Subaperture stitching interferometry : playing jigsaw puzzles in three-dimensional space

59 Pages·2016·15.936 MB·English

by  Chen, Shanyong, Dai, Yifan, Li, Shengyi

Subaperture Stitching
Interferometry:
Jigsaw Puzzles
in 3D Space
Shanyong Chen, Shengyi Li,
and Yifan Dai
Subaperture Stitching Interferometry: Jigsaw
(cid:45)(cid:76)(cid:74)(cid:86)(cid:68)(cid:90)(cid:3)(cid:51)(cid:88)(cid:93)(cid:93)(cid:79)(cid:72)(cid:86) (cid:76)(cid:81)(cid:3)(cid:22)(cid:39) (cid:54)(cid:83)(cid:68)(cid:70)(cid:72)
by Shanyong Chen, Shengyi Li, and Yifan Dai
doi: http://dx.doi.org/10.1117/3.2242227
PDF ISBN: 9781510602236
epub ISBN: 9781510602243
mobi ISBN: 9781510602250
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Spotlight vol. SL15
Last updated 18 May 2016
Table of Contents
1 Background: Wavefront Interferometry Limitations 1
2 History: Review of Related Work 3
3 Classification: Applications of Subaperture Stitching
Interferometry 6
4 Subaperture Aberrations: Methods for Subaperture Layout Design 8
4.1 Null subaperture layout design 9
4.2 Aberrations of off-axis aspheric subapertures 10
4.3 Non-null subaperture layout design 11
4.4 Near-null optics 16
4.5 Near-null subaperture layout design 20
5 Stitching: Jigsaw Puzzles in Three-Dimensional Space 21
5.1 Mathematical background 21
5.1.1 Surface error change related to the misalignment 21
5.1.2 Simultaneous stitching 23
5.2 Configuration space-based stitching model 25
5.2.1 Object–image mapping with ray tracing 27
5.2.2 Coordinate mapping with rigid body transformation 28
5.3 Iterative algorithm for subaperture stitching 29
5.3.1 Overlapping calculation subproblem 29
5.3.2 Configuration optimization subproblem 30
5.4 Special techniques dealing with a large number of subapertures 31
5.4.1 Sparse technique 31
5.4.2 Block-wise QR decomposition 31
5.4.3 Coarse–fine stitching strategy 32
6 Uncertainty: How Error Sources Affect the Stitching 32
6.1 Noise propagation during stitching 33
6.2 Decoupling induced aberrations from surface error 34
7 Case Studies 34
7.1 Large flats 34
7.2 Hyper-hemispheres 37
iii
TableofContents iv
7.3 Large convex spheres 40
7.4 Large convex aspheres 46
8 Conclusions 48
Acknowledgments 48
References 49
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Chen,Li,andDai:SubapertureStitchingInterferometry 1
1 Background: Wavefront Interferometry Limitations
Aspheres are widely used inmodern optical systems because of their advantage of
promising higher image quality with fewer elements compared with traditional
spherical systems. Surface figure metrology becomes more challenging as increas-
ingly higher performance demands aspheres with a larger aperture, higher accu-
racy, and even more complex surface forms, e.g., larger slope variation or a
freeform surface.
Wavefront interferometry is a standard solution for the measurement of opti-
cal surface error or wavefront aberration, which is usually at the submicron or
even nanometer scale. An interferometer fundamentally outputs a test beam and
records the fringes formed by interference of the reference beam and the test
beam. The reference beam is reflected by a well-polished reference surface, and
the test beam is modulated by the test surface or system. The surface error or
wavefront aberration is then obtained by analyzing the fringe pattern. Readers
are referred to Refs. 1–3 for details of the basic principle of wavefront interferom-
etry if required.
The standard form of the reference surface, which is the last surface of the
transmission flat (TF) or transmission sphere (TS) mounted at the exit pupil of
the interferometer, is either flat or spherical. Hence, flat or spherical surfaces
can be measured in the so-called null test configuration. The name “null test”
comes from the fact that a nominally null fringe pattern is obtained because the
test wavefront perfectly matches the test surface. However, for aspheres, non-null
fringes are, of course, observed, and the number of fringes depends on the
aspheric departure from the best-fit sphere of the surface. On the other hand, the
number of resolvable fringes of the interferometer is limited by the Nyquist fre-
quency. The dynamic range of measurement of a commercial interferometer is
typically restricted to only tens of fringes.
Worse still, the lateral range of measurement is also strictly limited by the
aperture of the interferometer when testing flat or convex surfaces as the test
beam is collimated or convergent. For example, convex spheres are null tested
at confocal positions, i.e., the center of curvature of the test surface coincides with
the focus of the TS, as shown in Fig. 1. The lateral range of measurement is obvi-
ously limited by the f/number (the ratio of the focal length to the diameter) of the
TS, which should be smaller than the R/number (the ratio of the radius of curva-
ture to the diameter) of the test surface. The smallest f/number of the TS commer-
cially available is f/0.65. As a matter of fact, we cannot test a convex hemisphere
or full sphere directly, no matter how small the diameter is.
This fact is really disillusioning because wavefront interferometry is restricted
to only a small group in the family of modern optics. Great efforts have been
made to extend both the lateral and dynamic ranges of measurement, including
null and non-null tests with auxiliary optics. Subaperture stitching interferometry
solves this problem by dividing the full aperture into a series of smaller
2 Chen,Li,andDai:SubapertureStitchingInterferometry
Figure1Null testofsphericalsurfaceattheconfocalposition.
subapertures that overlap with their neighbors, as shown in Fig. 2. The subaper-
tues are measured one by one and finally stitched together to give a full-aperture
surface error map. This method has three advantages. First, very large apertures
can be measured now because what we need to do is to measure small pieces of
subapertures. Second, aspheres with larger aspheric departure can be measured
because a subaperture’s departure is reduced to generate fringes resolvable with
a standard interferometer. Finally, as a small subaperture is spotlighted, more
details of the surface error can be resolved by the interferometer. Therefore, a sur-
face error of higher spatial frequency can be obtained by subaperture stitching
interferometry.
Figure2Schematicdiagramofsubaperturestitchinginterferometry.
Chen,Li,andDai:SubapertureStitchingInterferometry 3
2 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 Kwon5 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 subaper-
tures together with minimized uncertainty? Taking individual subaperture mea-
surements 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 sur-
face error. Therefore, a subaperture stitching algorithm is indispensable to recog-
nize 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 pre-
sented: the Kwon–Thunen method5 and the simultaneous fit method developed
by Chow and Lawrence.6 Both are based on Zernike polynomials, and a compari-
son 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 sys-
tems. 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, rela-
tively low-order surface or wavefront, insufficient for describing a wavefront con-
taining 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 ofsubapertures fromthediscrete pixel-based measurements. Thisistheidea 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 num-
ber 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
4 Chen,Li,andDai:SubapertureStitchingInterferometry
canbedescribed andretrievedbystitching, butinthiscase,some redundantinfor-
mation is required because we can no longer make use of the slope continuity of
the wavefront when stitching two neighboring subapertures. In contrast, polyno-
mial-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 sub-
apertures 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 stitch-
ing 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 interfer-
ometer for large plano optics in the National Ignition Facility and Laser
MégaJoule.14,15 It is also worth mentioning that Tang,16 Wyant, and Schmit17
extended subaperture stitching interferometry to high-resolution microsurface
measurements over a large field of view. The stitching algorithm developed by
Tang16 is special in that it takes the motion uncertainties of all six degrees of free-
dom (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 pis-
ton 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