Open Access
Add to BibSonomy
Abstract
In the adaptive freeform surface interferometer, the adaptive algorithms were equipped to find the required aberration compensation, making interferogram with dark areas (incomplete interferogram) sparse. However, traditional blind search-based algorithms are limited by convergence rate, time consumption, and convenience. As an alternative, we propose an intelligent approach composed of deep learning and ray tracing technology, which can recover sparse fringes from the incomplete interferogram without iterations. Simulations show that the proposed method has only a few seconds time cost with the failure rate less than 4‰. At the same time, the proposed method is easy to perform because it does not require the manual intervention of internal parameters before execution as in traditional algorithms. Finally, the feasibility of the proposed method was validated in the experiment. We believe that this approach is much more promising in the future.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The freeform surface, a promising optic, is usually used in frontier domains [1] due to its high freedom degree character. With the increasing demand for better-performing optical systems, large-scale production of freeform surfaces is pressing. However, it is limited by the high precision measurement on freeform surfaces. As the most popular method, interferometry provides elegant performance in traditional optical surface metrology. Various optical compensators were developed to generate suitable wavefronts to match the nominal surfaces or at least make the residual wavefront aberration in the measurable dynamic range of interferometers [2–4]. However, the freeform surface figure in fabrication changes gradually and has ambiguous parameters, which makes the design of traditional static compensators difficult to accomplish. Fortunately, adaptive optics elements (AOE) such as Liquid Crystal Spatial Light Modulator (LC-SLM) and Deformable Mirror (DM) can act as flexible compensators, enabling unknown freeform surface online measurement. The adaptive control algorithm is equipped to iteratively find the required aberration compensation to recover sparse fringes from the interferogram with dense fringes and even dark areas (incomplete interferogram) due to the large departure [5]. The stochastic parallel gradient descent (SPGD) algorithm [5,6] was first employed, which cost about 6 min for the ∼15µm surface departure. However, Monte Carlo simulation shows that the SPGD algorithm tends to fall into local minima and the non-convergence rate is about 21% [7]. Thus, Genetic algorithm (GA) based on global searching was employed and the non-convergence rate was reduced to about 0.6%. However, it takes about 10 times longer than the SPGD algorithm. It means that the metrology efficiency will be severely affected, as a larger departure can rapidly increase the time cost. In 2021, we proposed the simulated annealing-hill climbing (SA-HC) algorithm [8] to alleviate the contradiction between time cost and non-convergence rate to some extent. However, the speed advantage is still not obvious compared to the SPGD algorithm. In 2022, we proposed adaptive moment estimation (Adam) SPGD algorithm [9], which achieves about 1% non-convergence rate with a speedup of 37% compared to the SPGD algorithm. Despite the superior performance, the adaptation process is inconvenient as the performance metric needs to be modified at different steps. Crucially, the proposed Adam algorithm as a blind optimization algorithm still requires human intervention on the intrinsic parameters before the operation, and the time cost typically increases with the surface departure as well. Figure 1 shows the performance of these traditional blind optimization algorithms for ∼ 15 µm departure surfaces. It is clear that the aforementioned traditional adaptive control algorithms are limited by convergent rate, time consumption, and conveniences.
Convolution Neural Networks (CNN) as a data-driven technology provides an alternative manner with time-saving and high accuracy in recent years, such as object detection [10], semantic segmentation [11], denoising [12], phase unwrapping [13] and freeform surface misalignment prediction [14]. Some adaptive algorithms based on CNN also focus on wavefront correcting [15–17]. In [15], Ma et al. employ the revised AlexNet [18] to directly predict wavefront aberration due to turbulence from two intensity images, which simplifies the process of wavefront recovery in adaptive compensation. In [16], Jin et al. applied CNN to extract the low-order aberration from distorted PSF (point sparse function) images to achieve high-speed wavefront aberration corrections in biomedical imagining. In [17], Hu et al. proposed a learning-based SHWS (Shack-Hartman wavefront sensor) to improve the wavefront sensing ability of SHWS, which has good compatibility with an adaptive optics system. However, unlike adaptive freeform interferometer (AFI), these smart algorithms are typically faced with in- and out-of-focus images, PSF images, or SHWS data with minor aberration rather than incomplete interferograms caused by large aberration. Hence, it does not apply directly to AFI. Even if grafted onto the AFI, the additional optics required, such as beam splitter and CCD or SHWS, increase the cost and the system complexity.
Fortunately, the high accuracy phase demodulation for single interferogram based on CNN [19–21] inspires us, which can obtain accurate phase information almost in real-time from the complete null interferogram captured by Zygo interferometer. If the CNN can also extract the wavefront phase from a single incomplete interferogram with complex aberration, the unknown surface figure would be approximately estimated by ray tracing. The required aberration compensation of unknown surface can then be rapidly obtained using ray tracing. After it is loaded to AOE, the incomplete interferogram will be recovered to sparse, and even to near-null. Therefore, instead of relying on traditional blind optimization algorithms, we employ a combination of CNN and ray tracing technology to accelerate dark regions recovery in this paper, where the CNN is employed to predict the wavefront phase from a single incomplete interferogram for the first time. To compare the blind algorithms in AFI, the proposed method has unprecedented performance.
2. Principle
2.1 Adaptive process in AFI
Although several kinds of AFI [5,8,22–24] have been proposed in recent years, the adaptive progress is all the same. A relatively simple structure based on the reflection-type AOE is shown in Fig. 2 as a sample for illustration. This AFI is designed using the principle of polarization. It is composed of a commercial interferometer, polaroid, quarter wave plate (QWP), polarized beam splitter (PBS), reflection-type AOE, and partial null lens (PNL), in which the PNL can only compensate rotational symmetry aberration such as defocus or spherical aberration. In the first, the incident beam from the interferometer is changed to a p-polarizing one after passing through the polaroid. Next, p-polarizing will all transmit PBS, and it becomes circular polarizing via QWP, with a 45° between its polarizing direction and the fast axis of QWP. Subsequently, the circular polarizing is changed to s-polarized after the reflection by the AOE and successive passes through the QWP. It will all be reflected by the PBS and traveling through the PNL to reach the freeform surface. The beam carrying the surface information is then reflected by the freeform surface and travels back to the interferometer and interferes with the reference beam.
As we all know, the initial interferograms without aberration compensation are generally existing dark areas in large departure freeform metrology. For traditional adaptive algorithms, incomplete interferograms with real-time acquisition are fed to the algorithm and the output is then loaded into the AOE to iteratively find a suitable aberration compensation. Figure 2 bottom shows the interferogram change in the iterative procedure. It can be seen that the adaptive algorithm is executed in three steps to obtain sparse fringes. Each step has a different optimization objective. The first step was used to remove dark regions, which indicates the pixel number (Np) of dark areas down to 0 (Np = 0). The interferogram is very dense after this step, and it is still not demodulated by the interferometer due to the noise or external vibrations. In this way, the second step is used to sparse the dense fringes. In the end, the third step of the algorithm is operated to obtain near-null interferograms to reduce the effect of retrace errors. In contrast, the proposed method can directly obtain sparse fringes and even near-null fringes from the initial interferogram without iterations. It is worth emphasizing that the unknown freeform surface can already be tested in non-null after obtaining sparse fringes, and the near null compensation (third step) is easy to complete, such as loading the conjugate phase of the non-null test result onto the AOE [24]. Hence, in this paper we focus only on the process of going from incomplete interferogram to sparse fringes.
2.2 Principle of proposed method
In the case of unknown freeform surface metrology, the initial interferogram is usually incomplete in the full aperture before AOE provides aberration compensation. Therefore, the complete test wavefront ${W_{\textrm{real}}}$ is unavailable and it can be expressed by an implicit function as
The schematic flow chart of the proposed method is shown in Fig. 3(a). Real incomplete interferograms need to be preprocessed first before CNN prediction. The preprocessing includes dark areas (indistinguishable fringe areas) recognition by MV (Machine Vision) and dark areas gray value modification. According to [7], the dark areas mask can be obtained after median filtering, threshold segmentation, region filling and opening operation on the incomplete interferogram with real-time acquisition. Then the dark areas in incomplete interferogram would be recognized by using this mask, and subsequently its gray value was substituted to 1. The preprocessed interferogram as input is fed to the trained CNN. The corresponding 15 Zernike coefficients (excluding the first term) of test wavefront as output would be obtained. That is, the approximate test wavefront ${W_{\textrm{predicted}}}$ was acquired, i.e.,
Fig. 3. Diagram of the proposed method. (a) Proposed method process, (b) two conditions after compensation.
Combining Eq. (1) and Eq. (2), the estimated freeform surface figure $S{^{\prime}_{\textrm{UFS}}}$ can be gotten, as follows
In practice, the Eq. (3) was implemented by ray tracing, in which the optimized variate and objective are $S{^{\prime}_{\textrm{UFS}}}$ (fitted by Zernike Sag) and ${W_{\textrm{predicted}}}$, respectively. At the same time, the ${c_{\textrm{AOE}}}$ and ${E_{\textrm{system}}}$ should be tested in advance and stored in the model to improve the modeling accuracy. Noted that the “twice relation” cannot be used to estimate unknown freeform here because it is ignoring the effect of ray propagation. Subsequently, the $S{^{\prime}_{\textrm{UFS}}}$ was used to set up the function for obtaining the needed aberration compensation $c{^{\prime}_{\textrm{AOE}}}$, defined as follows
The $c{^{\prime}_{\textrm{AOE}}}$ was solved from Eq. (4) using ray tracing as well. In this ray tracing, the Zernike coefficients of AOE surface in the model are set to variable to make the system residual aberration down to 0 or minimum. After the compensation $c{^{\prime}_{\textrm{AOE}}}$ loading to real AOE, there are two possible outcomes condition A and condition B, as shown in Fig. 3(b). Condition A is that the incomplete interferogram was successfully recovered to sparse, which indicates the new pixel number Np’ of dark areas down to 0. The other one, condition B is that the dark areas get larger due to the wrong sign of compensation. It is well known that the wavefront signature cannot be determined from a single interferogram. If the sign of the predicted wavefront is opposite to the truth, the sign of the obtained compensation will be wrong. In this problem, the aberration compensation $c{^{\prime}_{\textrm{AOE}}}$ for loading to the AOE is modified to $- c{^{\prime}_{\textrm{AOE}}}$ to solve it. The time cost can be neglected in the modified compensation procedure.
According to the principle of the proposed approach, the accuracy of the estimated aberration compensation determines whether the incomplete interferograms can be successfully recovered. Theoretically, it suffers from the modeling error and the CNN prediction error. Modeling errors are imposed by element structure parameter errors and misalignment errors. But it can be tested in advance and stored in the model, either for DM- or SLM-based AFI. It is worth noting that the relatively large compensation error is allowed in the fringe sparse task. The modeling error after calibration can be ignored as it is less than 0.01 λ rms [22,25–27]. Hence, the other one, CNN prediction error becomes the dominating factor affecting the incomplete recovery. It will be tested and analyzed in Section 3.3.
2.3 Neural network
2.3.1 Dataset
The following mathematical expression is applied to generate complex interferograms with large aberration
where v denotes the intensity ratio of reference beam to testing beam, ${I_{\textrm{ref}}}(x,y)$ is the reference beam intensity, $\varphi (x,y)$ is the wavefront phase. These parameters all need to be set to variables for the numerous samples producing. In the first, the intensity ratio $v$ was random selected from the range of [0.2, 1] to guarantee the clearly contrast fringe acquired.The Zernike polynomials is applied to simulate testing wavefront as follow
where ${Z_i}(x,y)$ is the description of aberration types in a unit circle, ${C_i}$ is the corresponding amplitude. The first term is usually not considered because it represents a constant. The 2-4ord as the low-frequency component denotes alignment aberration. The 5-9th order is the low- and mid-frequency component, which is the dominant component in test wavefront of large departure freeform [24] because the commonly used freeform surfaces are low order aberration surfaces in reality. The 10-16th order is the mid-frequency aberration. The 17-37th order are considered to be mid- and high-frequency aberrations, which are usually only a small amount in the wavefront. These coefficients are set to be variable to randomly generate testing wavefront distribution, and the amplitude range of each Zernike coefficient is listed in Table 1. In this amplitude range, the maximum wavefront PV can be up to ∼ 70 λ which is sufficient for the large local deviation encountered in practical processing. In addition, it is worth noting that $\varphi (x,y)$ and $- \varphi (x,y)$ has the same interference fringe. The training loss of the neural network will not converge in the face of this ambiguity sign. From this, we set the defocus range to be greater than 1 to avoid this problem.
Table 1. Range of Zernike coefficients
In addition, the Gaussian beam was used to simulate various reference beams, which can be described as follows
where the a represents amplitude, w is the waist radius, ${x_0}$ and ${y_0}$ represents the offset of beam center in x and y directions respectively. These parameters were well studied in [21,28]. Hence, w, ${x_0}$ and ${y_0}$ are randomly selected from the range [0.8, 1], [-0.15, 0.15] and [-0.15, 0.15], respectively. And the a randomly selected from the following expression [21] asAt the same time, we add Gaussian noise with about 20 signal-to-noise ratio (SNR) to the reference beam and use the Gaussian filter subsequently to make the simulated fringe very close to real ones [20].
In the end, the dark region should be simulated. Empirically, the interference region in practice becomes indistinguishable when the Nyquist frequency exceeds 0.45 λ/pixel. However, these areas are still distinguishable or become moire fringe in simulation [24]. This means that it is very difficult for these regions to be generated exactly like reality. Fortunately, these regions in experiment are easily found by MV. This allows us to generate samples that are consistent with the real preprocessed interferogram. Therefore, regions with more than 0.45 λ/pixel in the simulated interferograms are assigned a value of 1 to simulate these regions.
In summary, sample production can be divided into two parts. One is the generation of distinguishable areas (fringe areas). We employ studied variables [20,21,28] including wavefront phase $\varphi (x,y)$, reference light intensity ${I_{\textrm{ref}}}(x,y)$, light intensity ratio v and noise to simulate the fringe very close to the real one. Another is the generation of indistinguishable regions. The gray value of indistinguishable regions is fixed and set to 1. These settings ensure that the samples are similar to the real preprocessed interferograms. There are 55000 pairs of data including the preprocessed interferograms and the corresponding 2∼16 Zernike coefficients were generated for CNN training, in which 50000 and 5000 pairs were used as the training set and validate set.
2.3.2 Network architecture and training
As the “Extreme Inception” model, Xception [29] was employed to obtain the dominant aberration from an incomplete interferogram, which had better performance in feature extraction recently. The size of input and output was revised to 256 × 256 × 1 and 15 × 1 to adapt our task, as shown in Fig. 4. The framework backbone consists of 14 models, which was divided into entry layers, middle layers and exit layers. To differentiate previous Inception (V1-V4) [30–33], the depth wise separable convolution was introduced in 2-14 modules so that cross-channel correlation and spatial correlations of feature map can be entirely decoupled. As a result, its reasoning power continues to improve with almost no increase in parameters. Meanwhile, the residual connections between models can accelerate model convergence.
During the CNN training, mean squared error (MSE) was set as loss function as follows
where ${N}$ is the number of items, in there $N = 16$, ${C_i}^\prime$ and ${C_i}$ are the $i\textrm{ th}$ coefficients of prediction and truth respectively. In backpropagation, the Adam optimizer is used to constantly update the network parameters. The CNN was trained for a total of 9 epochs, with 3125 iterations per epoch. Simultaneously, the variable learning rate strategy was adopted, and the learning rate decreases by 0.1 every 4 epochs and initial is 1 × 10−4. The CNN training code was implemented in MATLAB 2021b in Linux 7.5, using an Intel Xeon Gold 6348 CPU with RAM 512GB and two GPU cards with NVIDIA A100 for speedup.3. Simulation
After CNN training, the performance of trained CNN and proposed method were estimated in simulation, which are running in Desktop PC with 11th Gen Intel Core i5-11400 @ 2.60 GHz and 8GB of RAM.
3.1 Network performance
Four simulated interferograms with different amounts of aberration were generated to validate the performance of the trained CNN. The total of aberration PV of interferograms in Figs. 5(a)–5(d) are 35.6 λ, 44.7 λ, 54.7 λ and 66.8 λ, respectively. The trained CNN was used to extract 2∼16 Zernike coefficients from four samples and the corresponding output were plot (orange line) in Figs. 5(a)–5(d), respectively. As a comparison, we also employ the method that direct fitting incomplete wavefront after interferogram demodulation (DFAD) to obtain corresponding zernike coefficients, and the results were also plotted (green line) in Figs. 5(a)–5(d), respectively. It is apparent that the network output is almost the same as the truth (purple line), but direct fitting outcome differs greatly from the truth. At the same time, the mean time cost of CNN prediction is only 0.1 s. Hence, the CNN introduced is essential to extract aberration information from an incomplete interferogram.
Fig. 5. Simulated interferograms and corresponding Zernike coefficients with truth, CNN prediction and DFAD. (a) Sample 1(PV = 35.6 λ), (b) Sample 2(PV = 44.7 λ), (c) Sample 3(PV = 54.7 λ), (d) Sample 4(PV = 66.8 λ).
3.2 Proposed method validation
We employed the combination of Matlab and Zemax to verify the proposed method. The “Simulation model” was first built in Zemax, the same as Fig. 1 but without PNL. There are two incomplete interferogram caused by unknown freeform suface with different departures, in which the NP are 8914 and 14005, respectively.
Figure 6(a) shows the initial interferogram of the first unknown surface. These dark areas in the initial interferogram were assigned to 1 as the preprocessed interferogram, as shown in Fig. 6(b). Then the trained CNN was used to extract the corresponding 2∼16 Zernike coefficient form Fig. 6(b), which was executed in Matlab. Figure 6(c) shows the predicted wavefront. For estimating the freeform figure, the 5∼9 Zernike coefficients of tested surface and CNN predicted result are the optimized variate and objective in ray tracing, respectively. The obtained freeform surface figure was shown in Fig. 6(d). Subsequently, the optimized variable and objective were modified the 2∼16 Zerinke coefficients of AOE surface and the system residual rms = 0, respectively. The approximated aberration compensation can be effortless determined in ray tracing again, as shown in Fig. 6(e). The incomplete interferogram was successfully recovered to sparse after compensation, as shown in Fig. 6(f). In this procedure, the time cost is only 4.21 s including CNN predicting 0.1 s and ray tracing 4.11 s. As a comparison, the Adam-SPGD that most fast in traditional algorithms was also employed to sparse the interferogram in Fig. 6(a). First and second steps were shown in Fig. 6(g), and the J denotes the number of pixels in the highly dense fringe region in the operation of second step [9]. The time consumed by this procedure is up to 337.70 s (5.63 min).
Fig. 6. Sample 1 recovery procedure. (a) Initial interferogram without compensation, (b) preprocessed interferogram, (c) predicted testing wavefront, (d) the estimated unknown freeform surface figure, (e) the estimated aberration compensation, (f) the obtained sparse interferogram using the proposed method, (g) the Adam-SPGD algorithm process.
Another incomplete interferogram caused by the second unknown surface is shown in Fig. 7(a). The recovery procedure is the same as the last. The preprocessed interferogram is shown in Fig. 7(b) and CNN can extract the corresponding Zernike coefficient from it. The predicted wavefront is shown in Fig. 7(c). The unknown surface and aberration compensation can be approximately obtained through ray tracing, as shown in Fig. 7(d) and Fig. 7(e), respectively. The incomplete interferogram was successfully recovered after compensation, as shown in Fig. 7(f). This recovery process only cost 4.61s containing CNN predictions of 0.1 s and twice ray tracing of 4.51 s. The incomplete interferogram in Fig. 7(a) was also sparse by the Adam-SPGD algorithm, as shown in Fig. 7(g). J here has the same meaning as J in Fig. 6(g). The time consumption of the first two steps is up to 413.05 s (6.88 min).
Fig. 7. Sample 2 recovery procedure. (a) Initial interferogram without compensation, (b) preprocessed interferogram, (c) predicted testing wavefront, (d) the estimated unknown freeform surface figure, (e) the estimated aberration compensation, (f) the obtained sparse interferogram using the proposed method, (g) the Adam-SPGD algorithm process.
In summary, the feasibility of the proposed method is verified in this section. According to the mean time cost in two samples recovery, the proposed method is about 84 times faster than the fastest Adam-SPGD in traditional algorithms. Also, the time consumption of the proposed method is insensitive to the growth of Np. Meanwhile, the interferogram recovered by the proposed method is only a few fringes. Table 2 illustrates the comparison between the Adam-SPGD algorithm and the proposed method for recovering two samples with different Np. It can be seen that the advantage of the proposed method is more attractive. Noted that in the operation of the proposed method, Matlab and Zemax were linked by humans. Of course, writing macro commands for Matlab to communicate with Zemax allows the proposed method to be automated in the future.
Table 2. Comparison between Adam-SPGD algorithm and the proposed method
3.3 Uncertainty analysis
As mentioned in Section 2.1, the success rate of the proposed method is determined by the CNN prediction accuracy. Therefore, we used the Dataset method to generate a total of 5000 samples to test the trained CNN performance for estimating the uncertainty of the proposed method. The Np distribution of samples is shown in Fig. 8(a), in which the mean Np is about 1/5 of the total pixel number. The residual wavefront root means square error (rmse) was recorded and plotted in Fig. 8(b). Noted that the interferogram recovery task has a high tolerance to the error of the estimated aberration compensation. Hence, the relatively large predicted error about the CNN are permitted. We define the recovery of samples with prediction rmse over 1 λ as unsuccessful because the rmse less than 1 λ is enough to recover incomplete interferogram in our experience. Of course, samples with prediction rmse over 1 λ were still be recovered possible in practice. From this, the failure rate of the proposed method is less than 4‰.
4. Experiment validation
An experimental system identical to the simulation is set up to validate the proposed method, as shown in Fig. 9. The reflected AOE is DM (Alpao 97-25), which indicates 97 actuators behind the 25 mm mirror. The laser interferometer is Zygo verifire with a wavelength of 632.8 nm. The unknown freeform surface was formed by slightly squeezing a low-order aberration metal reflector of 25 mm aperture, which has a relatively medium departure from a flat sheet. In fact, the dominant aberration of the unknown freeform surface (pressed surface) is still low-order aberration, which is very similar to the unknown freeform surface in process. The accuracy modeling was implemented in advance.
The experimental validation process was shown in Fig. 10. Figure 10(a) shows the initial interferogram without compensation, in which the Np is about 11267. Firstly, the initial interferogram in Fig. 10(a) was processed as the trained CNN input, as shown in Fig. 10(b). The predicted wavefront and corresponding interferogram were presented in Fig. 10(c) and Fig. 10(d), respectively. Similar to simulation, the freeform surface figure and the required aberration compensation can be effortless estimated sequentially by ray tracing in the model system, as shown in Fig. 9(d) and Fig. 10(e). Then the calculated compensation was loaded to DM by voltage matrix and the incomplete interferogram in Fig. 10(a) is successfully recovered, as is shown in Fig. 10(g). The time cost in running the code and ray tracing is only about 5.50 s in total. We believe that previous traditional algorithms could not achieve such a rapid recovery speed.
Fig. 10. Experiment validation. (a) Initial interferogram without compensation, (b) preprocessed interferogram for input into CNN, (c) predicted testing wavefront, (d) the corresponding interferogram of predicted wavefront, (e) the estimated unknown surface using ray tracing, (f) the estimated aberration compensation using ray tracing, (g) the sparse interferogram after compensation.
5. Discussion
The proposed method makes the unknown freeform measurement from “Adaptive” to “Intelligent”. In the past, ray tracing as an efficient method was not suitable to calculate required aberration compensation due to the complete testing wavefront being unavailable. Today, the introduced CNN makes it possible. The proposed method has been validated in simulation and experiment. Its advantages over blind algorithms are fascinating. However, this approach still has several limitations that required further improvement, as follows.
Firstly, the production of the training datasets is a compromise because the dark areas of real interferograms are nearly impossible to simulate. Thus, these regions need to be recognized and reassigned before CNN prediction, which lead to a little bit cumbersome. At the same time, the deviation between the real preprocessed interferogram and the training set will also cause the prediction error, which can increase the estimated compensation error. Hence, it would be helpful to use real incomplete interferograms as datasets to train CNN to further improve the efficiency and performance of this method. Secondly, it can be seen that the time cost of the proposed method is mainly concentrated on the twice ray tracings. The ray tracing time also generally increases with the system complexity. Hence, deep learning-based ray tracing is required to continue reducing time consumption. Finally, in order to avoid extreme conditions in optical shops, such as local departure over large or CNN prediction faults, the combination of the proposed method and the Adam-SPGD algorithm can be considered to guarantee measurement efficiency. Of course, this issue is a small probability.
In the future, a more intelligent neural network in DM-based AFI is needed. The network input and output are incomplete interferogram acquired in real-time and actuators voltage of DM, respectively. It means that the incomplete interferogram can be recovered instantaneously. This is our future research direction.
6. Conclusion
Finding the required aberration compensation is necessary in adaptive freeform interferometer. Different from the traditional blind algorithm, we propose an intelligent approach to obtain approximate aberration compensation without iterations, making the indistinguishable interferograms rapidly sparse. Simulations show that the proposed method requires only a few seconds to obtain sparse fringes from incomplete interferogram, which is much faster than the fastest Adam-SPGD in the traditional blind algorithm. The time cost is also insensitive to the growth of the dark areas. Meanwhile, the failure rate of the proposed method is less than 4‰ in simulation analysis. What is more, the proposed method also eliminates the need for human intervention in the internal parameters before operation. In the end, the feasibility of the proposed method was further validated in the experiment. We believe that the proposed intelligent method has a self-evident potential in unknown freeform measurement.
Funding
National Natural Science Foundation of China (61705002, 52275515).
Acknowledgments
The authors appreciate the AHU-HPC for providing the computational resources.
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.
References
1. J. P. Rolland, M. A. Davies, T. J. Suleski, C. Evans, A. Bauer, J. C. Lambropoulos, and K. Falaggis, “Freeform optics for imaging,” Optica 8(2), 161–176 (2021). [CrossRef]
2. A. Offner, “A Null Corrector for Paraboloidal Mirrors,” Appl. Opt. 2(2), 153–155 (1963). [CrossRef]
3. J. C. Wyant and P. K. O’Neill, “Computer generated hologram; null lens test of aspheric wavefronts,” Appl. Opt. 13(12), 2762–2765 (1974). [CrossRef]
4. J. E. Greivenkamp and R. O. Gappinger, “Design of a nonnull interferometer for aspheric wave fronts,” Appl. Opt. 43(27), 5143–5151 (2004). [CrossRef]
5. L. Huang, H. Choi, W. Zhao, L. R. Graves, and D. W. Kim, “Adaptive interferometric null testing for unknown freeform optics metrology,” Opt. Lett. 41(23), 5539–5542 (2016). [CrossRef]
6. Y. Zhang, X. Tian, and R. Liang, “SPGD and Newton iteration mixed algorithm used in freeform surface metrology,” Optics and Lasers in Engineering 129, 106050 (2020). [CrossRef]
7. S. Xue, W. Deng, and S. Chen, “Intelligence enhancement of the adaptive wavefront interferometer,” Opt. Express 27(8), 11084–11102 (2019). [CrossRef]
8. L. Zhang, R. Liu, J. Wu, Z. Cheng, S. Zhou, J. Li, and B. Yu, “Freeform surface adaptive interferometry assisted with simulated annealing-hill climbing algorithm,” Measurement 181, 109597 (2021). [CrossRef]
9. J. Wu, C. Hu, R. Liu, S. Wu, J. Cao, Z. Cheng, B. Yu, and L. Zhang, “Adam SPGD algorithm in freeform surface in-process interferometry,” Opt. Express 30(18), 32528–32539 (2022). [CrossRef]
10. J. Redmon, S. Divvala, R. Girshick, and A. Farhadi, “You only look once: Unified, real-time object detection,” in Proceedings of the conference on computer vision and pattern recognition (IEEE, 2016), pp. 779–788.
11. O. Ronneberger, P. Fischer, and T. Brox, “U-Net: Convolutional Networks for Biomedical Image Segmentation,” (Springer International Publishing, 2015), pp. 234–241.
12. K. Yan, Y. Yu, T. Sun, A. Asundi, and Q. Kemao, “Wrapped phase denoising using convolutional neural networks,” Optics and Lasers in Engineering 128, 105999 (2020). [CrossRef]
13. J. Zhang, X. Tian, J. Shao, H. Luo, and R. Liang, “Phase unwrapping in optical metrology via denoised and convolutional segmentation networks,” Opt. Express 27(10), 14903–14912 (2019). [CrossRef]
14. L. Zhang, C. Li, S. Zhou, J. Li, and B. Yu, “Enhanced calibration for freeform surface misalignments in non-null interferometers by convolutional neural network,” Opt. Express 28(4), 4988–4999 (2020). [CrossRef]
15. H. Ma, H. Liu, Y. Qiao, X. Li, and W. Zhang, “Numerical study of adaptive optics compensation based on Convolutional Neural Networks,” Opt. Commun. 433, 283–289 (2019). [CrossRef]
16. Y. Jin, Y. Zhang, L. Hu, H. Huang, Q. Xu, X. Zhu, L. Huang, Y. Zheng, H. L. Shen, W. Gong, and K. Si, “Machine learning guided rapid focusing with sensor-less aberration corrections,” Opt. Express 26(23), 30162–30171 (2018). [CrossRef]
17. L. Hu, S. Hu, W. Gong, and K. Si, “Learning-based Shack-Hartmann wavefront sensor for high-order aberration detection,” Opt. Express 27(23), 33504–33517 (2019). [CrossRef]
18. A. Krizhevsky, I. Sutskever, and G. E. Hinton, “ImageNet Classification with Deep Convolutional Neural Networks,” in International Conference on Neural Information Processing Systems (2012), 1097–1105.
19. X. Liu, Z. Yang, J. Dou, and Z. Liu, “Fast demodulation of single-shot interferogram via convolutional neural network,” Opt. Commun. 487, 126813 (2021). [CrossRef]
20. S. Yuan, Y. Hu, Q. Hao, and S. Zhang, “High-accuracy phase demodulation method compatible to closed fringes in a single-frame interferogram based on deep learning,” Opt. Express 29(2), 2538–2554 (2021). [CrossRef]
21. Y. Sun, Y. Bian, H. Shen, and R. Zhu, “High-accuracy simultaneous phase extraction and unwrapping method for single interferogram based on convolutional neural network,” Optics and Lasers in Engineering 151, 106941 (2022). [CrossRef]
22. L. Zhang, S. Zhou, D. Li, Y. Liu, T. He, B. Yu, and J. Li, “Pure adaptive interferometer for free form surfaces metrology,” Opt. Express 26(7), 7888–7898 (2018). [CrossRef]
23. L. Zhang and Y. Zhang, “Freeform surface interferometry with an adaptive ring-cavity compensator,” Surf. Topogr.: Metrol. Prop. 8(2), 025036 (2020). [CrossRef]
24. S. Xue, S. Chen, Z. Fan, and D. Zhai, “Adaptive wavefront interferometry for unknown free-form surfaces,” Appl. Opt. 26(17), 21910–21928 (2018). [CrossRef]
25. D. Liu, T. Shi, L. Zhang, Y. Yang, S. Chong, and Y. Shen, “Reverse optimization reconstruction of aspheric figure error in a non-null interferometer,” Appl. Opt. 53(24), 5538–5546 (2014). [CrossRef]
26. L. Zhang, D. Liu, T. Shi, Y. Yang, and Y. Shen, “Practical and accurate method for aspheric misalignment aberrations calibration in non-null interferometric testing,” Appl. Opt. 52(35), 8501–8511 (2013). [CrossRef]
27. L. Zhang, D. Liu, T. Shi, Y. Yang, S. Chong, B. Ge, Y. Shen, and J. Bai, “Aspheric subaperture stitching based on system modeling,” Opt. Express 23(15), 19176–19188 (2015). [CrossRef]
28. Q. Zhang, S. Lu, J. Li, D. Li, X. Lu, L. Zhong, and J. Tian, “Phase-shifting interferometry from single frame in-line interferogram using deep learning phase-shifting technology,” Opt. Commun. 498, 127226 (2021). [CrossRef]
29. F. Chollet, “Xception: Deep learning with depthwise separable convolutions,” in Proceedings of the conference on computer vision and pattern recognition (IEEE, 2017), pp. 1251–1258.
30. C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich, “Going deeper with convolutions,” in Proceedings of the conference on computer vision and pattern recognition (IEEE, 2015), pp. 1–9.
31. S. Ioffe and C. Szegedy, “Batch normalization: Accelerating deep network training by reducing internal covariate shift,” in International conference on machine learning, PMLR37 (2015), 448–456.
32. C. Szegedy, V. Vanhoucke, S. Ioffe, J. Shlens, and Z. Wojna, “Rethinking the inception architecture for computer vision,” in Proceedings of the conference on computer vision and pattern recognition (IEEE, 2016), pp. 2818–2826.
33. C. Szegedy, S. Ioffe, V. Vanhoucke, and A. A. Alemi, “Inception-v4, inception-resnet and the impact of residual connections on learning,” in Thirty-first conference on artificial intelligence (AAAI, 2016), pp. 4278–4284.
