# History: Review of Related Work

The idea of subaperture stitching interferometry was first proposed in an Optical Society of America (OSA) Annual Meeting by Kim and Wyant,^{4} and an SPIE Conference by Thunen and Kwon^{5} in the early 1980s. In their proposal, the large reference flat, used as either a reference surface for the testing of large flats or an autocollimator flat for the testing of large telescope wavefronts, is replaced by an array of smaller flats. Kim and Wyant also mentioned that the subaperture test can be applied to fast aspheres because the wavefront slope for each subaperture can be greatly reduced.

Testing a large aperture by stitching subapertures sounds reasonable. However, a serious problem must be solved, i.e., how can we stitch all subapertures together with minimized uncertainty? Taking individual subaperture measurements is only a small part of the larger picture. It is impractical to align all subaperture reference flats perfectly to remove the relative piston and tip-tilt. As a result, all subaperture measurements are unavoidably “polluted” by mechanical positioning error that may be a few orders higher than the magnitude of the surface error. Therefore, a subaperture stitching algorithm is indispensable to recognize and then reduce the influence of the positioning error. It is critical and determines the performance of the subaperture stitching interferometry.

In the early version of subaperture testing, subapertures did not overlap with each other, and the full-aperture aberrations were reconstructed from subaperture aberrations by the least-squares (LS) fitting method. Two approaches were presented: the Kwon-Thunen method^{5} and the simultaneous fit method developed by Chow and Lawrence.^{6} Both are based on Zernike polynomials, and a comparison of them was given by Jensen et al.^{7} Zernike polynomials are a special set of polynomials generally used to describe the wavefront aberration of imaging systems. Unlike a power series expansion, Zernike polynomials are orthogonal over a circular pupil, and the low-order terms are physically related to the primary aberrations such as coma and astigmatism. For imaging systems with annular or even rectangular pupils, Zernike polynomials are redefined to keep orthogonality over the pupil.^{8} However, Zernike polynomials basically describe a smooth, relatively low-order surface or wavefront, insufficient for describing a wavefront containing localized irregularities.^{9} Therefore, both stitching approaches suffer from problems in the reconstruction of a real wavefront with middle-high frequency irregularities.

A straightforward approach is to estimate and then remove the piston and tip- tilt of subapertures from the discrete pixel-based measurements. This is the idea of the discrete phase method developed by Stuhlinger.^{9} The wavefront is represented not by Zernike polynomials but by optical phase values measured at a large number of discrete points across the aperture. The optical phase corresponds to the optical path traveled by the test beam. Differences between the measured and the nominal phase values reveal the aberrations of the system. Local irregularities can be described and retrieved by stitching, but in this case, some redundant information is required because we can no longer make use of the slope continuity of the wavefront when stitching two neighboring subapertures. In contrast, polynomial-based stitching extracts the full-aperture wavefront by assuming high-order continuity. Therefore, the pixel-based stitching requires first that overlapping regions exist among subapertures as proposed by Stuhlinger.^{9} The phase values measured as the surface heights or wavefront aberrations at two neighboring subapertures are ideally equal to each other in the overlapping region. Consequently, the relative piston and tip-tilt are still estimated by the LS fitting to the phase differences at overlapping points. The idea of an overlapping subaperture test is a new milestone in the development of stitching interferometry. It was further extended by Chen et al.^{10,11} in their multiaperture overlap-scanning technique (MAOST) and by Otsubo et al.^{12,13} with detailed discussions on the stitching measurement uncertainty. Except for providing redundant information for stitching optimization, the overlap is also necessary to cover the full aperture. From then on, the overlapping subaperture test becomes a standard form of stitching interferometry.

In the following decade, subaperture stitching interferometry has been mostly restricted to testing planar optics where only two-axis translations are nominally required to position the subaperture. The stitching algorithm simply removes the relative piston and tip-tilt of each subaperture without considering the influence of lateral shift or positioning error. Actually, for a real surface that is nominally flat, lateral shift at the pixel or subpixel level will not introduce remarkable phase error to the subaperture because the wavefront slope is sufficiently small. As an outstanding example of this application, Bray built a two-in-one stitching interferometer for large plano optics in the National Ignition Facility and Laser MegaJoule.^{14,15} It is also worth mentioning that Tang,^{16} Wyant, and Schmit^{17 }extended subaperture stitching interferometry to high-resolution microsurface measurements over a large field of view. The stitching algorithm developed by Tang^{16} is special in that it takes the motion uncertainties of all six degrees of freedom (dofs) into account. These uncertainties are estimated by chi-square fitting to the deviations at overlapping points. It is claimed to be insensitive to both the piston and tilt changes of each subaperture and the lateral shift and/or rotation between the overlapping maps. Considering that the overlapping points are prone to vary with different motion uncertainties, Day et al.^{18} proposed an iterative stitching model. An LS problem is solved by singular-value decomposition (SVD) to obtain optimal estimation of the six motion parameters, and then the overlapping points are updated in each iteration. A new objective function is obtained and again optimized. This step is repeated until the algorithm converges to an acceptable tolerance.

Stitching of a curved surface including aspheres was one of the goals right from the beginning of the proposal.^{1} Day et al.^{18,19} first extended subaperture stitching interferometry to full-sphere measurement, where Zernike polynomials were replaced by spherical harmonic functions. Griesmann et al.^{20} further applied it to measure the form and radius of full spheres. Chen and Wu^{21} extended the MAOST to 360-deg profilometry for cylinders, and recently the group made new progress in stitching interferometry for cylindrical surfaces with auxiliary null optics.^{22,23}

In the early 2000s, QED Technologies announced an automated subaperture stitching interferometer workstation capable of testing flat, spherical, and moderate aspheric surfaces.^{24,25} It is another significant milestone in the history of subaperture stitching interferometry because commercial products are now available. The company continues to make progress in inventing and commercializing new products of stitching interferometers, such as a subaperture stitching interferometer with aspheric capacity (SSIA) and an aspheric stitching interferometer. The latter can even measure steep aspheres with as much as 1000 waves (more than 600 pm) of aspheric departure thanks to the novel variable optical null (VON) technique.^{26} The VON is, in fact, a pair of counterrotating Risley prisms with an adjustable overall tilt. It can generate variable astigmatism, coma, and trefoil (not completely independent) for aberration correction and thus enables subaperture testing of various aspheric shapes.

Basically, the shape of subapertures is not restricted to circular. However, attention must be paid to the orthogonality over noncircular pupils for Zernike polynomial-based stitching algorithms. Annular subaperture stitching interferometry was first proposed by Liu et al.^{27} and then by Tronolone et al.^{28} with overlapping regions. It is a special case as it requires only a single translation along the optical axis. The defocused spherical test wavefront best matches the annular asphere at different surface heights, which effectively reduces the number of fringes. It extends the dynamic range of measurement and is able to measure aspheres without null optics. However, it is inherently applied to rotationally symmetric aspheres and does not contribute to the extension of lateral measurement range and enhancement of lateral resolution. As mentioned earlier, Zernike annular polynomials are required for polynomial-based stitching.^{29} Based on the idea of an annular subaperture test, Zygo Corp. commercialized the VeriFire Asphere product for measuring rotationally symmetric aspheres without null optics.^{30} But it does not require any overlap between the neighboring annules, and all annules are stitched together by using the precise axial position measured with distance measuring interferometers.

Aiming to develop a unified stitching algorithm for general optical surfaces, we first proposed an iterative algorithm based on the theory of configuration space.^{31} It treats the stitching optimization as a multiview surface registration problem. Different subapertures are related to each other by rigid body transformation that depends on the configuration space of the geometric feature of the test surface. The overlapping point pairs are recognized by transforming all measuring points into the global frame attached to the full aperture and then checking the enclosure with the convex hull algorithm in the *OXY* plane. Configuration parameters are optimized to obtain the minimal overlapping inconsistency. This algorithm was then revised with consideration of the lateral coordinate mapping determined by the test geometry of a spherical interferometer.^{32} Experimental verifications of the algorithm were published for cases of flat, spherical, and aspheric surfaces^{33} including special techniques required for large flats,^{34} spherical surfaces,^{35,36} quasiplanar freeform wavefronts,^{37,38} and hyper-hemispheres.^{39} The algorithm was further extended to test for convex aspheres with null^{40} or near-null optics.^{41} At the same time, methods for aspheric subaperture layout design were discussed on the basis of subaperture aberration calculation.^{42,43}

In the last five years, stitching interferometry was again discussed hotly for the testing of aspheres with circular^{44,45} or annular subapertures.^{46,47} A stereovision positioning technique and fiducial marks are employed to help with the alignment of subapertures. Liang et al.^{48,49} proposed vibration-modulated subaperture stitching interferometry, which enables subaperture testing with a very high overlapping density. Subapertures are measured on the fly, and the high overlapping density averages and reduces the measurement uncertainty.

Although so many contributors have made great efforts to advance stitching interferometry, the examples in this Spotlight come primarily our own research for the sake of convenience. Regardless, this book is a general methodological overview toward a systematical frame of stitching interferometry including classification, subaperture layout design, stitching algorithms, uncertainty analysis, and application cases.