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Abstract
Most tested surface calibration methods in interferometers, such as the direct coefficients removing method, the sensitive matrix (SM) method, and deep neural network (DNN) calibration method, rely on Zernike coefficients. However, due to the inherent rotationally non-symmetric aberrations in a non-null freeform surface interferometer, the interferograms are usually non-circular even if the surface apertures are circular. The Zernike coefficients based methods are inaccurate due to the non-orthogonality of Zernike polynomials in the non-circular area. A convolutional neural network (CNN)-based misalignment calibration method is proposed. Instead of Zernike coefficients, the well-trained CNN treats the interferogram directly to estimate the specific misalignments. Simulations and experiments are carried out to validate the high accuracy.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
One of the most popular approaches of optics surface metrology is interferometry, which has elegant performance in testing flat, spherical, aspherical and freeform surfaces [1–10]. The tested surface misalignment is the least desirable factor in interferometry. The real tested surface figure error would be covered by the uncalibrated misalignment aberrations. In general flat and spherical interferometry, the misalignments are easy to identify by the tested wavefront aberration evaluation [11], such as the first four terms of the Zernike coefficients. Direct coefficients removing would complete calibration. For aspherical and freeform surfaces, the calibration is relatively difficult, especially in non-null interferometers. Several calibration methods have been proposed. The most classic method in computer-aided alignment (CAA), sensitive matrix (SM) method [12–16], also known as the sensitive table method, establishes a corresponding relation between Zernike coefficients of the tested wavefront and surface misalignments. It has been the embedded algorithm in several commercial optical design software such as Code V. However, it works well only in the linear area owing to its matrix properties [17]. That is only slight misalignments in the linear area can be predicted accurately, which is not applicable for freeform surfaces because the complex surface figure and six degrees of freedom create a non-linear issue. Some other improved methods such as the merit function regression method [17] and differential wavefront sampling method [18] have been developed to treat the non-linear problem. However, thus far, these methods have been validated only for spherical optics. Some other methods address specific misalignments sequentially in a virtual interferometer (VI) with the assumption that the six misalignments were independent of each other [19–21]. However, this assumption is not always true in the case of general freeform surfaces. Recently, we employed a deep neural network (DNN) [22] to address the non-linear and error coupling issue, in which the misalignment estimation accuracy is validated higher than those in the SM and VI methods. However, as the same as most of the methods mentioned above, the DNN method relies on wavefront Zernike coefficients as inputs, which are accurate only in the circular aperture owing to the orthogonality of Zernike polynomials. As the input parameters of DNN, Zernike coefficients fitted in non-circular aperture would lose uniqueness and thus make the DNN inaccuracy. Most of optics are with circular aperture and thus result in circular interferograms, where the DNN-based calibration is still effective. However, in a non-null freeform surface interferometer, the circular aperture usually results in a non-circular interferogram due to the residual inherent aberrations. Several rotationally non-symmetric aberrations such as astigmatism and coma would introduce shape distortion of resulting interferograms. Therefore, Zernike coefficients-based DNN calibration would work in unsatisfactory accuracy and even result in misestimation of surface misalignments. Not only that, other Zernike coefficients-based calibration methods face the same dilemma.
In this paper, we propose a convolutional-neural-network (CNN)-based method to perform freeform surface misalignment aberration calibration in a non-null interferometer. Instead of Zernike coefficients, the well-trained CNN treats interferograms directly to estimate the specific misalignments with pixel-level accuracy. The induced aberrations are predicted and removed by model-based ray tracing with the estimated misalignments. The performances of CNN in estimating surface misalignments are discussed in general cases of non-circular interferograms. Simulations and experiments are carried out for validation.
2. Principle
In traditional non-null interferometers for the rotational symmetric aspherical surface test, the resulting interferograms are always circular even though in the case of large inherent aberrations. However, for freeform surfaces, the resultant interferograms are often non-circular due to the non-rotational symmetric aberrations. These aberrations would distort the interferogram shape, as is illustrated in Fig. 1. Figures 1(a) and 1(b) present two resulting interferograms of two freeform surfaces with circular apertures in a non-null interferometer. The sags of the two surfaces are in the form of astigmatism and trefoil aberrations, respectively. The non-rotational symmetric aberrations in the non-null interferometer make the resulting interferograms non-circular obviously. In these non-circular areas, Zernike coefficients lose orthogonality. A sufficient and necessary condition for linear independence of a group of polynomial basis functions in a special region is that the autocorrelation matrix of the polynomials is a real symmetric positive definite matrix [23]. For standard Zernike polynomials in a unit circle region, the autocorrelation matrix is a unit diagonal matrix. The autocorrelation matrices of the first 15 terms standard Zernike polynomials in the non-circular areas in Figs. 1(a) and 1(b) are evaluated. The results are shown in the form of images in Figs. 1(c) and 1(d), respectively. As is shown, both the autocorrelation matrices deviate from the unit diagonal matrix. The orthogonality is lost and thus the Zernike fitting solution for the corresponding phase would not be unique. Figures 1(e) and 1(f) illustrate the corresponding fitted Zernike coefficients in several operations, validating the description above. Therefore, the DNN-based calibration [22], with Zernike coefficients as inputs in training and working, would become imprecise or even wrong in misalignment prediction. Some methods for acquiring orthogonal polynomials in variable noncircular pupils [24–27] have been developed. However, for the region whose shape cannot be described by elementary functions, the orthogonal polynomials derivation is troublesome. Especially, with different misalignments, the shape of the interferogram may be various. It would be an impossible mission to acquire a vast accurate sample for DNN training. Therefore, a misalignment calibration method independent of Zernike coefficients is necessary for non-null freeform surface interferometers. The CNN rises in response, which has made important achievements in image-based classification, recognition, prediction, and decision-making [28–30]. It also has been widely employed in optics research such as optical lithography [31], imaging resolution enhancement [32], feature localization in holograms [33], light-field display [34], piston alignment [35,36], phase aberration compensation [37], and wavefront sensing [38]. In the CNN method, the misalignments would be estimated with interferograms directly.
Fig. 1. Non-circular interferogram characterizations for freeform surfaces in non-null interferometry. (a) and (b) are non-circular interferograms, (c) and (d) are autocorrelation matrixes of Zernike polynomials in the corresponding non-circular area, (e) and (f) are the different Zernike coefficients fitting results in the corresponding non-circular area.
The principle of the CNN-based estimation process is illustrated in Fig. 2. The CNN is trained with the tested surface misalignments as sample outputs and the induced interferograms as sample inputs. Then the actual interferogram is entered into the trained CNN to estimate specific misalignments. These misalignments are then substituted into the ray-tracing model to predict misalignment aberrations. The CNN generally includes the convolution layers, pooling layers, and fully connected layers. The convolution layers employ several convolution kernels, which characterize several features of the input map, to acquire feature maps with the image convolution operation. The elements in convolution kernels, as the operation weight, are just the training objects. That is the different map features are “learned” by the CNN itself. The pooling layers perform subsampling to reduce the computation burden, which mainly includes max-pooling and mean-pooling [30,39]. The fully connected layer is the parameter based neural network addressing the reshaped vector of the final feature maps. It is generally at the end of the CNN, where the outputs are misalignments. Compared with DNN proposed previously [22], CNN learns different local features by itself rather than the full aperture Zernike coefficients. CNN has better generalization ability because the sharing of convolution kernels reduces the number of parameters as well as the risk of over-fitting [40].
The CNN performance is affected by the convolution kernel sizes and numbers, convolution layer numbers, epochs, and fully connected layer numbers. There is no theory to determine the optimal structure of CNN. In this paper, we construct the CNN based on the famous VGG Net [39]. The original designed VGG Net has 5 groups of convolution layers and 3 fully connected layers, in which the 5 groups contain 8-16 convolution layers with the 3*3 convolution kernel. The convolution kernel number multiplies as the number of convolution layer groups with an initial 64. The ReLU function, acting as the active function, is performed in the final step of each convolution group. The max-pooling with 2*2 kernel is performed after each convolution group in the original VGG Net. Due to the large size of input interferograms, pooling kernels with 4*4 and 2*2 sizes are employed in turn. We expand these structures and evaluate its performance in different cases of convolution layer numbers and epochs, to find a relative optimal structure.
3. Simulation
For performance comparison, the tested surface in the simulation was the same as in DNN training in our previous work [22]. The difference is that the inputs are Zernike coefficients and interferograms in DNN and CNN, respectively. A sample dataset with 10,000 samples from the ray-tracing model was imported into the CNN for training while another test dataset with 10,000 samples was used to evaluate the performance of the trained CNN. Note that the interferograms in the sample dataset and test dataset are all circular. To simulate the actual situation, a map of figure error has been attached to the tested surface. Specifically, CNN was trained in the case of the nominal surface figure, and the misalignment estimation was executed with the surface figure error in the test dataset. With different convolution layer numbers and epochs, we evaluate the CNN performance. The total average relative error is defined as $\frac{\textrm{1}}{{\textrm{6n}}}\mathop \sum \nolimits_{i = 1}^{\textrm{6n}} |{({{y_{i\_\textrm{test}}}\textrm{ - }{y_{i\_\textrm{result}}}} )/{y_{i\_\textrm{test}}}} |, $ where ${y_{i\_\textrm{test}}}$ and ${y_{i\_\textrm{result}}}$ are the misalignments in the sample dataset and corresponding one estimated by CNN. The n is the sample number in the sample dataset (here n = 10,000). Figure 3(a) shows the performance of CNN in different convolution layer numbers (5-30) and epochs (1-100). Note that the structure of convolution layers grouping in the CNN follows the VGG Net when the total convolution layers number is smaller than 16, the largest layer numbers in the traditional VGG Net. If the total convolution layers number is larger than 16, newly added convolution layers would be inserted at the end of each convolution layer group in turn. From Fig. 3(a), we conclude the relative best performance occurs in CNN with 21 convolution layers and 43 epochs. The CNN training codes were implemented in MATLAB, which were executed on a computer with Intel i7-7700 CPU running at 3.6GHz and 16 GB RAM. The training time is about 51 hours in this case. The structure of the five convolution groups (the convolution layer numbers in five groups) is [3,3,5,5,5]. With the total 21 convolution layers, the performances of different grouping structures were evaluated, which is presented in Fig. 3(b). The average relative errors are all less than 0.45% in different structures with a total 21 convolution layers. Therefore, the structure [3,3,5,5,5] was proved an effective CNN structure with relatively high accuracy (0.37% relative error). Figure 3(c) presents the specific illustration of the structure in the CNN, with the sizes of the layers.
Fig. 3. The CNN performance in the sample dataset and selected structure. (a) Average relative errors in different convolution layer numbers and epochs, (b) Average relative errors with different convolution layer grouping. (c) The relative optimal CNN structure.
Then, the performance of the well trained CNN was evaluated, with the comparison with DNN. In the test dataset, specific misalignments were predicted by the well trained CNN and DNN, in both noise-free and noise cases. The average relative error of each kind of misalignment is shown in Fig. 4, which is defined as $\frac{\textrm{1}}{\textrm{n}}\mathop \sum \nolimits_{i = 1}^\textrm{n} |{({{y_{{i}\_\textrm{test}}}\textrm{ - }{y_{{i}\_\textrm{result}}}} )/{y_{{i}\_\textrm{test}}}} |$. For comparison with DNN, the noise-free dataset and noise dataset employed in CNN are the same as those in DNN. In the DNN method, the input parameters were Zernike coefficients and thus the noise was added by the Zernike coefficients as well [22]. That is the noise in the noise dataset is the low-frequency one. Figures 4(a) and 4(b) present the performance of CNN and DNN in the noise-free and the low-frequency noise cases, respectively. CNN has a good performance in misalignment estimation in these test datasets, as well as the DNN method. However, the inputs on CNN are interferograms and thus the most common noise is the high-frequency one. Therefore, Gaussian noises with about 0.1 NSR (noise-to-signal ratio, here σ/rms) were added on each interferogram in the test dataset, where σ is the standard deviation of the Gaussian noise and rms is the root-mean-square value of the interferogram. Figure 4(c) presents the performance in the Gaussian noise case, which illustrates the similar superior ability of CNN and DNN. Figure 4(d) shows the error variation of CNN with the increasing NSR of noise, where five results per noise level are plotted. Less than 1.2% relative error would be induced by the high-frequency noise even in the case of 0.5 NSR, which shows the ability of CNN in noise immunity.
Fig. 4. The average relative errors of DNN and CNN estimations in the test dataset from [20]. (a) Noise-free case, (b) Low-frequency noise case, (c) High-frequency noise case, (d) Error variations of CNN with the increasing NSR of noise, where five results per noise level are plotted.
Subsequently, the well trained CNN was employed for misalignment estimation in the case of non-circular interferograms in a non-null interferometer. Another group of datasets extracted from a non-null interferometer model for bi-conic surface metrology was entered into the CNN with above structure and DNN with a structure in [22] for training. The sizes of the example dataset and test dataset are the same as mentioned above. In the sample dataset and test dataset, the interferograms with elliptical shape acted as inputs in CNN and corresponding Zernike coefficients acted as inputs in DNN. After the training in the sample dataset, the performances of CNN and DNN in the corresponding test dataset are presented in Fig. 5 Fig. 5(a) presents average relative errors of specific misalignments in DNN and CNN, which shows that the average relative error of specific misalignments in DNN has a 5% increase while CNN has a 1% increase compared with those in circular interferograms (Fig. 4). That is CNN is 4% more accurate than DNN in misalignment prediction in non-circular interferograms. Figure 5(b) illustrates the regression analysis between the output in test dataset (test output) and DNN output while Fig. 5(c) illustrates those between the test output and CNN output. The CNN has a better performance than DNN.
Fig. 5. The performances of DNN and CNN in misalignment prediction. (a) The average relative errors of specific misalignments in DNN and CNN, (b) The regression analysis between the output in test dataset (test output) and DNN output. (c) The regression analysis between the test output and CNN output.
Even though the average relative error of the misalignment estimation of the DNN and CNN are both no more than 6%, it does not mean a real accurate result for a specific example because the 6% is an average value. The starting point of this thinking is that we found an additional potential error factor: misalignment aberrations coupling. The misalignment aberrations coupling means that the different misalignments combination induces similar aberration coefficients. Moreover, plus with the inaccurate Zernike coefficients of no-circular interferograms, there would be extremely close Zernike coefficients induced by two completely different misalignments, such as the tilt and decentration. In this case, the misalignment would be misestimated completely in DNN owing to the misalignment aberration coupling. Therefore, we evaluate the single misalignment identification performance in coupling cases in DNN and CNN, respectively. 600 samples were extracted with a single kind of misalignment, in which each kind of misalignment refers to 100 samples. We defined an estimation of more than 10% relative error as a misjudgment. The misjudgment ratios of the DNN and CNN for the six misalignments are illustrated in Fig. 6. The misjudgment ratio (relative error > 10%, black color piece) of CNN is much less than DNN in non-circular interferograms with misalignment aberrations coupling.
Fig. 6. The ratios of samples of different estimated errors in the total samples in the dataset with misalignment aberration coupling. (a) In the DNN method, (b) In the CNN method.
4. Experiment
Experiments were carried out to validate the CNN performance in misalignment calibration. The tested freeform surface, a deformable mirror (Alpao 97-25), was deformed to several freeform shapes to create different non-null interferograms in circular and non-circular areas. Note that the deformable mirror surface would creep in the experiment process. A closed-loop control was provided by a wavefront sensor, which ensures the surface stability over hours with no more than 5 nm rms variation. The system layout is illustrated in Fig. 7, with five tested freeform sags after retrace error correction. These sags are modeled in ray-tracing software to create the corresponding sample datasets. A near-flat lens was employed in the beam path to produce a predictable aberration (about 5 λ wavefront bending) or else the DM surface would be tested in a planar wave which makes the misalignments dx, dy, and dz have little influence to the resultant interferograms.
Fig. 7. Experiment design. (a) Experiment layout, (b) Freeform sags, where five sample datasets came from.
The resultant five interferograms in the non-null interferometer, responding to the five surface sags, are presented in Figs. 8(a)–8(e). Interferograms in Figs. 8(c)–8(e) are affected by rotational non-symmetric aberrations and presented in the non-circular region. Five corresponding misaligned interferograms are listed in Figs. 8(f)–8(j). Five groups of sample datasets with 10000 misaligned interferograms in each group extracted from the ray-tracing model were employed for CNN and DNN training. The specific misalignment ranges in the training dataset are listed in Table 1. The ranges of θx and θy are small than those of other misalignments because misalignment aberrations are very sensitive to θx and θy. The interferograms and corresponding Zernike coefficients were entered in CNN and DNN as training inputs, respectively. The total average relative error of the five trained CNNs and DNNs are listed in Table 2. It is obvious that the DNN and CNN both provide small training errors in case of a relative optimal structure.
Fig. 8. Aligned interferograms, misaligned interferograms, and misalignment estimation results by DNN and CNN, (a)–(e) are five aligned interferograms, (f)–(j) are five misaligned interferograms, (k)–(o) are misalignment estimation results by DNN and CNN.
Table 1. The specific misalignment ranges in the training dataset
Table 2. The total average relative error of the five trained CNNs and DNNs in sample datasets
With the well-trained CNNs and DNNs, Figs. 8(f)–8(j) and corresponding Zernike coefficients were imported as inputs, respectively. The estimation results are illustrated in Figs. 8(k)–8(o). It can be found that the misalignment estimation results by the DNN and CNN for Figs. 8(f) and 8(g) are almost the same, with little differences. It is because that the Figs. 8(f) and 8(g) are circular interferograms. Even the inherent aberration in Fig. 8(f) is relatively large, the resultant interferogram is still circular due to the rotational symmetry characteristic. However, relative large rotational non-symmetric aberrations in the non-null condition made the interferograms deviate from the circular shape, as is shown in Figs. 8(c)–8(e). Figures 8(m)–8(o) validate the different accuracy in misalignment estimation by the DNN and CNN.
To determine the misalignment estimation accuracy in Figs. 8(k)–8(o), we present two comparison forms. The first form is calibrating the freeform surfaces in the experiment by the precision adjusting mechanism. With the misalignments estimated [Figs. 8(k)–8(o)], the corresponding calibrated interferograms by DNN and CNN are presented in Figs. 9(a)–9(e) and Figs. 9(f)–9(j), respectively. These interferograms provide an intuitive illustration of the high calibration accuracy of CNN because Figs. 9(f)–9(j) show greater similarity with original interferograms [Figs. 8(f)–8(e)].
Fig. 9. Calibrated interferograms by DNN and CNN in the actual experiment. (a)–(e) are results by DNN and (f)–(j) are results by CNN-based calibration.
Limited by the accuracy of the adjusting mechanism in the experiment, Fig. 9 provides qualitative rather than quantitative comparison. Another more convincing quantitative comparison is presented in Fig. 10. The misalignments estimated by DNN and CNN, which are presented in Figs. 8(k)–8(o), were substituted into the system ray-tracing model to simulate the real misaligned interferograms. Then we compared the misaligned interferograms in the model and those in the experiment [Figs. 8(f)–8(j)]. The phase deviations are presented in Fig. 10, as the residual errors after calibration by the DNN [Fig. 10(a)–10(e)] and CNN [Fig. 10(f)–1(j)]. The specific parameters of these errors are listed in Table 3. In circular interferograms, the PV and rms differences of the residual errors by CNN and DNN are less than 0.14 λ and 0.004 λ, respectively. However, in non-circular interferograms, the residual errors by CNN-based calibration are much less than DNN-based calibration. We conclude that the CNN-based calibration has a similar performance in circular interferograms while higher accuracy in non-circular interferograms compared with the DNN-based calibration.
Fig. 10. The residual calibration errors (the phase deviations of real misaligned interferograms and simulated misaligned interferograms) by DNN [(a)–(e)] and CNN [(f)–(j)] based calibrations.
Table 3. Specific parameters of the residual error after calibration in Fig. 10
5. Conclusion
We proposed a CNN-based method for freeform surface misalignment calibration in non-null interferometers. Due to the inherent aberrations in non-null configurations, the interferograms usually are non-circular, which are distorted by non-rotational symmetric aberrations. The Zernike coefficients characterizing aberrations in these interferograms are thus inaccurate, which creates the inaccuracy of the DNN based calibration, as well as most other Zernike coefficients dependent methods. The CNN method addresses the interferograms directly and thus has higher accuracy and wider versatility. Simulations and experiments validate that the CNN-based calibration has similar accuracy in misalignment estimation in general cases but higher accuracy in non-circular interferograms than the DNN. As an enhanced calibration method, the CNN shows great potential in misalignment calibration for any freeform surface in interferometry.
Funding
National Natural Science Foundation of China (61705002, 61905001, 41875158); Natural Science Foundation of Anhui Province (1808085QF198, 1908085QF276); Opening project of the Anhui Province Key Laboratory of Non-Destructive Evaluation (CGHBMWSJC05); Opening project of the Key Laboratory of Astronomical Optics and Technology in Nanjing Institute of Astronomical Optics and Technology of the Chinese Academy of Sciences (CAS-KLAOT-KF201704); Doctoral Start-up Foundation of the Anhui University (J01003208); National Program on Key Research and Development Project of China (2016YFC0301900, 2016YFC0302202).
Disclosures
The authors declare no conflicts of interest.
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