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Abstract
During the thinning process of plate-shaped optical parts (PSOP), the release of internal stress would cause the deformation of ultra-thin PSOP, which deteriorates the processed surface figure. The stress-release-induced deformation is hard to be predicted and controlled due to the difficulty in measuring the tiny internal stress in ultra-thin PSOP. In this paper, an analytical model is established to depict the variation of internal stress and deformation during the thinning process. It can be used to calculate the initial internal stress distribution along the thickness according to the deformation and the residual thickness of the sample. Meanwhile, the model can predict the residual stress distribution and deformation in the whole thinning process. The prediction results obtained from the proposed model agree well with the experimental results, and the prediction error is less than 13%. The presented model has great significance for the analysis of the tiny internal stress and then guide the process of making ultra-thin PSOP.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Ultra-thin plate-shaped optical parts (PSOP), whose diameter-to-thickness ratio is in the range of 50-100, have extensive application in various fields such as the optical observation system and the laser system. The surface figure error of PSOP would seriously deteriorate the performance of the optical system. Take the high-energy laser system as an example, the figure error would modulate the laser beam and affect the energy-concentration-rate on the target. Therefore, the surface figure error should be kept lower than a certain value in the precision machining of ultra-thin PSOP.
Some high-precision figuring technologies have been developed to reduce the surface figure error of PSOP after full aperture polishing, such as Ion Beam Figuring (IBF) [1], Plasma Jet Machining (PJM) [2], Computer-Controlled Optical Surfacing (CCOS) [3] and Magnetorheological Finishing (MRF) [4]. In the figuring process mentioned above, the figure error can be corrected to be low enough by controlling the removing function and the machining path. However, the error of surface figure is required to be lower than hundreds of nanometer before high-precision figuring, since the removal rate of these existing technologies is very small [5–7].
For thick parts, the surface figure error can be effectively controlled by the full aperture polishing method. Suratwala et al. [8] fabricated 265 mm square and 50 mm thick fused silica flats by strictly controlling polishing parameters. The PV value of the surface figure is reduced to ∼λ/2 (∼330 nm) from 6.7 µm after polishing 4 hours with 0.6 psi applied pressure. Nevertheless, when the thickness is reduced to 8 mm, the smallest PV value can only reach ∼1λ (∼633 nm) under the same processing conditions. For the ultra-thin PSOP with high aspect ratio, it is difficult to keep the accuracy of the surface figure in the full aperture polishing. This is because the stiffness of the sample decreases as the thickness reduces, and various stresses (including the surface stress [9], the clamping stress [10], the internal stress [11], etc.) would deform the ultra-thin PSOP. The completed surface figure would be deteriorated by the stress-induced deformation, even if its accuracy were high enough before the parts are released from the fixture.
Some researchers investigated the effect of stress on the deformation of plate-shaped parts. Toshihiro Takeshita et al. [9] analyzed the wafer deformation caused by the stress in the thin-film deposition. Teixeira et al. [12] explored the relationship between the deformation of Si wafer and the residual stress of the damage layer. Forest et al. [13] analyzed the deformation of the thin optics affected by the stresses due to gravity, thermal expansion, and friction. However, the deformation caused by internal stress is seldom researched. Generally, the optical parts are annealed adequately before polishing to eliminate the large stress generated during molding and machining. According to the previous studies, internal stress could be released to a tiny amount after annealing [14]. However, the tiny stress would cause a deformation with hundreds of nanometers for the ultra-thin PSOP with extremely low stiffness. Therefore, the internal stress-induced deformation should be strictly controlled in the full aperture polishing to obtain a submicron surface figure.
At present, the most common method of measuring the internal stress of an optical part is based on the photoelasticity theory. It is motivated by the phenomenon of stress-induced birefringence [15]. However, the following shortcomings limit its applications in the high precision fields. 1) The measured values are the difference between two perpendicular principal stresses at a point in the optical parts. If the two principal stresses are equal, the total stress value measured is zero even if each stress is very large. This phenomenon often makes it impossible to measure the actual stress. 2) The stress cannot be measured when it is parallel to the direction of measuring laser. 3) The measured value is the algebraic sum of all the stresses at different positions on the optical path of the measuring laser. The actual stress is likely to be much greater than the measured value. 4) The measuring error becomes uncontrollable with the change of incident angle of the measuring laser when it is not perpendicular to the surface. 5) It can only measure transparent objects. Other methods of measuring stress include Micro-Hole Drilling, Micro-Slot Cutting, Nano-Indentation, X-Ray Diffraction [16], etc., which can only measure the local stress at one point on the surface but cannot display the value and the distribution of the internal stress inside. Therefore, the existing methods are not suitable for measuring the internal stress of ultra-thin PSOP. It is essential to develop a new measurement method.
In this paper, a bending deformation method is employed to obtain the residual stress distribution in PSOP. An analytical model of the internal stress is established and its accuracy is verified by comparing the predicted deflection data with the experimental results. The initial stress and the residual stress distribution in the thickness direction of fused silica sample can be obtained at any thinned thickness, as well as the stress-release-induced deflection. This work can provide guidance for the full aperture polishing process of the ultra-thin PSOP.
2. Theoretical model
2.1 Internal stress distribution in PSOP
PSOP is considered as a cylindrical and isotropic plate in the theoretical model. When the plate is cooled symmetrically from both end faces in the usual way, tiny stress with a parabolic distribution in the thickness direction will leave in the body after annealing. The stress direction is always parallel to the plate surfaces. The internal stress is compressive near the end faces and is tensile in the area near the middle plane [17–19]. Figure 1 shows the schematic of the internal stress distribution after annealing. The circular plate with the radius R and the thickness h0 is described in the cylindrical coordinates. The stress distribution is symmetrical to the middle plane of the plate. The compressive stress is set as negative and the tensile stress is set as positive. It can be expressed as a function of distance from the middle plane of the plate,
where σ0 is the internal stress remained in the plate after annealing, z is the coordinate along with the thickness, a, b and c are the constants in the parabolic equation.
Fig. 1. Distribution of the internal stress in the cylindrical coordinates. (a) The plate in the cylindrical coordinates. (b) The stress distribution in a section of the plate.
According to the symmetry of the stress distribution, the constant b in Eq. (1) should be zero. The internal stress satisfies the balance equation:
with dz an element parallel to the z direction. Equation (2) states that the sum of the forces in the r direction within the plate is zero. Combining Eqs. (1) and (2), the stress distribution can be expressed as:Fig. 2. Force and moment balance relation in the thinning process. (a) The force and moment of the removed part on the ultra-thin plate balance the internal stress remained after annealing. (b) The ultra-thin plate is deformed and its internal stress is redistributed after the material removed.
When the removed part is separated from the ultra-thin plate, its force and moment on the ultra-thin plate will disappear simultaneously. The ultra-thin plate will be deformed and produce a deflection w compared to its initial shape, see Fig. 2(b). Concurrently, the internal stress is redistributed as σ(z) from the initial σ0(z). Since the thickness and deflection of the ultra-thin plate are very small compared to the diameter, it can be considered as a thin plate in a small deflection bending problem. The basic assumptions are as follows [20]:
- 1) The deflection w is small enough compared with the thickness h*. Therefore, the slop of the deformed surface is very small and its square is negligible compared with 1.
- 2) There is no strain on the neutral plane of the bent ultra-thin plate.
- 3) The initial section perpendicular to the neutral plane is kept flat and perpendicular to the neutral plane after bending.
- 4) The stress perpendicular to the neutral plane is negligible compared to other stress components.
Combining Eq. (3) and Eqs. (6a)–(6c), the dimensionless initial stress is:
2.2 Deflection of PSOP in thinning process
The deflection of the ultra-thin plate is only related to the radial coordinate r because the deformation of the ultra-thin plate is considered to be symmetrical about the center of the plate. Therefore, the deflectional differential equation can be written as:
According to the following three boundary conditions, these constants can be determined.
- 1) The deflection of the center of the circular plate is zero,$$\mathop {\lim }\limits_{\mathop r\limits^{\sim } \to 0} \mathop w\limits^{\sim } = {c_1} + \mathop {\lim }\limits_{\mathop r\limits^{\sim } \to 0} {c_4}\ln \mathop r\limits^{\sim } = 0.$$From Eq. (22), we obtain that both c1 and c4 equal to zero.
- 2) The internal shear is zero,$$- \mathop D\limits^{\sim } \frac{\textrm{d}}{{\textrm{d}\mathop r\limits^{\sim } }}\left( {\frac{{{\textrm{d}^2}\mathop w\limits^{\sim } }}{{\textrm{d}{{\mathop r\limits^{\sim } }^2}}} + \frac{1}{{\mathop r\limits^{\sim } }}\frac{{\textrm{d}\mathop w\limits^{\sim } }}{{\textrm{d}\mathop r\limits^{\sim } }}} \right) = 0.$$From Eq. (23), c3 equals to zero.
- 3) The deformation is caused by the removal of ${\tilde{M}}$. Therefore, $- {\tilde{M}}$ is the equivalent of the inner moment:$$- \mathop M\limits^\sim = - \mathop D\limits^{\sim } \left( {\frac{{{\textrm{d}^2}\mathop w\limits^{\sim } }}{{\textrm{d}{{\mathop r\limits^{\sim } }^2}}} + \frac{\nu }{{\mathop r\limits^{\sim } }}\frac{{\textrm{d}\mathop w\limits^{\sim } }}{{\textrm{d}\mathop r\limits^{\sim } }}} \right).$$From Eq. (24), yields: Substitute Eq. (11) and Eq. (19c), yields:
Through the boundary conditions, all constants in the solution of the deflectional differential equation are determined. Substitute them into Eq. (21) and get the deflection:
Combining Eq. (15) and Eq. (27), the relation between the internal stress and the deflection is:
3. Experimental
3.1 Annealing process
The sample used in this work was a cylindrical shaped (φ30 mm×3 mm) fused silica. The SG-XQL1200 annealing furnace produced by Shanghai Institute of Optics and Fine Mechanics was used for the annealing process. Figure 3 shows the placement of the samples. Two Al2O3 ceramic plates covered the samples to equalize the heat transfer rate on both surfaces. The silica wool insulated the sample side from heat conduction. The sample was cooled strictly through the two end faces in the cooling stage. Thus its temperature gradient would be distributed along with the axis.
Fig. 3. Placement of the samples in the annealing process. The left picture shows the exploded view, and the right picture shows the cross section of a single sample.
The annealing schedule is shown in Fig. 4. It consisted of the heating stage, the soaking stage, the slow cooling stage, and the furnace cooling stage. The previous stress was eliminated in the soaking stage. New and tiny stress was generated in the slow cooling stage because of the change of viscosity and would exist in the body permanently. The stress generated at furnace cooling stage would disappear when the temperature of whole sample dropt to the room temperature because the fused silica was completely elastic at this stage [17]. Therefore, at the furnace cooling stage, the sample just needed to be furnace cooled to room temperature.
3.2 Thinning process
The two end faces of the sample were divided into measuring and machining one. The annealed sample was thinned to several suitable thicknesses (1.09 mm, 0.61 mm and 0.34 mm) through the well-controlled lapping on the machining surface. MP-1B lapping machine produced by Laizhou Weiyi Experimental Machine Manufacturing Co., Ltd was used in the experiment. Figure 5 shows that the sample was thinned by the 1500# diamond abrasive disk with 22.50 N load, and matched 500 rpm and 50 rpm rotation rates of the abrasive disk and the carrier respectively [21].
The sample was polished with 80 nm silicasol and polyurethane pad after being lapped to the specified thickness and then etched with the etchant consists of 5% wt. HF, 15% wt. NH4F and 80%wt. deionized water for 270 s. The polishing and etching could remove the damage layer generated in the thinning process to eliminate the deviation caused by the stress in damage layer.
3.3 Deflection measurement
FlatMaster 200 instrument produced by Corning Tropel Co., USA, was utilized to measure the surface figure of the measuring surface in each thinning step. In order to eliminate the influence of the edge effect, the measuring radius was selected as 14 mm. A matrix consisting of 217×217 data points detailed the measurement result of surface figure. The difference in deformation between planes perpendicular to the thickness direction could be negligible because the thickness of the sample was very small compared to the diameter. Therefore, the deformation of the sample was consistent with the variation in surface figure of the measuring surface. The matrix of the initial surface figure measured before the sample deformed was set to be the reference matrix. The deflection defined as the amount of the variation in surface figure was calculated by the subtraction of the reference matrix from the measured matrixes at different thickness. The value of deflection at a certain radius was determined as the average of all deflections at the same radius.
4. Results and discussion
4.1 Variation of surface figure
Figure 6 shows the variation of surface figure during the thinning process. The measurement result in Fig. 6(a) indicates that the measuring surface is extremely flat (PV ≤ 100 nm) when the sample thickness is 3.00 mm. Figures 6(b)–6(d) demonstrate that the deformation of the sample continues increasing as the thickness is reduced. The sample deflection is estimated by the variation of surface figure using the method in Section 3.3. Figure 7 reveals that the deflection is symmetrical about the central axis of sample and its function of the distance from the sample center fits well with a quadratic parabola with zero intercept which satisfies Eq. 27.
Fig. 6. Surface figure of the measuring surface at different thickness, (a) 3.00 m, (b) 1.09 mm, (c) 0.61 mm, (d) 0.34 mm.
4.2 Calculation of internal stress
According to Eq. (27), the best fit value for ${\tilde{\Gamma }}$ is 5.28×10−6 for the deflection measurement at thickness of 0.61 mm (${\tilde{h}}$∼1/5), see Fig. 8(a). Thus, the distribution of initial internal stress can be obtained from Eq. (7). Furthermore, the deflection prediction model of the sample is established by Eq. (27) after the constant ${\tilde{\Gamma }}$ is determined. The theoretical and experimental results with thickness of 1.09 mm (${\tilde{h}}$∼1/3) and 0.34 mm (${\tilde{h}}$∼1/10) are compared in Fig. 8(b). The prediction errors are 12.16% and 12.83%, respectively. As the thickness is reduced, the actual deflection of the sample tends to be larger than the predicted result increasingly. In the theoretical analysis of this work, the stress in hydrolyzed layer and the edge stress of sample are ignored because their effects on the sample deformation are quite slight compared to that of internal stress. The residual stress in hydrolyzed layer caused by polishing may tension the sample surface and enhance the deformation [22]. Although the sample surfaces were etched by etchant for 270 s to eliminate the residual stress in hydrolyzed layer, it is difficult to ensure that the residual stress is removed completely. In addition, the edge stress of sample after annealing may be released with the material removed and contribute to the sample deformation [15]. The effects of these two factors are gradually more obvious with the thickness reduction. Therefore, the error between the theoretical model and the experimental results may be caused by the stress in hydrolyzed layer and the edge stress of sample. We will continue to investigate the effects of these factors on the sample deformation in subsequent work to reduce the prediction error.
Fig. 8. Theoretical and experimental results of the deflection with the thickness of (a) 0.61 mm, (b) 1.09 mm and 0.34 mm.
The distribution of internal stress at any thickness of the sample can be calculated according to Eq. (28). Figure 9 illustrates the calculations of internal stress at five different thicknesses. The stress distributions are in excellent agreement with the empirical model proposed by Adams and Williamson [17].
Consequently, the internal stress generated in the annealing process is the main factor of the deformation. In this work, both initial and residual internal stress can be inferred by using the sample thickness and its corresponding deflection. Thus, the stress-induced deflection of the ultra-thin PSOP can be well predicted in advance before the thickness reduction.
5. Conclusion
Stress-induced deformation is the main difficulty of making the ultra-thin PSOP. It is caused by the release of internal stress with material removal during the thinning process. The distribution of internal stress is hard to be accurately measured by existing methods. In this work, an analytical model is proposed to describe the relationship between internal stress and deflection. It reveals the stress release process and forecasts the deflection at each residual thickness in the thinning process. The conclusions are listed as follows:
- 1) The analytical model is established to determine the distribution of initial stress and residual stress in PSOP during thinning process. The model reveals the relationship between the stress distribution and the deflection of ultra-thin plate at any thickness.
- 2) The deflection of ultra-thin PSOP at different thickness is investigated. The expression of deflection is obtained by solving the deflection equation in the cylindrical coordinates. The sample deflection can be predicted before the processing.
- 3) By experiment verification, the analytical model error is less than 13%. The constant in the stress expression can be determined by experiment method. Compared with the experimental results, the prediction errors of the analytical model are 12.16% and 12.83% in the experiment.
Funding
National Key Research and Development Program of China (2016YFB1102205); Science Fund for Creative Research Groups of NSFC (51621064); National Natural Science Foundation of China (51875078, 51605079, 51775084).
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