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Abstract
The adaptive wavefront interferometer (AWI) we have reported recently is utilized to test in-process surfaces with severe surface figure error which is beyond dynamic range of conventional interferometers [S. Xue, S. Chen, Z. Fan, and D. Zhai, Opt. Express 26, 21910 (2018).]. However, it shows low intelligence when Monte-Carlo simulation is conducted to apply AWI on various surface figure error. In some simulation cases, the unresolvable fringes keep still or cannot be turned into completely resolvable fringes. To troubleshoot this issue, we studied AWIs in a general framework of global optimization for the first time. Under this framework, we explained that three optimization issues contribute to the poor performance of AWI. On this basis, we proposed a machine vision and genetic algorithm combined method (MV-GA) to control AWI to realize efficient and robust tests of various surface figure error. Monte-Carlo simulation and experiment verify the robustness has been greatly enhanced.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
During fabrication process of optical elements, the in-process (i.e., not-yet-completed) optical surfaces must be measured to guide iterative figuring process. Among various test techniques, interferometry is preferred for its higher accuracy and less measurement time compared with coordinate measurement [1] and shear interferometry [2], etc. However, surface figure error of in-process surfaces at the initial stages of polishing is often beyond dynamic range [3] of interferometers. The fringes of captured interferogram are too dense or unresolvable. The detector cannot resolve it to obtain surface figure error. To test in-process surfaces with severe surface figure, we recently proposed spatial light modulator (SLM) based adaptive wavefront interferometer (AWI) [4]. The principle is shown in Fig. 1. The SLM is controlled by algorithms to iteratively generate adaptive wavefronts. The SLM is close-loop controlled, i.e., stochastic parallel gradient descent (SPGD) algorithm is utilized to control it. The compensation effects are monitored real time to guarantee convergence of the iteration. Ultimately, unresolvable fringes turn into resolvable fringes.
Huang et al. also made pioneering contributions by developing an AWI system utilizing a deformation mirror (DM) [5]. Both their method and ours successfully demonstrated superb adaptive capability in a single experiment that measuring an in-process surface with unknown severe surface figure error. However, severe problems come to light when we generalize the search process of the both AWI modalities as a global optimization problem of
where Z = {z4, z5, …, zn} is the coefficients of orthogonal polynomial terms (e.g., Zernike terms) or influence functions of actuators, the linear combination of which constitutes the dynamic null phase of adaptive optics (AO) elements. Objective function f(∙) represents the compensation effect during the search process. In view of the trivial conversion of any maximization problem into a minimization one, we express the optimization problem as a global minimum problem.The first issue comes from the representation of objective function f(∙) existing in current available AWIs. The function value of f(∙) should be calculatable at any point of the feasible region A to guide optimization process. Pixel number fpn(∙) of unresolvable fringes sub-region maybe the simplest quantifiable objective function. Moreover, it can directly represent the compensation effects. However, its value is difficult to be obtained from interferograms because it is difficult for AWI/computer to segment interferograms into resolvable fringes sub-region and unresolvable fringes sub-region. Both the existing AWI modalities adopted a kind of image metric called SSD (sum of squared gray level differences between any two pixels of the interferogram) as objective function. Unfortunately, SSD is not a monotonic function of compensation effects [4]. This complicates the optimization process by breaking the whole process into subsection pieces [4]. This issue is referred as objective function evaluation issue hereafter.
The more serious issues come to light when we make Monte-Carlo simulations to investigate ability of AWIs to adapt surfaces with different unknown surface figure error. We discover that, in some simulation cases, the interferogram containing unresolvable fringes may keep nearly still from the beginning to the end as shown in Fig. 2. The full Media is given in Visualization 1. In view of optimization, this issue is referred as non-convergence issue. Another kind of issue is also common. The interferogram containing unresolvable fringes can be improved, however, the fringes can never be completely resolvable (see Fig. 3 or Visualization 2). In view of optimization, it seems the optimizer gets trapped in the local optimum. This issue is referred as local-convergence issue hereafter. For better understanding the non-convergence issue and the local-convergence issue, the variation of objective function value fpn(∙) with iteration number for the process of Visualization 1 and Visualization 2 is shown in Fig. 4.
Fig. 2 Fringes evolution in one simulation that non-convergence issue appears. (a) beginning of the search, (b)-(g) during the search, (h) end of the search.
Fig. 3 Fringes evolution in one simulation that local-convergence issue appears. (a) beginning of the search, (b)-(g) during the search, and (h) end of the search.
Fig. 4 Variation of the objective function value with iteration number for the non-convergence and local-convergence issue.
All the above three issues together lead to poor performance of AWIs. This poor performance manifests as low intelligence and poor robustness when utilizing AWIs to test surfaces with different surface figure error. It is an interesting issue that has never been disclosed to our knowledge. Digging deep into this issue is helpful to reveal the essence of the AWIs working mechanism. Troubleshooting this issue will enhance intelligence of AWIs to make it realize robust and efficient tests of in-process surfaces with different unknown surface figure error.
In this paper, we firstly analyzed AWI as a case of global optimization, and explained that three optimization issues, i.e., objective function evaluation, non-convergence, and local-convergence contribute to the poor performance of AWI (Introduction part). For troubleshoot these issues, we introduced machine vision basic techniques (image-processing and mathematical morphology) to solve the objective function evaluation issue and introduced a global optimization algorithm – GA to troubleshoot the convergence issues of AWIs. Then we synthesized these two techniques and proposed a genetic algorithm with fitness evaluated by machine vision intelligent optimization algorithm (MV-GA). Then, the MV-GA method was applied to SLM based AWI to conduct simulations and experiments. In Section 3, Monte-Carlo simulation was conducted to quantitatively evaluate the performance of the proposed method on various surface figure error. The results show it performs more robustly than the existing control method of AWIs. In Section 4, experiments were conducted on an in-process surface with severe surface figure error. Discussions of the proposed method are provided in Section 5.
2. Principle
2.1 Methods for giving AWI ability of vision to analyze interferograms and evaluate objective function for efficient optimization
If AWIs have the ability of vision like human to automatically segment the interferogram into resolvable fringes sub-region and unresolvable fringes sub-region, the pixel number of the unresolvable fringes sub-region, i.e., fpn(∙), can be easily calculated. Since fpn(∙) is a monotonic function of the compensation effect, the optimization will be conducted more efficiently. Thus, the objective function evaluation issue can be troubleshot. To fulfill this goal, we have developed a method as follows by introducing image-processing and mathematical morphology [6] techniques into AWI to analyze interferograms.
The typical interferogram during the search process is shown as Fig. 5(a). The interferogram can be divided into two sub-regions, i.e., the unresolvable fringes sub-region [indicated by purple arrows in Fig. 5(a)] and the resolvable fringes sub-region. Skillful metrologists can roughly distinguish these two sub-regions, but machines cannot. To make AWIs have the similar vison ability, following procedures are conducted.
Fig. 5 Machine vision method to automatically segment the interferogram into resolvable fringes sub-region and unresolvable fringes sub-region. (a) A typical 8-bit interferogram during the search process. (b), (c), and (d) are the binary image which can distinguish the unresolvable fringes sub-region (black region) from the resolvable fringes sub-region (white region) after threshold segmentation, region filling operation, and opening operation, respectively.
Firstly, median filter is applied on the automatedly acquired interferogram [Fig. 5(a)] to reduce random noise. Then threshold segmentation is conducted to roughly distinguish unresolvable fringes sub-region from resolvable fringes sub-region. It is conducted by transforming the 8-bit grayscale interferogram into a binary image shown as Fig. 5(b) according to
where is gray level of the transformed binary image, is gray level (0~255) of the 8-bit interferogram. (i, j) is index of the image. Tgl and Tggn mean artificially set thresholds for gray level, and gray gradient norm, respectively. 0 means black for identifying unresolvable fringes sub-region, and 1 means white for identifying resolvable fringes sub-region. The basic idea of the threshold segmentation process is that the pixels of unresolvable fringes sub-region have a relatively lower grayscale value and lower grayscale value variation compared with pixels within the resolvable fringes sub-region. Proper values of Tgl and Tggn can be easily obtained by several trials of applying Eq. (2) with different Tgl and Tggn values on the 8-bit interferogram. Based on this idea, the unresolvable fringes sub-region can be roughly distinguished from the resolvable fringes sub-region. As shown in Fig. 5(b), the unresolvable fringes sub-region has been roughly identified as black pixels. However, two kinds of undesired objects exist in Fig. 5(b). The first kind of undesired objects is black skeletons of fringes [indicated by blue arrows in Fig. 5(b)]. The second kind is the countless small white objects [indicated by green arrows in Fig. 5(b)] within the roughly identified unresolvable fringes sub-region.To identify and remove these two kinds of undesired objects, mathematical morphology [6] is introduced by us to further analyze the transformed binary image. Mathematical morphology has been continuously receiving a great deal of attention in image-processing applications. It can provide a quantitative description of geometric structure and shape for images. To remove the black skeletons of fringes, region filling operation [6] is conducted. Region filling is a basic morphological operation. The objective of region filling is to fill value 1 (white) into the object region which contains all boundary pixels labeled 1 (white) and non-boundary pixels labeled 0 (black). The effects after region filling is shown in Fig. 5(c). Note, the skeletons are removed. Then, opening operation [6] is performed to remove the countless small white objects. Opening is also a basic morphological operation. Opening operator can remove all objects that are too small to contain a user-defined probe (structuring element). After opening operation, the countless small white objects disappear as shown in Fig. 5(d).
Finally, the number of black pixels of Fig. 5(d) is counted to obtain the pixel number of unresolvable fringes sub-region, i.e., fpn(∙). The whole process can be incorporated to AWIs to realize real time evaluation of optimization effects (objective function values).
2.2 Methods for making the AWI robust for better adapting different unknown surface figure error
When we say the AWI system behaves robustly for adapting different unknown surface figure error, we mainly mean the AWI can always turn unresolvable fringes into resolvable fringes. In view of global optimization, the term ‘always’ implies the odds of non-convergence and local convergence should be minimized. To realize this goal, we will formulate our method for troubleshooting the convergence issues as follows starting with analysis of the global optimization problem of Eq. (1).
When confronted with a global optimization problem to be solved, one important thing we should do is try to understand the characteristics of the global optimization problem and, possibly, its complexity. Complexity of a global optimization problem is mainly determined by properties of the objective function [7]. Hence it is reasonable to study objective function, i.e., f(Z) in Eq. (1).
As stated in Section 2.1, fpn(Z) is a basic objective function. Other objective functions are composite functions of fpn(Z). Hence, the properties of fpn(Z) is studied. Considering that it is hard to deduce mathematical expression of fpn(Z), we plot its map as a demonstration. The map is plotted when severe surface figure error is represented by a standard Zernike (Noll’s ordering) [8] sag surface (with sample grid of 128 × 128) with Z11 = −5λ (λ = 632.8nm). Note, standard Zernike is Root-Mean-Squared (RMS) normalized. Hence, the RMS value is 5λ for the severe surface figure error. The null (or partial null) phase provided by AO elements is represented by standard Zernike polynomials with coefficients equaling Z. Due to the difficulty of plotting high dimensional function, we set Z = {z4, z11}, i.e., only power (Z4) and primary spherical aberration (Z11) terms are selected as active null phase terms. The range of variation for z4, z11 are both from −20λ to 20λ with 0.5λ increment. For every value of Z, the unresolvable fringes sub-region is determined by identifying the pixels that have wavefront slope larger than a half wave per pixel. The idea of this operation is based on the definition of dynamic range of CCD. Dynamic range is limited by the pixel density of CCD. Theoretically, wavefront error with slope larger than a half wave per pixel [3] is beyond dynamic range. The corresponding fringes region is unresolvable. Considering the phase map shown in Fig. 6(a), the pixels with wavefront slope larger than a half wave per pixel are identified as shown in Fig. 6(b). The corresponding interferogram of Fig. 6(a) is shown in Fig. 6(c) or Fig. 6(d) [unresolvable fringes region identified by black]. By comparison of Fig. 6(b) with Fig. 6(c), it can be seen that the region that have wavefront slope higher than a half wave per pixel matches the unresolvable fringes region. Since the wavefront can be easily obtained by simulations, we can easily identify the unresolvable fringes region without using the technique shown in Section 2.1.
Fig. 6 Relation of wavefront error slope with the unresolvable fringes sub-region. (a) wavefront error, (b) pixels with wavefront slope larger than a half wave per pixel are identified, (c) and (d) are interferogram corresponding to (a).
Then the value of fpn(Z) can be easily obtained by counting pixel number of the unresolvable fringes sub-region. The map of fpn(Z) can be plotted by calculating the values of fpn(Z) on points set of {(z4, z11) ∣ z4 = (−20λ: 0.5λ: 20λ), z11 = (−20λ: 0.5λ: 20λ)}. Figure 7 shows the map of −fpn(Z), note that −fpn(Z) is plotted other than fpn(Z) because peaks are thought by us to be more visualized than valleys. For better showing the function map, a visualization (Visualization 3) is made to demonstrate the objective function from different point of view.
Two important properties of fpn(Z) must be noted from Fig. 7. Firstly, fpn(Z) is a multimodal function. This is concluded since more than one local extremums (marked by pink triangular in Fig. 7) exist. It can also be reasonably imagined that if the high-dimension (more than two-dimension) function is visualized, more local extremums will be seen. Since objective function fpn(Z) is multimodal, then the application of an optimization algorithm poor in global search will lead to find a local minimum [9]. Unfortunately, the available two AWI modalities both adopt SPGD algorithm. The poor performance in global search of SPGD leads to the local-convergence issue shown in Visualization 2. A robust global optimization algorithm is required to fix this issue. Secondly, fpn(Z) is a step function. The gradient within the region far from the global optimum (e.g., the region within green wireframe in Fig. 7) is zero or near zero. Therefore, if the optimization algorithm belongs to gradient-based algorithms, the optimization can hardly proceed when the initial value of optimization variables is blindly set within the zero-gradient region. Again, SPGD, unfortunately belongs to gradient-based algorithms. That is why the non-convergence issue shown in Visualization 1 occurs. To solve this problem, a wise way is to utilize non-gradient-based or/and multi-start optimization algorithms [9]. Multi-start technique means to repeatedly perform an optimization method with randomly generated starting points.
Summarized from the above analysis, the possible method to troubleshoot the local-convergence issue and non-convergence issue of the existing SPGD controlled AWIs lies in finding a robust non-gradient-based or/and multi-start global optimization algorithm. GA [10] is this kind of algorithm meeting these requirements.
GA simulates natural evolution to solve optimization problems [10]. Some special tools are introduced by GA to allow a global exploration of the feasible region. These are multi-start technique (GA always uses population of solutions rather than a single solution for searching); selection operators allowing backtracking (i.e., acceptance of non-improving moves); crossover and mutation operators for creating new individuals for enlarging the search region with in the feasible region. All these are helpful to escape from local optimum and to end up in global optimum. Moreover, GA uses fitness function for evaluation rather than gradients. It means that GA does not belong to gradient-based algorithms. As a result, GA performs well for optimization problems with step objective function. Considering these merits, GA is introduced by us into AWIs to troubleshoot the convergence issues of the current SPGD controlled AWIs.
The flowchart of control process of using GA in AWI is shown in Fig. 8. It begins by creating an initial population of M individuals (i.e., chromosomes). M is the population size. The dynamic null phase control vector Z = {z4, z5, ⋯, zn} is considered as an individual to be evolved. For our SLM-based AWI, Z is the coefficients of standard Zernike terms. Direct value encoding method is utilized, i.e., the value of zi is the gene which constitute a chromosome. The initial population is generated randomly within the feasible region A. A is defined as
where Bu and Bl are upper bound and lower bound of zi, respectively, n is the max Zernike term used for nulling the severe surface figure error. If the Zernike polynomials are standard Zernike (i.e., Noll’s ordering), the absolute values of Bu and Bl can be set around the estimated RMS value (about one fifth of PV value) of the unknown surface figure error. If the Zernike polynomials are fringes Zernike (i.e., Wyant’s ordering), the absolute values of Bu and Bl can be set around the estimated amplitude (about one half of PV value) of the unknown surface figure error. Too large absolute values of Bu and Bl will make the search region too large. Too small absolute values of Bu and Bl will make the search region too small to include the optimal solution. Secondly, the SLM is driven by individuals of the initial population sequentially. At the same time, the interferograms are stored, and the fitness value of each individual in the population is calculated. Note, fitness function is defined as 1/fpn(Z), and fpn(Z) ≠ 0. The fitness corresponds to an evaluation of how good the candidate solution is. The larger of fitness value is, the better the individual is. The best individual of the current population can be easily found. Then the best fitness value is tested whether it reaches the maximum threshold [i.e., 1/fpn(Z) = 1 because fpn(Z) ≠ 0]. If not, breeding process is conducted to create a new population by selection, crossover, and mutation operators. Then go to the ‘Drive SLM’ step and iterate this loop until the best individual fitness passes. Finally, the passed best individual is used to drive the SLM. Fringes can be completely restored as resolvable fringes.The breeding process of creating new population based on the current population is the heart of GA. It consists of selection, crossover, and mutation shown in Fig. 8. It is in this process that the search process creates new and hopefully fitter individuals. In [10,11], various types of selection, crossover and mutation are reviewed. For our application, the property of AWI is considered to make the appropriate choice among these different operators. Simulation trials are also conducted to direct choosing proper operators and tuning the parameters of these operators. One proper combination of operators which behaves well in our simulation trials is briefly formulated as following.
The purpose of selection is to emphasize fitter individuals in the population in hopes that their off springs have higher fitness. One key difference among different selection schemes is the magnitude of selection pressure [10]. The selection pressure is defined as the degree to which the better individuals are favored. In our application, tournament selection [10] is chosen. Tournament selection strategy provides selection pressure by holding a tournament competition among some individuals. The simulation trials show it can supply proper selection pressure. Proper selection pressure can favor better individuals and meanwhile preserve population diversity by allowing backtracking.
After selection, crossover takes two parent solutions and produces from them a child with the hope that the offspring is better. With many simulation trials conducted, arithmetical crossover is adopted as the crossover operator in our application. Arithmetical crossover [11] is defined as a linear combination of two vectors. If and are to be crossed, the resulting offspring are
where t and t + 1 means the generation number, v and w means the individual order in the population. a is a random generated factor of proportionality between 0 and 1.Crossover makes clones of good genes but does not create new ones for enlarging the search region with in the feasible region. To change the genes of the offspring and to increase the diversity of the population, the populations are subjected to mutation after crossover. Mutation helps escape from local minima’s trap and maintains diversity in the population. With the same simulation trials method, non-uniform mutation [11] scheme is finally chosen. Non-uniform mutation performs uniformly at the beginning and very locally towards the end of the search. Non-uniform mutation is defined as follows: is the individual, where t is the current generation number, v is the individual order in the population, is one single Zernike term and also the gene constituting an individual. The half (user defined number) genes (i.e., , where [ ∙ ] is the round operator) of all genes of the individual are selected for mutation. Then the genes after mutation are
where, Bu and Bl are upper bound and lower bound of, respectively. The function is defined aswhere r is a random number in range of [0, 1], T is the max generation number.The basic parameters for crossover and mutation are the crossover probability (Pc), and mutation probability (Pm), respectively. Crossover/mutation probability decides how often crossover/mutation will be performed on an individual [10]. The proper probability values should be determined by off-line parameter sweep method. Off-line parameter sweep method means to tune the values of the parameters in simulations, and obtain the proper values according to the simulation results.
2.3 Intelligence enhancement of AWI by combing machine vision and GA
We incorporate the machine vision method for calculating fpn(∙) shown in Section 2.1 into the fitness evaluation step of GA shown Section 2.2. It constitutes a GA with fitness evaluated by machine vision intelligent optimization algorithm (MV-GA) for AWIs. Flowchart of the control process of using MV-GA in AWIs is shown in Fig. 9. The MV-GA has a monotonic objective function and has the merits of GA for reaching global optimum. Hence it is theoretically to perform robustly in AWIs. To verify the performance of applying MV-GA in AWIs, simulation is conducted as the following Section.
3. Simulation
To statistically evaluate the performance of using MV-GA in AWIs, Monte-Carlo simulation was conducted by Matlab. The Monte-Carlo simulations consists 500 trials. The severe surface figure error is represented by a standard Zernike (Noll’s ordering) sag surface with coefficients of , where (i = 4, 5, 6, ⋯, 11) is the coefficient of ith standard Zernike term. The max Zernike term for the surface figure error is Z11 (i.e., primary spherical aberration term) because the dominate frequency component of severe surface figure error caused by fabrication is usually low-spatial error. The dynamic null phase control vector (i.e., the individual) is represented by standard Zernike phase with coefficients of , where (i = 4, 5, 6, ⋯, 11) is the coefficient of ith standard Zernike term, t means the tth generation, v means the vth individual of tth generation population.
For every trial, surface figure error (with sample grid of 128 × 128) is randomly generated with random value of (i = 4, 5, 6, ⋯, 11) between −10λ and 10λ (λ = 632.8nm). Note, standard Zernike is RMS normalized, hence the Zernike standard sag surface with coefficient up to ± 10λ corresponds to surface figure error with peak-to-valley (PV) up to ~50λ. 50λ is usually high enough for PV value of surface figure error of practical in-process optical surfaces before polishing. Then, the initial population with population size of 200 (i.e., 200 individuals) is randomly generated within the feasible region A. The expression of A is shown in Eq. (3), where Bu and Bl are 10λ and −10λ, respectively. The following procedures are similar with the flowchart of the control process of the GA-based AWI shown in Fig. 8. However, some steps are simplified. Firstly, the step of ‘Drive SLM by individuals sequentially’ was performed in Matlab just by (v = 1, 2, ⋯, 200), which means null or partially null the surface figure error by dynamic null phase with coefficient of . Note, the calculation of omits the light propagation effect which is negligible compared with the severe surface figure error. Secondly, the step of ‘Acquire interferograms of the individuals’ was performed in Matlab by generating the interferogram of wavefront 2(). The wavefront is the wavefront reflects from the dummy test surface and double passes the dummy SLM. The parameters of the breeding operators for all trials are as follows. Tournament selection: tournament among 20 individuals. Arithmetical crossover: crossover probability Pc = 0.8. Non-uniform mutation: mutation probability Pm = 0.4, and the T = 200 which is max generation number in Eq. (6). The max generation number for every trail is set as 200. The trials were conducted 500 times for probabilistic analysis.
For comparison, the Monte-Carlo simulation with 500 trials was also conducted using SPGD algorithm with objective function J (SSD) which is not a monotonic function of null effect. This algorithm is used by the current existing AWIs [4,5]. This algorithm is referred as SSD-SPGD hereafter. It is defined as
where optimization variable Z = {z1, z2, ⋯, zn} is the coefficients of Zernike terms, the combination of which constitutes the phase of SLM. γ is the gain constant. δZ(k) = {δz1, δz2, ⋯, δzn }(k) are small random perturbations having identical amplitudes α and a Bernoulli probability distribution; i and j are two pixel index numbers of the interferogram; gi and gj are the grayscale values of the two pixels. J is the objective to be maximize.The method for conducting one SSD-SPGD trial can follow the procedures in the Simulation Part (Section 3) of [4]. For every trial, surface figure error is also random with randomly generated (i = 4, 5, 6, ⋯, 11) value between −10λ and 10λ. The parameters for SSD-SPGD algorithm are as follows. The amplitude α of small random perturbations equals 0.2λ, gain constant γ = 0.006. The max iteration time for one trail is set as 4000.
The Monte Carlo simulation results are demonstrated as follows from three perspective. From an overall perspective, the pixel number fpn(Z) of the unresolvable fringes sub-region of the final interferogram after search is completed for every trail of MV-GA and SSD-SPGD is shown in Fig. 10. The result of MV-GA is represented by red square, and SSD-SPGD by blue circle. It can be roughly seen, almost all trials of MV-GA converge. However, many trials of SSD-SPGD fail to converge. By calculation, the probability of non-convergence and local convergence is only 6‰ for MV-GA. By contrast, it is as high as 21% for SSD-SPGD.
Fig. 10 The objective function values when search is finished for every trail of MV-GA and SSD-SPGD.
To compare the error propagation process of the two algorithms, variation of the mean value and standard derivation of the 500 trials’ objective function values with generation (or iteration) number is shown in Figs. 11 and 12. Figure 11 is the results of MV-GA, and Fig. 12 for SSD-SPGD. It can be seen from Fig. 11, for MV-GA method, the standard derivation values of objective function values become smaller as generation number. The standard derivation value is nearly zero when generation number is larger than about 150. For SSD-SPGD, it can be seen from Fig. 12 that the standard derivation values of objective function values almost keep constant as iteration number varies. The standard derivation values when iteration is finished can be used to evaluation the odds of non-convergence and local-convergence of the two method. Therefore, Figs. 11 and 12 also demonstrate that the MV-GA performs more robustly than SSD-SPGD does. Robust implies MV-GA behaves more intelligent than SSD-SPGD does on various surface figure error.
Fig. 11 Variation of the mean value & standard derivation of the 500 trials’ objective function values with generation number for MV-GA method.
Fig. 12 Variation of the mean value & standard derivation of the 500 trials’ objective function values with iteration number for SSD-SPGD method.
To observe the fringes evolution, the typical search process of a MV-GA trial in which fringes are completely restored is shown in Visualization 4. The initial fringes, restored fringes, and some fringes during the search process are captured from Visualization 4 as snapshots. The snapshots are shown in Fig. 13. The typical search processes of two SSD-SPGS trials in which non-convergence and local-convergence appear are shown in Fig. 2 (or Visualization 1), and Fig. 3 (or Visualization 2), respectively.
Fig. 13 The initial fringes (a), restored fringes(p), and some fringes during the search process (b-o) captured from Visualization 3.
4. Experiment
To further verify the performance of MV-GA, MV-GA was integrated into our SLM-based AWI system [4] to conduct practical experiments. The experiment apparatus of the MV-GA controlled AWI for testing an in-process surface with severe surface figure error was set up as shown in Fig. 14. The test surface is an in-process Φ61mm flat aluminum mirror with unknown severe surface figure error within the central Φ26mm circular region. The interferogram corresponds to the central Φ26mm circular region is shown in Fig. 15. The experiment apparatus works like a wavefront sensor-less AO system, which is mainly constituted by aberration source, AO correctors, detectors, and control software. The surface figure error of the in-process surface is the source of the aberration introduced to the optical system. Note, it is dynamic during different fabrication processes while keeps static during one measurement process. The SLM (HoloeyeTM LC 2012) acts as the AO corrector to null or partially null the unknown severe surface figure error of the test surface. The Zygo 6” GPI interferometer acts as the detector to record the interferogram generated by the test wavefront and the reference wavefront. Note, the interferometer is utilized to record interferograms, not to measure the wavefront which is beyond the dynamic range of the interferometer. That is why the system is termed as ‘wavefront sensor-less AO’. The control software in the blue dotted box of Fig. 14 is integrated to the computer. The core of the control software is MV-GA which acts as wavefront sensor-less algorithm.
The whole system works following the flowchart of the control process of using MV-GA in AWIs as shown in Fig. 9. The input (not the start shown in Fig. 9) of MV-GA is the interferograms which are the nulling effect feedbacks from the interferometer. The output of the MV-GA is dynamic null phase control vector Z. The vector Z is utilized to generate the dynamic null phase. Then the dynamic null phase is transformed to a gray scale map with the interferogram-type computer generated holograms (ICGH) encoding method [4,12]. In [12], it shows SLM can generate low-order aberration with PV about 40λ and with accuracy about 0.04λ RMS. Compared with the amplitude of the severe surface figure error, 0.04λ RMS is small enough. So, the SLM control error has no obvious affection on the convergence of the process. However, the accuracy of wavefront measurement is about 0.04λ RMS since the phase control accuracy of SLM is about 0.04λ RMS. The analysis about the control of SLM can be referred to [12]. Loading the gray scale maps corresponding to the individuals of the MV-GA, the SLM is controlled to generate dynamic null phase. That is how the test surface, SLM, interferometer and MV-GA are linked to constitute a wavefront sensor-less AO system.
The parameters of MV-GA were set the same with the simulation. The experiment was conducted 50 times for probability analysis. The result shows 0 failure occurs, i.e., the MV-GA can always completely turn unresolvable fringes into resolvable fringes. The typical evolution process of interferograms corresponding to the best individual of different generations is shown in Visualization 5. Some interferograms during the search are show in Fig. 16. For comparison, the 50 times experiments were also conducted by using SSD-SPGD algorithm. The result shows the non-convergence and local-convergence occur 7 times in the 50 times experiments. The typical failed search process is shown in Visualization 6. It shows the local-convergence occurs. In summary, the experiment shows MV-GA performs more robustly than SSD-SPGD does. In other word, MV-GA behaves more intelligent than SSD-SPGD does in statistical sense. To obtain the surface figure error, phase conjugation step and reverse optimization step are required after the fringes are restored. It can be referred to [4]. The reconstructed surface figure error is shown in Fig. 17(a) with 9.129λ RMS. For comparison, the surface figure is tested by LuphoScan 260 [13]. LuphoScan 260 can provide a measurement accuracy of better than ± 50 nm (3σ). The LuphoScan test result is shown in Fig. 17(b) with 9.154λ RMS. The point-to-point difference between the two results is shown in Fig. 17(c) with 0.034λ RMS.
Fig. 16 Fringes evolution in one experiment using MV-GA. (a) beginning of the search, (b)-(g) during the search, and (h) end of the search.
Fig. 17 The surface figure error measurement results by (a) MV-GA AWI, and (b) LuphoScan 260. (c) shows the point-to-point difference between (a) and (b).
5. Discussion
In both above simulations and experiments, the MV-GA has shown excellent robustness for AWI compared with the current existing SSD-SPGD method. However, there still exist several issues required to be clarified. Discussions of the presented method are also provided as follows.
- (1) Compared with SPGD and almost all other conventional optimization algorithms, GA uses population of solutions rather than a single solution for searching. This plays a major role to the robustness of GAs. It improves the chance of reaching global optimum and also helps in avoiding local stationary point. However, it also brings some drawbacks. This meets the No Free Lunch theorems for optimization [14]. One major drawback is that it usually takes longer time for convergence. Experiments show the interferogram acquisition and objective evaluation are the most time-consuming operations of AWIs. However, GA has to conduct these operations several hundred times (i.e., the number of individuals in one generation) for every generation. Commonly, GA requires several tens generations to reach global optimum. That means several thousands of these operations have to be conducted. In comparison, SPGD commonly requires only several hundred iterations if the search process works well. That means MV-GA usually takes ten times longer than SSD-SPGD does (for SSD-SPGD, typically, several minutes). This looks like a significant limitation for MV-GA. However, it is not so bad in practical application. That is because time is not a serious issue for AWI. AWI is a time-independent AO system. Time-independent AO system means the AO system does not suffer from the highspeed perturbations seen in AO astronomical/atmospheric applications. For AWI, the surface figure error is static during the test process. The variation of error only occurs during different fabrication processes. Therefore, requirements on convergence rate of control algorithms are relaxed. This makes it possible to use MV-GA in AWIs. After all, speeding up convergence rate of MV-GA is no harm for the practical application. Hence, further research can be focused on how to speed up the convergence rate of MV-GA to realize robust and fast search for AWIs.
- (2) The experiment verifies the MV-GA performs more robustly than SSD-SPGD does when 50 trails are conducted on the same test surface. However, it is more persuading to investigate the experimental performance of these two algorithms on various surface figure error. Therefore, it is worth to design several kinds of test surfaces with typically different surface figure error to conduct the Monte-Carlo experiments. Further research is required.
- (3) Finding a proper parameters setting is essential for our GA method, however, it is also important for other optimization algorithms including SPGD. In fact, it is a permanent topic of optimization field. There are mainly two types of parameter setting: parameter tuning (i.e., offline parameter search) and parameter control (i.e., online parameter tweaking). In our presented method, we adopted the simple off-line parameter sweep method to find a proper parameter setting for AWI. However, the simple off-line parameter sweep method is experience-dependent and tiring. Developing online parameter tweaking methods such as adaptive parameters control method [15] may liberate metrologists from such kind of tedious labor. Further research on this issue is worthy.
- (4) One thing should also be clarified. Our presented method is developed for testing in-process surface with unknown severe surface figure error that beyond dynamic range of interferometers. For enlarging the dynamic range of interferometers, there exists a lot of methods such as high density detector array method [16], longer wavelength [17] or two-wavelength techniques [18], Sub-Nyquist interferometry [3], and Tilt-Wave-Interferometry [19]. These methods work well. Unfortunately, they are a little complicated/costly to be integrated into commercial interferometers. The prevailing variable null methods [20–25] which only compensate the intrinsic aberration of test surfaces are not applicable. Because surface figure error of in-process surfaces is unknown free-form surface varying during fabrication. That means the compensation target is unknown. Our AWI method introduces wavefront sensor-less AO technique into conventional interferometers. Due to the wavefront sensor-less algorithm, the adaptive null correctors are given the intelligence to speculate the unknown compensation target, and the flexibility to null it. This paper is focused on enhancing the intelligence of AWIs. This is where the title of this paper comes from. However, researches about improving the flexibility are worthy to be conducted to keep pace with the intelligence enhancement. Surveys on AO correctors and the driving methods for the AO correctors would be very helpful to start this future work.
Conclusion
By regarding AWI as a case of global optimization, we explained that three optimization issues, i.e., objective function evaluation, non-convergence, and local-convergence contribute to the poor performance of AWI. We proposed MV-GA method to troubleshoot these issues. The method has a monotonic objective function and has the merits of GA for reaching global optimum. Hence it is theoretically to perform robustly in AWI systems. The Monte-Carlo simulation on various surface figure error show the probability of non-convergence and local convergence is only 6‰ for MV-GA. By contrast, it is as high as 21% for SSD-SPGD. Monte-Carlo experiments on an in-process surface also verify the robustness has been enhanced by applying MV-GA in AWI. The AWI can be more intelligent to adapt unknown surface figure error of in-process surfaces when the proposed MV-GA is used as the control algorithm.
Funding
Science Challenge Program of China (TZ2018006); Hunan Provincial Natural Science Foundation of China (2016JJ1003); National Natural Science Foundation of China (51835013).
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