Subaperture Stitching Interferometry Research Articles

Circular Subaperture Stitching Interferometry Based on Polarization Grating and Virtual–Real Combination Interferometer

This paper presents a polarization grating based circular subaperture stitching interferometer. The system can be used for small F/# concave surface tests with a large F/# transmission sphere, where F/# is the ratio of focal length to aperture. A polarization grating was employed to deflect the incident beam for subaperture scanning by its axial rotation instead of a multi-axis motion-control system. Compared with the traditional subaperture stitching interferometric system, the system proposed in this paper is smaller in size and reduces the measurement error introduced by mechanical adjustment. Using a virtual interferometer model and a virtual-real combination algorithm to remove the retrace error, the full-aperture figure error can be directly obtained without the need for a complex stitching algorithm. The feasibility of the algorithm was verified, and the measurement error caused by the modeling error was analyzed by simulation. The capability of the polarization grating to scan subapertures was experimentally confirmed, and possible solutions to some engineering challenges were pointed out. The research in this paper has pioneering and guiding significance for the application of polarization grating in interferometry.

Stitching method based on dual quaternion for cylindrical mirrors.

In this study, a stitching method based on dual quaternion is proposed. The application of a dual quaternion in sub-aperture stitching interferometry is analyzed in detail, and a calculation method for sub-aperture stitching based on a dual quaternion is deduced. The experimental results demonstrate the accuracy of the stitching method proposed in this study (residuals of overlapping area approximately 0.22 nm RMS). Finally, the residual differences of 0.79 nm RMS between the figure errors are acquired with a stitching by parts algorithm based on the dual quaternion and long trace profiler (FSP at HEPS). The high-accuracy and high-efficiency stitching method proposed in this study will expand its application in the metrology and manufacture of long cylindrical mirrors.

Hybrid self-calibration method for reference surface elimination in subaperture stitching interferometry

Subaperture stitching interferometry (SSI) is an essential method for the map testing of large-aperture optical components. The surface map of a reference surface is a typical error source. In this study, we propose a hybrid self-calibration method to eliminate the reference surface error in SSI by combining the modified shift-rotation method and the maximum likelihood method. The traditional shift-rotation method is a general full-aperture absolute interferometric measurement that can retain localized irregularities. The shift-rotation operations are leveraged to generate a couple of subapertures covering the surface under test, whereby a ring of subapertures and a central subaperture are acquired by rotations and a lateral shift, respectively. The modified shift-rotation method is proposed to obtain the rotationally asymmetric components of the test surface in the ring of subapertures. The same components within the central subaperture are retrieved using the maximum likelihood method. Then, the rotationally symmetric components of the test surface are acquired using the least squares method, utilizing the measured data before and after the shift. Reference surface maps are sufficiently eliminated from the measured data. High-frequency components of the test surface are also retained, which engender high-accuracy SSI. Simulations are conducted to verify the proposed method. The positioning errors of the proposed method are analysed and discussed. Subaperture testing experiments of a 100-mm aperture flat are performed and compared with full-aperture absolute measurement results. The stitched errors with 0.018 λ PV and 0.003 λ RMS are obtained.

Sub-aperture Stitching Interferometry Based on Down-sampled Particle Swarm Optimization

Sub-aperture Stitching Interferometry Based on Down-sampled Particle Swarm Optimization

Optical form measurement employing a tiltable line scanning, low coherence interferometer for annular subaperture stitching interferometry

This contribution presents experimental and simulation results of a tiltable line scanning low coherence interferometer applied for form measurement of spherical and aspherical objects with a diameter of up to 300 mm. The region of interest is sampled by multiple annular subapertures that are realigned employing stitching algorithms based on Cartesian- and Zernike polynomial fittings. The paper addresses common challenges in the reduction and modeling of displacement errors associated with the motion of the interferometric sensor between subaperture measurements and compares the topography deviations of the experimental results with those simulated by a Monte Carlo based model.

Form deviations caused by lateral displacement errors in annular subaperture stitching interferometry

Annular subaperture stitching interferometry (ASSI) is a common approach for the measurement of aspherical surfaces. A common obstacle of ASSI is the occurrence of lateral displacement errors when the sensor or specimen is repositioned between the subaperture measurements. Our contribution focuses on modeling of the statistical displacement errors. A virtual experiment is presented simulating the propagation of the displacement errors through a cumulative and a global stitching algorithm to the retrieved surface form. For the considered experimental setup, the uncertainty in lateral position depends on the positioning uncertainties of the employed motion system and the uncertainty in the absolute distance measurement between the sensor and specimen. The lateral displacement uncertainty is determined experimentally employing a calibratable lateral grating. Thus, it is traceable to the SI unit of the length (meter). The experimental results show that the lateral displacement errors may be modeled by a normal distribution, and the results of the virtual experiment indicate that the statistical lateral displacement errors transfer linear through the stitching procedure and also cause a normal distributed topography error. This enables the assignment of an expanded uncertainty to each individual sample point employing the Zernike polynomial expression of the topography measurement.

Accurate localization method for subaperture stitching interferometry in aspherical optics metrology.

Aspherical optics have superior performance in many applications. However, precision metrology for an aspherical surface is a key stage in its manufacture. Subaperture stitching interferometry (SSI) is one of the mainstream testing methods for aspherical metrology. In many cases, Annular Stitching Subaperture Interferometry (ASSI) is more suitable for components having rotational symmetry because, compared to SSI, the number of motion directions that need to be adjusted is less. Although motion only along the optical axis is of concern in ASSI, inaccurate localization for subapertures may also occur due to the machining error at the optical center of the optics under test and errors in Cat's eye location, thereby leading to measurement errors. To overcome these problems, an accurate localization method for subaperture stitching is proposed. It is based on the maximum asphericity information between the reference spherical wavefront and the test aspherical surface. First, by using the test aspherical formula and theoretical spherical wavefront, we can calculate the minimum peak-to-valley (PV) value of maximum asphericity. The theoretical location of an optimal reference sphere, which corresponds to this value, can be obtained. Then, we perform a practical test, which starts at the initial zero position, and find an actual minimum PV value near its theoretical location. The difference between the theoretical location and the actual one is the compensation quantity. Finally, we execute ASSI measurement to aspherical optics. The location coordinate of each subaperture is compensated with the acquired quantity. Through the experiments, it can be concluded that the proposed method can improve the measurement accuracy of ASSI in terms of error elimination. The results produced by the new method are more desirable than those of the conventional one.

Subaperture stitching interferometry based on the combination of the phase correlation and iterative gradient methods.

Subaperture stitching interferometry (SAS) is an important method for map testing of large aperture optical components, in which a mechanical structure is often employed for the testing of each subaperture. By eliminating the phase deviation of the corresponding points in the overlapping regions of every adjacent subaperture, the whole aperture map can be obtained. Accurate subaperture positioning is an important guarantee for precise stitching. In this paper, a hybrid optimization algorithm is proposed to realize subpixel-level positioning accuracy in SAS based on the combination of the phase correlation and iterative gradient methods. The phase correlation method is adopted to calculate the pixel-level positioning deviation first, and the subpixel deviation is derived and then corrected by iterative optimization through the gradient method. The subpixel-level positioning accuracy of the proposed optimization algorithm is verified by simulations and a 76.2 mm off-axis parabolic mirror is chosen as an experimental testing sample. The surface map obtained from the proposed hybrid optimization method is consistent with the full aperture testing result, which also verifies that the proposed optimization algorithm is a powerful tool with subpixel-level positioning accuracy in SAS testing.

Interferometric measurement of freeform surfaces using irregular subaperture stitching

Irregular subaperture stitching interferometry (ISSI) is proposed for non-null optical freeform surface testing. Compared with circular and annular subapertures, irregular subapertures have a better performance in matching the locally resolvable interferograms due to the rotational asymmetry of freeform surfaces. A modified multi-aperture simultaneous reverse optimization reconstruction algorithm is employed for the subaperture stitching, allowing not only non-complementary subapertures but also no subaperture overlaps. Numerical simulations with error analyses display the accuracy of our method. Experiments showing the effectiveness of the ISSI are presented by testing a Φ20 mm biconic surface with cross-validation.

Reconstruction of reference error in high overlapping density subaperture stitching interferometry.

The vibration modulated stitching interferometry acquires many subaperture phases timely forming the high overlapping density subapertures for asphere phase stitching. The large number of overlapping subapertures had been proven effective in suppressing the reference error. In this research, we propose a pixel-by-pixel reference calibration method by using the averaged difference between the stitched phase and compensated phase within the overlapping subapertures. The measurement for both tested optics and calibration of the reference optics are accomplished in a single phase stitching process. The requirement for a high-quality reference optics or dedicated reference calibration procedure for subaperture stitching interferometry is therefore significantly eliminated. Both the simulation and experimental results shows the feasibility of the proposed method for high frequency reference error and most of the form error in the third order Zernike aberrations.

Sensitivity coefficient for monitoring residual surface error in plane subaperture stitching interferometry

Subaperture stitching interferometry is widely used in large-aperture plane surface measurements. The reference surface error forms a primary reason leading to erroneous stitching surface figures. The methods to remove this error include the normal stitching algorithm with the absolute test and a special algorithm with interlocked compensators without the absolute test. Here, we perform a series of subaperture stitching experiments to demonstrate that the reference surface error can easily be calculated wrongly using the algorithm with interlocked compensators. Consequently, we propose the use of a sensitivity coefficient to describe the sensitivity of the stitching algorithms of the two methods. Next, we perform simulations of pairwise subaperture stitching to illustrate the variations in the sensitivity coefficient with different overlapping area ratios. Finally, two plane surfaces with size of 200 mm × 200 mm are stitched with the reference surface error as reduced by the two methods. Comparisons of the residual surface errors of the two methods show that the algorithm using interlocked compensators is less accurate and less stable, which could be monitored by the sensitivity coefficient. Thus, for reducing the reference surface error in plane subaperture stitching interferometry, the absolute test method should be considered first.

High-precision spherical subaperture stitching interferometry based on digital holography

A novel subaperture stitching interferometry is proposed to measure the high-numerical-aperture sphericity for workshop testing, which combined digital holographic technology, subaperture stitching algorithm and systemic aberration calibration. Each subaperture is fleetly measured by the digital off-axis holography with a single shot so that the influence of measurement environment can be effectively reduced. The high-precision full aperture shape is obtained by stitching these subapertures, meanwhile, the subaperture alignment errors and the systemic aberration can be correctly compensated by the method of least-squares with the Zernike polynomial fitting. The approach is verified by testing three spherical surfaces, including a concave sphere, a convex sphere, and a hemisphere, in general environment. Moreover, the measurement results are compared with full aperture results by using a large aperture interferometer of Zygo and stitching result by using a subaperture stitching interferometer of QED. The results based on the proposed approach are consistent with the commercial interferometers. Specifically, the relative deviations of RMS are 0.009 λ and 0.02 λ for testing a concave and a convex spheres, and that is 0.011 λ for testing a hemisphere. Our method not only has the comparable measurement accuracy with the commercial interferometers but also is more robust, feasibility and inexpensive than other subaperture stitching interferometry. We provide an anti-interference way to test spherical surfaces in workshop environment.

平面子孔径拼接干涉测量精度分析

平面子孔径拼接干涉测量精度分析

Validation of simultaneous reverse optimization reconstruction algorithm in a practical circular subaperture stitching interferometer

We demonstrate the validity of the simultaneous reverse optimization reconstruction (SROR) algorithm in circular subaperture stitching interferometry (CSSI), which is previously proposed for non-null aspheric annular subaperture stitching interferometry (ASSI). The merits of the modified SROR algorithm in CSSI, such as auto retrace error correction, no need of overlap and even permission of missed coverage, are analyzed in detail in simulations and experiments. Meanwhile, a practical CSSI system is proposed for this demonstration. An optical wedge is employed to deflect the incident beam for subaperture scanning by its rotation and shift instead of the six-axis motion-control system. Also the reference path can provide variable Zernike defocus for each subaperture test, which would decrease the fringe density. Experiments validating the SROR algorithm in this CSSI is implemented with cross validation by testing of paraboloidal mirror, flat mirror and astigmatism mirror. It is an indispensable supplement in SROR application in general subaperture stitching interferometry.

DEVELOPMENT OF NEW STITCHING INTERFEROMETRY FOR THE SPICA TELESCOPE

The telescope to be onboard SPICA (Space Infrared Telescope for Cosmology and Astrophysics) has an aperture diameter of 2.5 m and its imaging performance is to be diffraction-limited at a wavelength of <TEX>$20{\mu}m$</TEX> at the operating temperature of <8 K. Because manufacturing precise autocollimating flat mirrors (ACFs) with sizes comparable to the SPICA telescope is not technically feasible, we plan to use sub-aperture stitching interferometry through ACFs for optical testing of the telescope. We have verified the applicability of the sub-aperture stitching technique to the SPICA telescope by performing stitching experiments in a vacuum at a room temperature, using the 800-mm telescope and a 300-mm ACF. We have also developed a new method to reduce uncertainties possibly caused by cryogenic and gravitational deformations of ACFs.

Subaperture stitching algorithms: A comparison

With the research focus of subaperture stitching interferometry shifting from flat wavefronts to aspheric ones, a variety of algorithms for stitching optimization have been proposed. We try to category and compare the algorithms in this paper by their modeling of misalignment-induced subaperture aberrations which are of low orders. A simple way is to relate the aberrations to misalignment by linear approximation with small angle assumption. But it can not exactly model the induced aberrations of aspheres. In general, the induced aberrations can be fitted to free polynomials and then removed from subaperture measurements. However, it is at the risk of mixing up the surface error and the induced aberrations. The misalignment actually introduces different terms of aberrations with certain proportions. The interrelation is then determined through analytical modeling or ray tracing. The analytical model-based algorithm and the ray tracing-based algorithm both are tedious, aperture shape-related and surface type-specific. While the configuration space-based algorithm we proposed numerically calculates the surface height change under rigid body transformation, it is generally applicable to various surface types and different aperture shapes. Simulations and experiments are presented to compare the stitching results when different algorithms are applied to null cylindrical subapertures measured with a computer generated hologram. The configuration space-based algorithm shows superior flexibility and accuracy.

最大似然估计法去除子孔径拼接检测中的支撑误差

最大似然估计法去除子孔径拼接检测中的支撑误差

子孔径拼接干涉的快速调整及测量

子孔径拼接干涉的快速调整及测量

Subaperture stitching interferometry based on digital holography

A novel subaperture stitching interferometry based on digital holography is developed to measure the deformation of spherical surfaces. The subaperture measurement is performed by off-axis digital holography on single exposure. Then, the subaperture phase maps are obtained by digital holographic reconstruction, in which the phase aberration caused by position errors of each subaperture measurement is effectively compensated by the method of numerical parametric lens. After that, the full aperture phase map is retrieved by a subaperture stitching algorithm, in which the relative alignment errors of adjacent subapertures are eliminated with an iterative process of stitching optimization. The experiments demonstrate the feasibility and effectiveness of the proposed interferometry, which provides a rapid and robust way to measure spherical surfaces with high resolution and precision. A practical example is given to demonstrate the performance of this method. The stitching result shows good agreement with the full-aperture result.

环形子孔径拼接检测的中心偏移误差补偿

环形子孔径拼接检测的中心偏移误差补偿