Optical Glass Lens Molding Experimental and FEA Results

Because the lens shape is primarily determined in the molding stage, the material model introduced is critically related to shape prediction accuracy. The Newtonian flow behavior in the molding stage can be verified by observing the time-displacement relationship between the experimental and simulated results. Figure 4.10 shows a good agreement between these results, indicating that the Newtonian flow indeed accurately describes the deformed shape before annealing.

Because the stresses generated in the molding stage are relaxed instantly owing to high molding temperature (568°C) and low viscosity (10^8.35 Pa∙s), the residual stresses in the lens product are all induced owing to the thermal strain in the annealing stage.

The FEA predicted lens shape and residual stress were obtained by incorporating the structural relaxation and stress relaxation properties in the annealing stage. Figure 4.11 shows the simulated final lens shape together with predicted residual stresses.

Residual stress results in stress birefringence. According to ISO 10110-2 [37], the birefringence produces a difference in index of refraction in the glass for light polarized parallel or perpendicular to the residual stress. This can affect the wavefront quality or optical path difference of the light transmitted through the optical element. The residual stress induced birefringence is specified in terms of OPD of retardation. This is given by the equation:

OPD = a s c⋅ ⋅ (4.3) where OPD is the optical path difference in nm, a is the sample thickness in cm, s is the residual stress in units of N/mm² and c is the difference in the photoelastic constants in units of 10^-7mm²/N. A retardation of more than 10nm/cm sample thickness generally corresponds to “coarse” annealed glass while a retardation of less than 10nm/cm sample thickness refers to “fine” annealed glass, typical for precision optical elements.

For L-BAL42, c is not specified by the manufacturer, and we assume c is 0.2 by

referring to glass with similar compositions. According to the simulated results, the max OPD was about 1.60nm/cm sample thickness in the outer lens and the OPD was about 1.16nm/cm sample thickness in the center, indicating that the cooling process was a fine annealing process. Because the birefringence measurement on the molded lens to measure the residual stress in the lens is difficult to achieve with current apparatus, future studies can keep searching for a suitable apparatus to perform this measurement for experimental verification.

In the FEA prediction, a larger residual stress appears close to the upper surface of the lens. This is because the upper mold cools slightly faster than the lower mold, inducing a larger temperature difference close to the upper surface and resulting in a larger residual stress in this area.

Predicted thickness in the center of the lens is 4.843mm, and the final lens diameter is 21.551mm. Figure 4.12 shows the lens formed in the molding experiment. The central thickness was 4.838mm (measured using Mitutoyo IDC digimatic indicator (543-251) with 0.003mm accuracy). The average diameter was 21.665mm which was calculated by averaging the measured values in four equally divided directions (values were measured using Mitutoyo dial caliper (505-666) with 0.01mm accuracy).

Deviations between the verification experimental results and the simulated results are 0.103% in thickness and -0.526% in diameter. Table 4.3 summarizes these shape differences. These results indicate that incorporating structural relaxation allows an accurate prediction of the thickness and diameter.

Figure 4.13 compares the experimental and simulated surface curve results and Figure 4.14 shows the deviations between the simulated and experimental surface curves. Table 4.4 shows the root mean square and absolute values of deviations.

Deviations on the upper surface (RMS: 0.559μm and absolute max: 1.972μm) are slightly larger than deviations on the lower surface (RMS: 0.290μm and absolute max:

1.167μm). This implies that the real temperature distributions on each lens surface may not be uniform, unlike those in the FEA. The temperature distributions were inputted uniformly on each lens surface in the FEA and the inputted temperature histories were measured by single thermo-couples embedded in each mold. Figure 4.15 shows the FEA inputted temperature distribution and temperature history. With the uniform temperature input, the variations of the surface curves predicted by FEA before and after annealed are shown in Figure 4.16. The directions of variation reductions for each surface curves during and after annealing are shown in Figure 4.17.

The directions of variation reductions change suddenly at a point about 5mm from the center in the radius direction for the upper surface and about 5.3mm for the lower surface. By comparing both the experimental and simulated surface curves after annealing to the designed ones, as shown in Figure 4.18 and Figure 4.19, deviations are found to be increased after the above mentioned changing points of the reduction directions. For the upper surface, this may indicate that the cooling rate is slower near the outer lens than that in the lens center, thus results in smaller values of reduction than those predicted by the FEA with uniform temperature distribution. For the lower surface, the experimental and simulated results are very close. Small deviations near the outer lens may indicate that the cooling rate is faster than that in the lens center, thus results larger values of reduction than those predicted by the FEA with uniform temperature distribution. Future study is expected to conduct multi-point measurements of the temperature distribution on the lens and molds surfaces to obtain the real temperature distributions. Also, a more complex thermal model that considers heat transfer between the molds, lens, and the atmosphere inside the furnace can be constructed in the future by referring to related studies and conducting multi-points temperature measurements to enhance the FEA prediction capability on the optical glass molding process.

Figure 4.20 and Figure 4.21 show the deviations of the experimental and simulated surface curves from the designed ones after annealing. FEA results with and without stress and structural relaxation properties are shown in these two figures. Table 4.5 shows the RMS values of the surface curve deviations between experimental and FEA results. These results show that the simulated results with stress and structural relaxation properties are more close to the experimental ones than the simulated results without these two properties. Moreover, Figure 4.22(a) and Figure 4.22(b) show that the residual stresses in the lens after annealing can be predicted only by incorporating stress and structural relaxation properties. The simulated surface curves and the residual stresses all indicate that incorporating the stress and structural relaxation into FEA in the annealing stage enhances the prediction accuracy and is necessary for the FEA on the optical glass molding process.

Despite the small discrepancy on surface curve prediction, the lens shape was accurately predicted in this study. Thus it can be concluded that the FE model of this study is useful for providing industrial design references.

4.6 Further Discussions on the Forming Parameters