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Abstract
Laser beam figuring (LBF), as a processing technology for ultra-precision figuring, is expected to be a key technology for further improving optics performance. To the best of our knowledge, we firstly demonstrated CO2 LBF for full-spatial-frequency error convergence at negligible stress. We found that controlling the subsidence and surface smoothing caused by material densification and melt under specific parameters range is an effective way to ensure both form error and roughness. Besides, an innovative “densi-melting” effect is further proposed to reveal the physical mechanism and guide the nano-precision figuring control, and the simulated results at different pulse durations fit well with the experiment results. Plus, to suppress the laser scanning ripples (mid-spatial-frequency (MSF) error) and reduce the control data volume, a clustered overlapping processing technology is proposed, where the laser processing in each sub-region is regarded as tool influence function (TIF). Through the overlapping control of TIF figuring depth, we achieved LBF experiments for the form error root mean square (RMS) reduced from 0.009λ to 0.003λ (λ=632.8 nm) without destroying microscale roughness (0.447 nm to 0.453 nm) and nanoscale roughness (0.290 nm to 0.269 nm). The establishment of the densi-melting effect and the clustered overlapping processing technology prove that LBF provides a new high-precision, low-cost manufacturing method for optics.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the development of modern optical technology, fused silica optics are widely used in high-precision extreme ultraviolet lithography systems [1], high-power laser systems [2], and astronomical telescopes [3], while surface quality demands are continuously increasing. Sub-aperture polishing is an essential way to realize ultra-precision optics, however, the common sub-aperture polishing ways such as magnetorheological polishing [4], bonnet polishing [5], and small-tool polishing [6] unavoidably introduce polishing fluid impurities. Ion beam polishing is also an ultra-precision processing technique for optics, but this technology similarly introduces ion contamination and requires operation in vacuum [7], which restricts the development of fused silica optics to higher precision and efficiency. Therefore, the non-contact, low requirements in the environment of laser is highly advantageous for ultra-precision optical processing [8].
In the 1980s, the Lawrence Livermore National Lab firstly used continuous-wave (CW) CO2 laser for polishing fused silica based on melting flow without material removal, and they proved that CW CO2 laser polishing could predominantly smooth fused silica surfaces by surface tension [9]. In recent years, the Fraunhofer Institute for Laser Technology optimized the CW CO2 laser polishing parameters to reduce the roughness to the single-digit-nanometer level [8]. To further reduce the roughness, He et al. established a transient numerical model for laser polishing and revealed that the Marangoni effect plays a dominant role in reducing the roughness of fused silica. Based on this theory, they achieved the reduction of surface roughness to sub-nanometer by carefully controlling the parameters [10]. However, form error is destroyed during CW CO2 laser polishing on account of the thermal-gradient-induced stresses, and the deterioration of form error remains the main challenge for achieving full-spatial-frequency error convergence [8,11].
To reduce the form error, LBF becomes the key technology, and high-precision laser ablation based on evaporation is proposed to achieve LBF experiments. Heidrich et al. applied pulsed ultra-short laser to reduce heat accumulation and investigated the feasibility of pulsed ultra-short laser for LBF experiments, but the achieved ablation depth of 135 nm was too large to meet the resolution of high-precision form correction [12]. Subsequently, Weingarten et al. used pulsed CO2 laser to avoid thermal effect and carried out LBF experiments to demonstrate the possibility of obtaining an ablation depth of 4 nm for meeting high-precision form correction requirements [13]. However, laser ablation will inevitably deteriorate the roughness in theory. Zhao et al. established the laser ablation model according to the normal velocity of the gas-solid interface and explained that the residual ripple after laser ablation of more than 10 nm will destroy surface roughness [14]. Tan et al. developed a multi-physics coupled model to investigate the surface morphology after laser ablation, and they revealed that vaporization recoil pressure is the dominant factor for forming a raised rim feature to deteriorate the roughness [15]. Although Temmler et al. used the iteration strategy for pulsed LBF to retard the impact on roughness deterioration, the roughness is still more than 3 nm due to the laser scanning ripples (MSF error), which is challenging to meet the requirements of high-precision optics [16]. The above theoretical and experimental investigations indicate that the existing laser polishing and laser ablation techniques seem unable to ensure both form error and roughness, and it remains challenging for LBF research. Therefore, finding innovative laser processing physical mechanism and new processing technology is the key to solving the above problems.
In this paper, we found a densi-melting effect of fused silica to achieve full-spatial-frequency error convergence and revealed the nano-precision figuring mechanism according to this effect under special parameters. Subsequently, we devised a clustered overlapping processing technology through the parameters overlapping control of TIFs to avoid laser scanning ripples (MSF error). The experimental results indicate that the form error is controllably reduced without deteriorating microscale roughness and nanoscale roughness at low thermal stress.
2. Densi-melting effect of nano-precision figuring
Breaking through the existing figuring mechanism is the key to meeting LBF requirements. Therefore, we studied the fused silica modification mechanism during the laser radiation process. The schematic diagram is plotted in Fig. 1(a). Starting at a temperature of 1350 K, fused silica is densified, leading to nano-precision volume shrinkage subsidence [17]. When the temperature exceeds 1875K, the material is molten to smooth the fused silica surface [18]. When the temperature continues to rise above 2200 K [19], the material is removed due to evaporation, which leads to residual scanning ripples and destroys roughness [20].
Fig. 1. (a) Schematic diagram of material modification during laser radiation process (left); Densification process of material density changing with temperature (right). (b) Sketch map of microscopic ablation process and densification process.
Figure 1(b) illustrates the microscopic ablation process and densification process. The network structure of fused silica belongs to an unregular amorphous material. From the material structure view, fused silica consists of massive void space in addition to the volume occupied by silicon and oxygen atoms, and the reduction of the void space would induce the variation of the microscopic material properties. When the material undergoes densification process, the particle arrangement would be disarranged and the void space will decrease, leading to volume shrinkage [21]. Compared with traditional ablation principle of figuring process which achieves form correction by removing particles, densification process does not remove particles during the figuring process.
Moreover, by controlling the maximum surface temperature, we found that fused silica undergoes densification and melt process successively without ablation process in the “melt + densification” temperature range. Therefore, we proposed a densi-melting effect to meet both form error and roughness requirements. However, it is unclear whether densification process can achieve controlled figuring depth in theory. Thus, we established a simulated model based on the finite element method to analyse the influence factor of the physical process.
2.1 Model and theory
Densification is closely related to the thermal history of the material [22]. Therefore, a simulated model developed in COMSOL 5.6 is established to analyse thermodynamic and fictive temperature evolutions at different pulse durations. Fictive temperature is defined as the thermodynamic temperature at which the fused silica structure is in equilibrium [23]. The geometry and boundary conditions are shown in Fig. 2. The processing layer interacts directly with the laser compared with the base. Therefore, for precision results, the mesh of the processing layer is finer and the maximum mesh size is 10µm, while the base is divided into sparse mesh with a maximum mesh size of 50µm. The mathematical model in this paper is established based on the following assumptions: fused silica is isotropic and homogeneous, the effect of densification on heat diffusion is ignored, and the processes of solid-liquid phase transition and melting flow are ignored.
The thermodynamic boundary conditions are as follows. The Gaussian pulsed laser is applied on the top surface of fused silica and absorbed through the heat conduction inside the material, and heat dissipation occurs through natural convection and radiation. The nonlinear heat diffusion equations are expressed as Eq. (1). The side surfaces are subjected to natural convection, while the bottom surface is subjected to thermal insulation.
During the thermal cycle in the process of laser interaction with fused silica, fused silica has not been densified when the laser has not irradiated the material, and the original fictive temperature is Tf,0. When the laser beam is turned on, the temperature is higher than the structural transition temperature of 1350 K, and the structure of fused silica is rearranged to reach equilibrium in a short period. During the cooling process, the equilibrium state cannot maintain, and the final fictive temperature is Tf,∞ thanks to frozen fused silica structure. Due to thermal effects induced by laser radiation, the maximum thermodynamic temperature is higher than the initial fictive temperature, Tf,0. The subsequent fictive temperature Tf,i + 1 can be calculated using the recursive function [17]:
This model determines the fused silica parameters of thermal conductivity, specific heat capacity, and viscosity according to Refs. [24–26]. The rest fused silica parameters used in this model are listed in Table 1.
Table 1. Relevant parameters in this model
2.2 Trigger condition of densi-melting effect for fused silica
To ensure fused silica is in the densi-melting effect under laser radiation, we need to calculate the relationship between the fused silica density and thermal history to find the parameter boundary [30]. Through the model simulation, the fictive temperature and density of fused silica at different pulse durations are calculated in Fig. 3(a). When the pulse duration increases from 40µs to 50µs, the fictive temperature increases linearly from 1399 K to 2000K. The relationship between fictive temperature and the density of fused silica is given in the study as Eq. (4) [29]. According to Eq. (4) and the simulated fictive temperature results, the density of fused silica at different pulse durations changes linearly from the initial 2.2003g/cm3 to the final 2.2064g/cm3.
In addition, the relationship among thermodynamic temperature, fictive temperature, and time with a pulse duration of 50µs is shown in Fig. 3(b). When the laser is turned on, the thermodynamic temperature is below the structural transition temperature of 1350 K. Due to the long relaxation time, the fictive temperature is constant. As the thermodynamic temperature rises above the structural transition temperature, the internal structure of fused silica is changed [19], and the relaxation time decreases rapidly according to Eq. (3). The fictive temperature begins to increase so that the structure reaches equilibrium. When the laser is turned off, the material starts to cool, and Tf decreases with the substrate temperature. When the rate of structural change cannot keep up with the cooling rate, the material is frozen, and the fictive temperature Tf ultimately plateaus to a value of Tf,∞, driving dTf/dt to zero. Therefore, the final fictive temperature after laser processing is obtained. The thermodynamic temperature and cooling rate varied with different pulse durations are exhibited in Fig. 3(c). The maximum thermodynamic temperature varies from 1888K to 2148 K, and all maximum thermodynamic temperatures are between 1875K and 2200 K, which ensures that the material undergoes densification and melt process successively. The thermodynamic temperature after laser radiation is lower than 1350 K, so the material is cooled down without heat accumulation. In addition, the cooling rate is a key factor in affecting fictive temperature [31]. The cooling rate k increases from 12.84 K/s to 15.42 K/s, and Tf goes up with an increasing cooling rate leading to a more densely fused silica structure.
Fig. 3. Thermodynamic temperature and fictive temperature distribution at different pulse durations. (a) Fictive temperature and density of fused silica at different pulse durations; (b) Relationship among thermodynamic temperature, fictive temperature, and time with a pulse duration of 50µs; (c) The thermodynamic temperature varied with different pulse durations during laser processing in simulation (left). The red region is the evaporation, melt and densification temperature range, the pink region is the melt and densification temperature range, and the orange region is the densification temperature range. The partially enlarged figure of the maximum temperature (right). The cooling rate k increases with the increase in pulse duration.
2.3 Quantitative control of figuring depth
To realize the controllable figuring depth, we ought to obtain the distribution of densification region. On account of the densification of fused silica occurring in the heat-affected zone (HAZ), the HAZ is considered as the densification region. When the laser heating is turned off, the region where the temperature exceeds the structural transition temperature of 1350 K is defined as the HAZ [18]. According to this principle, the HAZ irradiated by the single laser pulse is simulated at different pulse durations. The HAZ distribution is plotted in Figs. 4(a1)-(a6). Based on the distribution, the HAZ shape is considered as half of a spheroid, and the HAZ thickness and HAZ radius are gained. The thickness and radius of 40-50µs pulse duration are plotted in Fig. 4(b). As the pulse duration increases, the thickness varies from 6.6µm to 8.8µm and the radius changes from 100µm to 114.6µm. Larger thickness and radius prove that more material is densified in the HAZ.
Fig. 4. Influence of pulse duration on HAZ distribution and figuring depth. (a1)-(a6) The calculated HAZ distributions at the end of different pulse durations from 40µs to 50µs; (b) Variation of HAZ thickness and HAZ radius with respect to pulse duration; (c) Figuring depth at different pulse duration. The dotted lines represent the pulse duration of 39µs and 51µs to distinguish different interaction processes. The blue region is the densification region when the pulse duration is below 39µs. The orange region is the melt and densification region when the pulse duration is between 39µs and 51µs. The red region is the evaporation, melt and densification region when the pulse duration exceeds 51µs.
Based on the simulated model, the fused silica density, HAZ thickness, and HAZ radius are obtained from the above results at different pulse durations. These factors will affect the degree of densification-induced subsidence. Thus, we proposed the densi-melting effect to predict the nano-precision subsidence depth, which is controlled by adjusting the pulse duration. Assuming that there is no mass change with the densi-melting effect. Then, we obtained the volume before and after LBF by dividing the mass by the density. Change in the volume of fused silica in the densification region can be written as Eq. (5). Subsequently, we assumed that the shape of the HAZ and volume shrinkage region is half of a spheroid. The figuring depth h is written as Eq. (6).
According to Eq. (6), the figuring depth is closely related to HAZ thickness and material density, and the predicted figuring depth is shown in Fig. 4(c). When the pulse duration increases from 40µs to 50µs, the figuring depth increases from 1.37 nm to 24.34 nm with the increase of HAZ thickness and material density. The results show that this effect can achieve controlled subsidence of nano-precision depth for form correction. Plus, we calculated the pulse duration corresponding to the transition temperature among the densification, melt and evaporation processes. When the pulse duration is controlled between 39µs and 51µs, the maximum thermodynamic temperature is in the range of 1875-2200 K. By controlling the material temperature and pulse duration in the “melt + densification” interval, the form error is reduced by controlled nano-precision subsidence which is related to fused silica density change, and the roughness is smoothed by melting flow. Therefore, the densi-melting effect is fundamental to revealing the figuring mechanism and guiding parameters selection of LBF experiments.
3. LBF with clustered overlapping processing technology
3.1 Principle
Except for the controllable figuring depth realized by the densi-melting effect, the laser scanning ripples (MSF error) caused by power fluctuation [16] and the huge control data volume are two additional challenges in macroscale LBF. To address these two problems, clustered overlapping processing technology is proposed to be applied in LBF experiments. Figure 5(a) displays the procedural principle of the clustered overlapping processing technology. Many single Gaussian lasers are clustered as processing sub-region (which is considered as TIF), and the form correction is achieved by overlapping control of TIF figuring depth and processing path. Each sub-region is overlapping and processed according to the path marked by the white line. This scanning path can avoid the convex slits between each sub-region, which is caused by the clustered process. In each sub-region, a unidirectional scanning strategy is used to avoid heat accumulation at reversal points. The shape of TIF is shown as square for easier achieving the clustered overlapping technology, but the shape of TIF is not limited to square.
Fig. 5. (a) Schematic diagram of clustered overlapping processing technology. (b) Region division and processing path. The Blue region is the target processing region. The white region is the additional region of measurement. Other color regions with triangular symbols are the processing sub-regions.
Figure 5(b) displays the procedural principle of the region division and processing path with the clustered overlapping processing technology. The size of the sub-region is adjusted according to the size of the target processing region from 1 × 1mm2 to n × n mm2. This method requires a measurement size of (m + n-1) × (m + n-1) mm2 to ensure that each 1 × 1mm2 sub-region in the target processing region is repeatedly processed the same number of times. Compared with the method of Fraunhofer Institute for Laser Technology (a software that sets the pulse duration on each position depending on the surface height) [13], this method dramatically improves processing efficiency. Taking processing the area of 5 × 5mm2 as an example, if all heights are converted to pulse durations, about 40,000 points are input to the signal generator while using the clustered overlapping processing only needs 49 points with a sub-region of 3 × 3mm2. The control data volume is greatly simplified by three orders of magnitude, and the processing efficiency is improved for meeting form correction requirements.
3.2 Process
The schematic diagram of the whole form correction process is shown in Fig. 6. First, the form error of fused silica is measured with laser interferometer. Through the data of measurement, the image of height points $z(x,y)$ is obtained. Then, the average height is calculated in each sub-region to get the average height points $z(\overline x ,\overline y )$. The laser processing in each sub-region is regarded as the TIF $f(\bar{x},\bar{y})$, which is obtained by the experiments. This process is considered as a clustered process. Subsequently, the average height points $z(\overline x ,\overline y )$ is converted into the dwell time $t(\bar{x},\bar{y})$ according to the Preston equation [32]. To simplify the calculation of the Preston equation, the average height points $z(\overline x ,\overline y )$ is expressed as the convolution of the TIF $f(\bar{x},\bar{y})$ and the dwell time $t(\bar{x},\bar{y})$:
According to the analysis of densi-melting effect, the relationship between the dwell time $t(\bar{x},\bar{y})$ and the pulse duration points ${\tau _p}(\bar{x},\bar{y})$ is fitted to:
Then, the pulse duration points ${\tau _p}(\bar{x},\bar{y})$ in each sub-region are calculated and input into the signal generator, following the scanning path of clustered overlapping processing technology to conduct the LBF experiments. In LBF experiments, different height is selectively processed by controlling pulse duration to achieve different figuring depth for form correction. Finally, the surface after LBF is measured and characterized to gain the form error results.
4. Experimental demonstration
4.1 Experimental setup
The experiments are carried out on a CO2 laser processing equipment independently built by ourselves shown in Fig. 7. The Gaussian beam with a wavelength of 10.6µm is used to adjust the output of the CO2 laser (Diamond Cx-10, Coherent). The CO2 laser with a maximum power of 100W is operated at a frequency of 95kHz. The acousto-optic modulator is used to work synchronously at the frequency of 1kHz, and the peak of the CO2 laser output waveform is deflected into the first-order diffracted beam to obtain rectangular laser pulses with a pulse duration of 10-150µs. The pulse duration is precisely set in discrete, temporal steps of less than 10 ns, representing the minimum pulse duration error. The F-theta lens with a focal length of 100 mm is used to focus the laser beam and resulting in a focus spot radius of 150µm (at 1/e). The focused CO2 laser beam moves on the fused silica surface according to the F-theta lens with a maximum speed of 250 mm/s. The sample is positioned on the precision three-dimensional mobile platform, which could be removed entirely from the processing position to measure the sample. The initial samples are ground and polished with small-tools before LBF. Before every experiment, all samples (Heraeus Suprasil 311) need to be ultrasonically cleaned with deionized water and alcohol.
Fig. 7. Schematic diagram of LBF experiment platform. The target processing region is located by the cross symbol to ensure the accuracy of measurements.
To characterize the processing results, the form error is measured by laser interferometer (VeriFire XPZ, Zygo). The microscale roughness is measured by white light interferometer (NanoCam Sq,4D Technology) and the nanoscale roughness is measured by atomic force microscope (Dimension 3100, Veeco). The residual stress birefringence is measured by an imaging high-precision polarizer (StrainMatic M4/150.10, ILIS Gmbh).
Plus, to verify the reliability of the densi-melting effect and the clustered overlapping processing technology, we carried out a series of experiments. All the parameters in the LBF experiments are consistently processed with those used in the simulation model (fixed laser power of 26.5W and scanning speed of 25.5 mm/s). In order to achieve homogeneous figuring, the laser pulses overlap in both scan and feed directions. The pulse distance dx in scan direction is determined by dividing the scan speed vscan (25.5 mm/s) by the pulse frequency frep (1kHz). The pulse spacing dy in feed direction describes the distance between two parallel scan vectors. To ensure that the spatial resolution is the same in scan and feed direction, pulse overlap in x and y direction was set to be the same in all cases (dx = dy = 25.5µm).
4.2 Effects of clustered processing technology on TIF figuring depth and microscale roughness
The relationship between pulse duration and figuring depth is essential to guarantee the accuracy of form correction. To obtain the relationship, we measured the TIFs at different pulse durations to gain the figuring depth using the form error results before and after laser processing, shown in Figs. 8(a)-(b). It can be seen from the results that there are no noticeable laser scanning ripples (MSF error) in the processed region. In Fig. 8(b), the figuring depth is defined as the average subsidence of the surface morphology after processing compared to the initial surface morphology. Plus, to obtain the error bars of figuring depth, we measured the figuring depth of TIF at three different positions randomly and averaged the figuring depth of the three positions. The error bars represent the standard deviation from the average figuring depth for different pulse duration.
Fig. 8. Results of TIF figuring depth at different pulse duration. (a1)-(a6) Form error results from 40µs to 50µs pulse duration before laser processing; (b1)-(b6) Form error results from 40µs to 50µs pulse duration after laser processing; (c) Principle of measuring figuring depth of TIFs; (d) Variation of the calculated and experimental depth concerning pulse duration at the same parameters. The error bars represent the standard deviation from the average figuring depth for different pulse duration.
According to the measurement principle of figuring depth, the simulated and experimental depths are plotted against pulse duration, shown in Fig. 8(c). The calculated results fit well with the experimental results, which can effectively demonstrate the rationality of the densi-melting effect. Plus, when the pulse duration increases from 40µs to 50µs, an exponential increase in experimental depths is observed from 3 nm to 25 nm. The discernable minimum experimental depth is down to 3 nm, determining the minimum spatial resolution of the pulsed CO2 LBF. Through this relationship between pulse duration and figuring depth, high-precision figuring of different heights is achieved by controlling pulse duration in LBF experiments.
Due to the size of laser scanning ripples (MSF error) in the microscale range, it will severely influence the microscale roughness. Although there are no noticeable laser scanning ripples (MSF error) in the form error images of TIFs, we measured the microscale roughness of TIFs with a 50× mirror white light interferometer to further verify that this clustered processing method can avoid laser scanning ripples (MSF error). The results in Figs. 9(a)-(b) illustrate that the microscale roughness shows an increasing trend from the initial 0.433 nm to 0.536 nm. A destroyed microscale roughness (>0.5 nm) is still observed in Fig. 9(c) when the pulse duration is above 48µs because of the limitation of the Marangoni effect and surface tension [10]. Since a higher depth will result in larger microscale roughness, the processing depth range between 3 nm and 20 nm is already sufficient in ultra-precision form correction without generating laser scanning ripples (MSF error) and maintaining microscale roughness.
Fig. 9. Results of TIF microscale roughness at different pulse duration. (a1)-(a6) Microscale roughness from 40µs to 50µs pulse duration; (b) Initial microscale roughness; (c) The initial microscale roughness compared with microscale roughness at different pulse durations. The blue dot region represents the destroyed microscale roughness.
4.3 Demonstration of clustered overlapping processing technology under densi-melting effect
To verify the effectiveness of the overlapping technology, we carried out clustered processing LBF experiments without overlapping technology and with overlapping technology. In the clustered processing experiments without overlapping technology, a 5 × 5mm2 area is selected for experimental verification, and the form error result before LBF is shown in Fig. 10(a), where the form error PV and RMS are 0.083λ and 0.015λ, respectively. The area of each sub-region is set as 1 × 1 mm2 without overlapping. The result without overlapping technology is presented in Fig. 10(b). The form error PV and RMS are slightly reduced to 0.063λ and 0.012λ, respectively. However, it leads to convex slits between the processing sub-region as indicated by the marked red dotted line and red circles, which significantly affects the form correction results.
Fig. 10. Laser interferometer measurement images of clustered LBF without overlapping technology. (a) Before LBF; (b) After LBF. The red dotted line and the red circles represent the convex slits.
In the clustered processing experiments with overlapping technology, a target processing region of 5 × 5mm2 is selected, and the area of each sub-region is set as 3 × 3mm2 with overlapping. In Fig. 11(a1)-(a2), the convex slits are significantly smoothed by overlapping technology. Compared to the initial surface of fused silica, the form error PV is significantly reduced from 0.049λ to 0.024λ with a decrease of 51.0% and the form error RMS is reduced from 0.009λ to 0.003λ with a decline of 66.7%. In Fig. 11(b1)-(b2), the microscale roughness changes slightly from 0.447 nm to 0.453 nm, while the nanoscale roughness in Fig. 11(c1)-(c2) decreases from 0.290 nm to 0.269 nm during the LBF process. The nanoscale roughness results prove that the innovative figuring mechanism can reduce roughness by melting flow in the nanoscale range. The above results successfully demonstrate the feasibility of pulsed CO2 LBF for full-spatial-frequency error convergence.
Fig. 11. LBF of full-spatial-frequency error convergence. (a1) Form error before LBF; (a2) Form error after LBF; (b1) Microscale roughness before LBF; (b2) Microscale roughness after LBF; (c1) Nanoscale roughness before LBF; (c2) Nanoscale roughness after LBF; (d) PSD curves of the material surface before and after LBF.
To further analyze the results, power spectrum density (PSD) analysis before and after LBF was calculated in Fig. 11(d). The results show that the PSD value of form error and nanoscale roughness is reduced through LBF, and the spectral peak of the periodic structure is not produced in the microscale range, which proves that the laser scanning ripples (MSF error) are suppressed. The PSD results further confirmed the effectiveness of full-spatial-frequency error convergence.
Plus, to investigate the influence of thermal stress on fused silica during LBF experiments, we measured the residual stress before and after LBF with an imaging high-precision polarizer, and the results are shown in Fig. 12. The maximum stress birefringence before and after LBF varies slightly from 2.79 nm/cm to 3.94 nm/cm. Typically normalized stress birefringence less than 5 nm/cm is required for most optical applications, which further proves the effectiveness of LBF for machining ultra-precision optics. Limited by the energy density and power stability of current laser equipment, the whole processing takes 4.5 hours for a 5 × 5mm2 region, and the larger scale surface processing is going to be developed by improving processing efficiency.
5. Conclusion
This study theoretically and experimentally demonstrated that pulsed CO2 LBF is feasible to achieve full-spatial-frequency error convergence at low thermal stress. The following conclusions are drawn.
- (1) We found a “melt + densification” interval to ensure both form error and roughness by controlling the material temperature and pulse duration. In this interval, the form error is reduced by volume shrinkage, and the roughness is smoothed by melting flow.
- (2) An innovative densi-melting effect in this paper is proposed to reveal the innovative figuring mechanism and analyze the form correction of controlled figuring depth. Plus, the simulation depths at different pulse durations agree with the experimental results to guide LBF experiments.
- (3) A novel clustered overlapping processing technology used in LBF is conducted through the parameter control of TIFs and overlapping processing to avoid the deterioration of laser scanning ripples (MSF error) and reduce the control data volume by three orders of magnitude.
During the LBF experiments with the innovative effect and technology, the form error RMS is reduced by 66.7% from 0.009λ to 0.003λ, the MSF roughness changes slightly from 0.447 nm to 0.453 nm, and the HSF roughness decreases from 0.290 nm to 0.269 nm. The maximum stress birefringence varies slightly from 2.79 nm/cm to 3.94 nm/cm. Although pulsed CO2 LBF has already been applied in small-sized optics, further studies on larger-sized curved optics will be the next task by the proposed LBF processing technology.
Funding
Member of Youth Innovation Promotion Association of the Chinese Academy of Sciences (2022246); Key projects of the Joint Fund for Astronomy of National Natural Science Funding of China (U1831211); Natural Science Foundation of Shanghai (21ZR1472000); National Natural Science Youth Foundation of China (62205352); Shanghai Sailing Program (20YF1454800); National Key Research and Development Program of China (2022YFB3403403).
Acknowledgments
The authors would like to thank the referees for their valuable suggestions and comments that have helped improve the paper.
Disclosures
The authors declare no competing interests.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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