis" shaped softening zone in the single section of the glass.

Experimental results
To conduct further experiments, we set the laser power at 230W, 250W and 300W, respectively, the spot radius at 1.5 mm, and the irradiation time at 5 minutes. We obtained the glass samples with bulges on the upper and lower surfaces, shown in Figure 11. It can be observed that the bulge widths of the upper surfaces are similar to those of the lower surfaces. Using a surface profiler with a measurement resolution of 0.5 μm, type Contracer CV-1000, we obtained the upper surface diameter profiles shown in Figure 12.
As the top and bottom surfaces of the glass sample are horizontal, the Y-coordinate was established on the horizontal surface during the glass sample measurement. The first position with a 0.001 mm change in height from left to right on the Y-axis was identified as the edge of the surface bulge. The coordinates of this position were indicated as Coordinate point 1 in Figure 12. At the  same time, the position at which the Y-axis coordinate reverts to 0.001 mm was determined to be the other edge position of the surface bulge, indicated as Coordinate 2 in Figure 12. At this point, the absolute value of the difference between coordinate points 1 and 2 on the X-axis was the bulge widths of the glass sample under laser irradiation. The width of the bulge on the upper surface were calculated to be 0.82 mm, 1.97 mm and 3.28 mm at 230W, 250W and 300W, respectively.

Comparison between the simulation and experiment
In the simulation model, the same spot radius and irradiation time were set to obtain the softening area of the cut surface of the glass sample when the laser power was 230W, 250W and 300W (as shown in Figure 12). The widths of the softened upper surface of the glass were calculated to be 0.78 mm, 1.95 mm and 3.26 mm at 230W, 250W and 300W, respectively. The deviations from this value compared to the experimental results were 0.04 mm, 0.02 mm and 0.02 mm, respectively. It can be seen that the widths of the softened area obtained in the simulation are similar to the widths of the surface bulge obtained in the experiment, results demonstrating the effectiveness of the method of predicting the width of the softened area from the width of the surface bulge. Figures 13 gives the temperature variation curves over time at fixed points obtained from the experimental acquisition and simulation calculations. In the figures, the red, blue and green curves represent the temperature variations at the measurement points 11 mm, 10 mm and 9 mm from the edge, respectively. The comparison shows that after 5 min of the laser irradiation, the temperature deviation between the simulated results and the experimental measurements was within 5°C。Errors in the simulation study were attributed to variations in certain material parameters with temperature, which had a negligible impact on the experiment. The temperature variation curves shown for both are the same, which demonstrates the validity of the prediction results in the temperature field simulation.

Influence of laser power density on the maximum temperature of the irradiated glass
In the simulation, after setting the laser power density to 27W/mm 2 , 30W/mm 2 , 33W/mm 2 , 36W/mm 2 , 39W/mm 2 and 42W/mm 2 , respectively, and setting the spot radius to 1.5 mm, we employed cylindrical soda-calcium glass (r = 15 mm, h = 5 mm) for the simulation, and calculated the maximum temperature in the irradiated domain of the glass sample under different power densities at regular intervals (0.1 min). After processing the calculated data, the changing temperature curves were obtained, as shown in Figure 14(a). In this figure, the dotted line represents the softening temperature of the glass (730°C). According to the curves, the temperature of the glass irradiated under different laser power densities rose with increases in time, followed by decreases in the temperature gradient, which tended gradually toward stability. When the laser power density was 27W/mm 2 and 30W/mm 2 , the maximum temperature of the glass was always lower than the softening temperature so that no softening zone could be formed on it.
In addition, with increases in the irradiation time, the variations in the softening zone width decreased and gradually became stable. This is due to the decrease in the temperature rise during irradiation and the glass's low thermal conductivity. Figure 14(b) shows the spacing between the curves from bottom to top growing smaller, indicating that the width of the softened area of the glass sample increases less and less as the power density increases. It is due to the relatively low thermal conductivity that the glass softens only in and around the laserirradiated zone. If the width of the softening zone needs to be further increased, therefore, the laser spot size needs to be adjusted to change the dimensions of the irradiated zone.

Influence of laser spot size on softening zone width
After setting the laser spot radius to 1 mm, 1.5 mm, 2 mm and 2.5 mm, respectively, and setting the laser density to 40W/mm 2 in the simulation, we obtained the calculated temperature distribution on the upper surface of the glass irradiated by laser for 5 min, shown in Figure 15(a). It can be observed from this figure that when the laser power density in the simulation model is determined, the corresponding heat-affected zone increases with increases in the laser spot size, resulting in a relatively higher glass temperature rise. When the spot radius is r = 1 mm, the upper glass surface does not reach the softening temperature (730°C). Figure 15(b) shows the variation curves of the softening zone widths with the irradiation time when the power density remained unchanged (40W/mm 2 ), and the laser spot radius was set to 1.5 mm, 2 mm and 2.5 mm. The upper surface of the glass softened at 0.3 min, 0.02 min and 0.05 min, respectively. At the three different spot radiuses, the softening zone widths after irradiation for 5 min were 2.93 mm, 6.63 mm and 9.99 mm, respectively. As mentioned above, when the power density increased to a certain extent, the further increases had little effect on the  softening zone width. Thus, increasing the spot radius can increase the softening zone width.

Variation law model of the influence of laser parameters on the size of the softening zone
To obtain a variation model of the influence of the laser power density and spot radius on the softening zone width, we set the variation range of laser power density to 33 ~ 40W/mm 2 with an interval of 1W/mm 2 and the variation range of the spot radius at 1.5 ~ 2.5 mm with an interval of 0.1 mm for the further orthogonal experiments. The irradiation time for each group was 5 min, and 88 groups of experimental data were obtained (see the attached appendix).
Afterwards, deviations between the experimental data and the fitted equation were calculated. The fitting equation was chosen to be a polynomial, and the results are depicted in Figure 16. Additionally, a residual plot was generated to evaluate the quality of the fit. The maximum residual between the experimental data and the fitted equation is 0.1969 mm, corresponding to a maximum error of 0.319%. The coefficient of determination (R^2) for the fit reaches an impressive value of 0.9986, while the fitting standard deviation is 0.0933.
The softening of glass under laser irradiation is a complex physical process. Linear models are not suitable for regression analysis of laser melting due to their poor fit. In this study, a quadratic model was used to analyze the regression law during the laser melting process. The quadratic model is represented by    Equation (7). The parameter values in the fitting model are listed in Table 2.
In this equation, W represents the softening zone width, I the laser power density, r the spot radius, anda i (i = 0-9) the mean correlation coefficient, respectively. More specifically, I 2 33; 40 ; r 2 � ½1:5; 2:5 ½ �. To further verify the correctness of the model, experiments were carried out on other non-fitting points besides the above-mentioned 88 groups of data. The measured data of the softening zone widths were extracted and compared with the calculated values in the model. We then compared the deviations with the experimental results. The relevant data we acquired is listed in Table 3.
Consideration of the above data, it can be concluded that the proposed model is suitable for the calculation of the softening zone width under laser irradiation in the range of I 2 33; 40 ; r 2 � ½1:5; 2:5 ½ �. Even for those experimental data outside the fitting points, the calculation results differ slightly.

Conclusion
This study has established prediction models for temperature ranges and softening zone widths of glass irradiated by semiconductor laser and then verified the generated models through multi-point temperature and softening zone width measurement experiments. The experimental results indicate that the maximum temperature deviation of the temperature range prediction model is 4.7°C, while the minimum deviation is 0.53°C; the maximum width deviation of the softening zone prediction model is 0.04 mm while the minimum deviation is 0.02 mm. Based on a large number of experimental data, this research has finally determined the proposed calculation model for softening zone width, and the fitting correlation coefficient r 2 reached 0.9986. This study has important significance for reference when analyzing the temperature distributions and softening zone widths for use in using a semiconductor laser to process glass materials and the subsequent thermal forming process.

Disclosure statement
No potential conflict of interest was reported by the author(s).