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Abstract
Limited by measurement methods, measuring the surfaces and thickness of large thin parallel plates has been challenging. In this paper, we propose a multi-dimensional stitching method using thickness alignment (MSuTA), which use the sub-aperture stitching method based on the phenomenon of parallel plate self-interference with wavelength-tuned interferometer (WTI) for measuring the surfaces and thickness of large thin parallel plates. We establish the stitching correction model based on Legendre polynomial to separate the aberrations caused by the elastic deformation of the thin plate in the unconstrained support tooling by analyzing the influence of the stress state of the thin plate with unconstrained three-point support. The stitching experiment has carried out on 6.3 mm thick, 6-inch parallel plates that the stitching residual is better than 0.35 nm RMS. Compared with 12-inch vertical interferometer, the surfaces and thickness deviation are better than 0.8 nm RMS, and the 36 standard Legendre polynomial coefficient deviation are better than 2.5 nm. Moreover, MSuTA can improves the lateral resolution of the measurement by nearly four times, allowing for a display of more comprehensive surface information. The stitching method proposed in this paper will be widely applied in the manufacture and measurement of large thin parallel plates, and provide reference for the elastic deformation analysis of the thin optical elements in the unconstrained support tooling.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Parallel plates are widely used in optical communication, semiconductor, and other fields. With the rapid development of modern optical technology, major optical engineering, such as semiconductors and precision measurement, put forward higher requirements for the optical surface, thickness and size of parallel plates in order to control the transmitted wave-front quality. At present, the figure error and thickness uniformity of large thin parallel plates is a great challenge for optical measurement [1].
The front and rear surfaces, as well as the thickness, together constitute the primary optical parameters of large-thin parallel plates. Since the front and rear surfaces of the parallel plate are nearly parallel, interference fringes will be observed between the two surfaces, which posing a significant challenge for measurement. The traditional method typically involves applying a protective layer to the rear surface to prevent the creation of unnecessary fringes. The four-step transmission method is achieved by making multiple measurements to achieve absolute thickness uniformity [2,3]. By using absolute test methods, such as three-flat test method [4–6], the figure error and the system error introduced by the reference and interferometer can be obtained. In our previous work, we also proposed the modified six-step method that enables the simultaneous measurement of surface and thickness uniformity [5,7]. However, the measurement steps of those methods are complicated, and it is easy to introduce errors in the measurement process. Furthermore, the above method requires additional optical elements, which makes it both costly and challenging to measure large parallel plates.
Wavelength tuned interferometer (WTI) [8–10] uses wavelength phase-shifting method and Fourier transform algorithm [11–13] to separate interference fringes on multiple surfaces, so as to measure the two surfaces and thickness uniformity of parallel plates simultaneously. At the same time, the stitching method can greatly expand the effective measurement range of WTI through guaranteeing high precision and resolution. The sub-aperture stitching method uses the sub-aperture data of single surface collected by the traditional interference method to stitching [14–18]. However, the sub-apertures obtained by WTI are multidimensional data, including surfaces and thickness uniformity. On the other hand, the alignment method of the multi-dimensional sub-aperture data is necessary. Sub-aperture alignment is to minimize the overlapping residuals after stitching. The figure error of sub-aperture will be affected by the measurement environment, which will reduce the accuracy of sub-aperture alignment, and further aggravate the stitching error. Hence, it is imperative to formulate a stitching model and algorithm that are compatible with the measurement characteristics of large-thin parallel plates with WTI.
In this study, we begin by analyzing the principle of the parallel plate measurement method to derive the stitching model. By summarizing the laws of surface changes in a thin parallel plate with unconstrained support tooling, a stitching corrected model is formulated to enhance the stitching accuracy. In addition, we introduce a multidimensional stitching algorithm using thickness alignment (MSuTA) in Section 2. In Section 3, a simulation analysis are performed to demonstrate the performance of the stitching method. In Section 4, comparative experiment was conducted to verify that MSuTA can achieve high accuracy of sub-aperture alignment through the parallel plate self-interference phenomenon. We also apply the proposed stitching method to measure 6.3-mm thick, 6 inches square thin parallel plate. Compared to the measurement results of the 12-inch vertical interferometer, the deviation of surface and thickness is less than 0.80 nm RMS, and the deviation of 36 standard Legendre polynomial coefficients is better than 2.5 nm. The conclusions are presented in the final section.
2. Theory
When using the sub-aperture stitching method to stitching the multidimensional data of parallel plates collected by WTI, the key lies in how to adjust the aberration relationship within the overlapping area of adjacent sub-apertures. There is a certain relationship between each group of sub-aperture, independent analysis of each group will destroy the integrity of multi-dimensional data. Therefore, we analyzed in detail the adjusting aberrations in each dimension in the process of using WTI to obtain multi-dimensional data, and established a multi-dimensional parallel plate stitching method based on the stitching corrected model to achieve the multidimensional stitching of parallel plates.
2.1 Adjustment aberration form of sub-aperture using WTI
The wavelength-tuning phase shifting method is to shift the phase of the interference signal by changing the wavelength of the source. In the process of wavelength phase shift, the interference fringes formed by different surfaces will have different frequency changes, which is related to the optical path difference of two interference beams. The reference surface reflects on the front and rear surfaces of the parallel plate and between the front and rear surfaces to form interference signals, as shown in Fig. 1.
The intensity of interference fringes collected by the image sensor can be expressed as follows:
It can be concluded that the adjusted aberration introduced by adjustment only has the front and rear surface, and their adjusted aberrations are the same. The adjusted aberration will not be introduced into the thickness data, although the spatial position of parallel plates has changed.
2.2 Surface change of thin parallel plate with unconstrained support tooling
We suppose that the parallel plate is planned into several sub-aperture. Let the three-dimensional area enclosed by the front and rear surfaces in the i-th sub aperture area of the parallel plate be represented as Si2, Oi,j2 represent the overlapping area between the i-th and j-th sub-apertures, as shown in Fig. 2.
For the sub-aperture data composed of front and rear surface, the front and rear surfaces need to be stitching at the same time. Based on the theory of sub-aperture stitching, the acquired sub-aperture meets the following relations:
Where, Sk and S′k respectively represent the k-th measured sub-aperture and the theoretical sub-aperture. ak and bk are the relative tilt, ck is the piston. In the process of sub-aperture data acquisition, the thin parallel plate under test (TPUT) needs to be placed on the unconstrained three-point support tooling, as shown in Fig. 3(a). The center of the circle formed by the three points overlaps with the center of the TPUT. Unconstrained means that the side edge of the support tooling has no constraint on the TPUT. The final surface is obtained by subtracting the deformation from the measurement results, which caused by the three-point support tooling is calculated by the finite element simulation method (Fig. 3(c)).
Fig. 3. The unconstrained three-point support tooling. (a) Actual placement of TPUT, (b) the stress distribution of TPUT, (c) the deformation caused by three-point support tooling
When the TPUT is placed on the supporting tooling for a long time, its own stress will be in balance (Fig. 3(b)). In the process of moving the sub-aperture, the TPUT itself will produce slight shaking, resulting in the change of the stress state on the supporting tooling and slightly deform until it is back in balance. The sub-aperture data collected in the process of stress balance will have additional aberrations. It is complex to analyze the analytical form for the stress balance process of TPUT on unconstrained three-point support tooling. In addition, the finite element method is used to simulate the transient change in large scale to achieve the simulation nano-accuracy are also great difficult. However, the deformation of TPUT under the stress balance process on unconstrained three-point support tooling can be analyzed by using WTI to measure it in real time. According to Eq. (3)-(5), it shows that the deformation of the TPUT is the three-dimensional change process, so we can explore the form of additional aberrations by analyzing the changes of the front surface. The unchanged thickness can be used as the standard to evaluate the correctness of the analysis results.
Using the wavelength tuning phase shifting method of a 12-inch vertical interferometer, we collected the front surface and thickness data of TPUT during 12 hours(Fig. 4(a)), which is 6.3-mm thick, 6 inches square thin parallel plate was placed on unconstrained three-point support tooling. On the other hand, we also collected the data after moving and rotating the TPUT 90 degrees, as shown in Fig. 4(b). The figure error caused by the three-point support tooling have be removed from the front surfaces. Figure 4(a) shows that after TPUT was placed on the unconstrained three-point support tooling, the stress balance state is reached after 2 hours, and finally stabilized to 2.02 nm RMS. Although the center of the three points almost coincides with the center of the TPUT (<0.4 mm), the support points are more inclined towards the edge of the TPUT rather than uniformly distributed on the TPUT (Fig. 3(b)), the corresponding deformation is a symmetrical three point distribution along the X direction (Fig. 3(c)). The distribution of the support points causes the TPUT to undergo elastic deformation according to the position of the support points until it reaches a stable state, which can be clearly observed from the surface change of the TPUT in Fig. 4. At the same time, the thickness data remained unchanged during this process, which verified the correctness of the analysis results. It can be observed that the TPUT will produce three-dimensional deformation during the process of stress balance, which affect the figure accuracy of sub-aperture.
Fig. 4. The change of the front and thickness using 12-inch vertical interferometer. (a) The measurement result after putting TPUT into the tooling, (b) The measurement result after moving and rotate TPUT.
Similar phenomena is also observed in the measured data after moving and rotating (Fig. 4(b)), indicating that moving the TPUT will also destroy its stress state and make it in the stress balance process again. The difference is that it takes longer to stabilize after the stress balance caused by movement is broken. In addition, the power of X and Y on the surface also rotates, which further proves that the change of the measurement surfaces is caused by elastic deformation of the TPUT.
We selected the surface of different stage and used the first 36 standard Legendre polynomials for fitting. Figure 5(a)-(b) shows the Legendre polynomial fitting results of the first 36 standards after TPUT placement and after moving and rotating respectively. Figure 5(c)-(d) shows the time-varying trend of the items 4, 5, 6, 8 and 9 of Legendre polynomials. It can be seen that only when the TPUT is in the stress balance state after a long time of stabilization, there will be no additional aberration in the sub-aperture. From the fitting results, we can use the items 4, 5, 6 and 8 of the standard Legendre polynomial to represent the elastic deformation of TPUT in the process of stress balance. These polynomials represent Power in the X and Y directions, Astigmatism and Coma, as shown in Table. 1.
Fig. 5. The fitting result of front surface using the first 36th standard Legendre polynomial. (a) TPUT after putting TPUT into the tooling, (b) TPUT after moving and rotate, (c) The items that changes drastically of Legendre polynomial within 12 hours after TPUT is placed on the tooling and (d) after moving and rotate TPUT
Table 1. The Legendre polynomial
2.3 Multidimensional stitching based on the stitching corrected model
According to the above analysis and experimental results, there will be no additional aberration in the sub-aperture after a long time of stability, which will lead to low measurement efficiency. From Fig. (4)–(5), the stress imbalance caused by moving the TPUT will introduce items 4, 5, 6 and 8 of Legendre polynomial into the sub-aperture. Therefore, we further modified the stitching model and added the additional aberration caused by stress imbalance,
The adjusted sub-aperture Sj + 2 and the thickness jointly form a multidimensional region Sj3 of the parallel plate in the sub-aperture.
2.4 Multidimensional stitching algorithm using thickness alignment
In the previous section, the multi-dimensional stitching method based on the stitching corrected model is introduced in detail, and another key is aligning the overlapping of adjacent sub-apertures to determine the location coordinates of sub-apertures on the parallel plates. We proposes a multidimensional stitching algorithm using thickness alignment (MSuTA). The algorithm is based on the feature that there is no adjustment error in the thickness data of adjacent sub-apertures, combined with the marking point method to accurately locate the overlapping of each sub-aperture, and then solve the relative spatial position of the multi-dimensional sub-aperture. The stitching process is shown in Fig. 6.
STEP 1: Planning the sub-aperture parameters and the measurement path according to the aperture of interferometer and the parallel plate.
STEP 2: Collecting the k-th sub-aperture Sk3={Fk,Rk,Tk}, wherein Fk, Rk and Tk respectively represent the front, rear surface and thickness.
STEP 3: Input the k-th and k + 1-th sub-apertures data, find the optimal overlapping position of adjacent sub-apertures according to the feature of marked points on the sub-aperture, and move the k + 1-th sub-apertures to corresponding spatial coordinates. Calculate the overlapping residual ΔTk,k + 1 of the k-th and k + 1-th sub-aperture thickness data. When ΔTk,k + 1 is less than the set threshold σ, STEP 4 is performed; otherwise, the optimized position is repeated until the residuals of the overlapping meets the set threshold.
STEP 4: Substitute the aligned sub-aperture data of overlapping into Eq. (7)-(9) to calculate Legendre coefficients, Δak,k + 1, Δbk,k + 1, Δck,k + 1, Δmk,k + 1, Δnk,k + 1, Δuk,k + 1 and Δvk,k + 1.
STEP 5: Substitute Legendre coefficients, Δak,k + 1, Δbk,k + 1, Δck,k + 1, Δmk,k + 1 and Δnk,k + 1, Δuk,k + 1 and Δvk,k + 1 to Eq. (10) to obtain the adjusted k + 1-th sub-aperture data, and synthesize sub-aperture Sk + 13={F’k + 1,R’k + 1,Tk + 1} with thickness.
STEP 6: Stitching the k-th and k + 1-th sub-aperture, and the stitching result is used as the basis sub-aperture for stitching with the k + 2-th sub-aperture. Determine whether the k + 1-th sub-aperture is the last one. If so, output the stitching result, otherwise return to STEP 3.
3. Simulation
To verify the feasibility and theoretical accuracy of the above stitching methods, corresponding simulation experiments are conducted. The 6-inch parallel-plate was used in the simulations, including experimental data with high frequency information as the original two surfaces F, R and thickness T (Fig. 7).
The parallel-plate was divided into nine sub-apertures, approximately 95*95 mm, with adjacent sub-apertures overlapping approximately 70%. Then we get the combined experimental results by introducing an adjustment aberration and measurement noise (with 2.0 nm PV and 0.2 nm RMS) for sub-aperture, so that the front surface of each sub-aperture is in the null-fringe. Table 2 shows the values of relative tilt added for each sub-aperture.
Table 2. Relative tilt of the sub-apertures
Figure 8(a)-(c) show the stitching results of the surfaces and thickness of the parallel plate, respectively. Figure 8(d)-(f) show the corresponding stitching errors, which are obtained by subtracting the stitching results from the original surface (Fig. 7). Figure 8 shows that the deviation of stitching results from the original surface are 0.16 nm RMS, which verifies the feasibility of MSuTA.
Fig. 8. Stitching results and residuals. (a) Stitching results of TPUT, The upper is the F, the center is R, and the lower is T. Stitching surface of (b) F, (c) R, (d) T. The stitching residual error, (e) F, (f) R, (g) T
Figure 9 shows the overlapping residuals of the surfaces and thickness. It can be observed that the overlapping residuals of the two surfaces (F, R) and thickness (T) are 0.2 nm RMS, almost the same as the added Gaussian noise. From the simulation results, MSuTA can accurately align the sub-apertures, and the simulation alignment accuracy is better than 0.2 nm RMS (Fig. 8(g)), thus enabling multi-dimensional high-precision stitching.
4. Experiment
4.1 Sub-aperture alignment based on the parallel plate self-interference
In MSuTA, accurate alignment is achieved by the phenomenon of parallel plate self-interference. Therefore, we conducted multiple groups of comparative experiments on the influence of adjusted aberrations on the surfaces and thickness of TPUT. In the experiment, the center area of the TPUT was adjusted for different tilt, and WTI was used to collect the surfaces and thickness data. Figure 10 shows the variation trend of the surfaces and thickness of TPUT under 12 different tilts. The front and rear surfaces exhibit the same amount of change under different relative tilts indicates that the displacement of the TPUT during the adjustment process is representative of rigid motion. Meanwhile, the distribution of thickness remains unchanged, which verifies that the thickness of the TPUT is not affected by adjustments.
Figure 11 shows the histogram of tilt parameters (a, b) in 12 measurements. It can be observed that the change of tilt calculated from the front and rear surface are the same. The tilt calculated from the thickness remains at 0.17/0.70 μrad, and the fluctuation range is less than 1.8/2.4 nrad. In conclusion, the above experimental results confirm the rigid rotation characteristics of TPUT during sub-aperture acquisition. They also verify the self-interference phenomenon of parallel plates in WTI, indicating that the thickness is not affected by adjustments. This feature can be utilized for high-accuracy alignment of sub-apertures.
4.2 Stitching experiment
The TPUT in the stitching experiment is 6.3-mm thick, 6 inches square thin parallel plate with three-point support tooling placed on a multi-dimensional motion stage with six degrees of freedom. The clear aperture is 144*144 mm. The TPUT is divided into 9 sub-apertures. Each sub-aperture has a width of 88 mm and an overlapping area of approximately 65% between adjacent sub-apertures. The front surface of each sub-aperture is adjusted to null-fringe through the manipulation of multi-dimensional stage, while a WTI with a resolution of 0.072 mm is utilized to gather surface and thickness data. Figure 12 shows the sub-aperture data without reference error.
Figure 13(a)-(d) shows the stitching results of the surfaces and thickness. The stitching results are smooth without obvious stitching error. Figure 13(e)-(h) shows the residual of the overlapping area of surfaces and thickness using MSuTA, which is obtained by stitching the residual of the overlapping area according to the actual position. It can be seen that the surfaces and thickness overlapping residuals of TPUT are 0.35 nm RMS and 0.14 nm RMS, respectively.
Fig. 13. Stitching results. (a) Three-dimensional sketch of the parallel plate, front, rear surface and thickness from top to bottom. (b)-(d) front, rear surface and thickness. (e) Three-dimensional sketch of the stitching residuals, front, rear surface and thickness from top to bottom. (f)-(h) the stitching residuals of front, rear surface and thickness.
The thickness stitching result is obtained by directly stitching the thickness in the sub-aperture data without calculation. The thickness stitching residual is only 0.14 nm RMS (Fig. 13(h)), which proves that MSuTA can achieve high sub-aperture alignment accuracy. MSuTA can achieves a stitching accuracy better than 0.35 nm RMS with high alignment accuracy, and significant aberration is not found in the residuals of overlapping region. In order to further compare the stitching accuracy, a 12-inch vertical interferometer with resolution of 0.3077 mm is used to obtain the full-aperture surfaces and thickness of TPUT by combining the improved six-step inversion method, as shown in Fig. 14.
Fig. 14. The measurement results using 12-inch vertical interferometer. (a) The measurement results of TPUT, The upper is the F, the center is R, and the lower is T. The surface of (b) F, (c) R, (d) T.
By comparing Fig. 13 and Fig. 14, the difference between the MSuTA and the full-aperture measurement using the 12-inch interferometer are less than 0.8 nm RMS, including surfaces and thickness. At the same time, we performed standard Legendre polynomial fitting on the data obtained by MSuTA and 12-inch interferometer respectively to compare the standard Legendre polynomial coefficients with different measurement methods. Figure 15(a) shows the first 36 standard Legendre polynomial coefficients obtained by fitting the surfaces and thickness of the TPUT, respectively. Figure 15(d) shows the difference between the two standard Legendre polynomial coefficients. It can be observed that the standard Legendre polynomial coefficients for both methods are similar (Fig. 15(a)). The biggest difference between the two methods is less than 2.5 nm (Fig. 15(b)), which may be caused by the system error of different interferometer, adjustment errors and support errors.
Fig. 15. Fitting results of standard Legendre polynomials. (a)The coefficient of standard Legendre polynomials, (b) the coefficient residual between MSuTA and 12inVI.
We also processed the surfaces in Fig. 13 and Fig. 14 to remove the figure error caused by the three-point support tooling, as shown in Fig. 16. From Fig. 16, we can more intuitively compare the differences of the two different methods. The surfaces of TPUT using the two methods are better than 3.2 nm RMS, and the thickness after removing the tilt is better than 5.10 nm RMS. The measurement results obtained using both MSuTA and the 12-inch interferometer are highly similar, both in value of the figure error and the distribution of the surface. The figure differences of 0.73 nm RMS between the surfaces acquired with the MSuTA and 12-inch interferometer. It is evident that MSuTA achieve high accuracy measurements of ultra-precision large-thin parallel plates with a surface accuracy superior to 3 nm and a thickness uniformity better than 5 nm. Moreover, compared to a 12-inch interferometer, MSuTA improves the lateral resolution of the measurement by nearly four times, allowing for a display of more comprehensive surface information.
Fig. 16. The measurement results after removing the deformation caused by the three-point support tooling. Three-dimensional sketch of the parallel plate, front, rear surface and thickness from top to bottom acquired with (a) the MSuTA and (e) 12-inch interferometer. The front, rear surface and thickness acquired with (b)-(d) the MSuTA and (f)-(h) 12-inch interferometer.
In order to evaluate the reproducibility of MSuTA, the measurement was repeated twice under the same conditions, in which the first and second measurement of the TPUT were rotated by 0°and 90° respectively. Figure 17 shows the results of two repeated measurements, including surfaces and thickness. From the results of the reproducibility experiment, the differences of 0.40 nm RMS between the surfaces and thickness acquired with the first and second measurement. Based on all the provided comparisons, it can be concluded that the proposed stitching method has the advantages of high accuracy and reproducibility.
Fig. 17. Stitching results of the TPUT were rotated by 0°and 90° respectively. The results of (a) front surface, (b) rear surface and (c) thickness after TPUT rotated 0 degrees are 2.81 nm RMS, 3.26 nm RMS and 5.05 nm RMS, respectively. The results of (d) front surface, (e) rear surface and (f) thickness of TPUT rotated by 90 degrees are 2.43 nm RMS, 3.11 nm RMS and 5.18 nm RMS, respectively
5. Conclusion
In this paper, we analyzed the principle of parallel plate measurement method using WTI in detail, studied the surface change during the measuring process under unconstrained support, and proposed stitching correction model of thin parallel plate. At the same time, a multidimensional stitching method using thickness alignment (MSuTA) is proposed, which can realizes high-accuracy alignment of sub-aperture through the feature that the thickness of the parallel plate without adjusted aberrations, and the multi-dimensional stitching measurement of the thin parallel plate combined with the modified stitching model. The stitching experiment was carried out on 6.3-mm thick, 6 inches square thin parallel plate, and the overlapping residuals was better than 0.34 nm RMS. Compared with the measurement results of 12-inch vertical interferometer, the deviation of surface and thickness is better than 0.8 nm RMS, and first 36 standard Legendre polynomial coefficients is better than 2.5 nm. Moreover, MSuTA improves the lateral resolution of the measurement by nearly four times, allowing for a display of more comprehensive surface information. In the future, we will further control the stability of the stitching process to improve the measurement accuracy.
Acknowledgments
The authors would like to thank Xi Wang, Liang Zhang and Bin Liu in the Optics and Metrology group of Institute of Optics and Electronics, Chinese Academy of Sciences for help in measuring advices.
Disclosures
The authors declare no conflicts of interest.
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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