## Abstract

Limited by the aperture and f/number of the transmission sphere (TS), large convex spheres with very small R/number (ratio of the radius of curvature to the aperture) cannot be tested in a single measurement with a standard interferometer. We present the algorithm and troubleshooting for subaperture stitching test of a half meter-class convex sphere with R/0.61. Totally 90 off-axis subapertures are arranged on 5 rings around the central one. Since the subaperture is so small, its surface error is comparable with that of the TS reference error. Hence a self-calibrated stitching algorithm is proposed to separate the reference error from the measurements. Another serious problem is the nonlinear mapping between the subaperture’s local coordinates and the full aperture’s global coordinates. The nonlinearity increases remarkably with the off-axis angle. As a result, we cannot directly remove power from the full aperture error map as we usually do. Otherwise incorrect spherical aberration will be generated. We therefore propose the sphericity assessment algorithm to match the stitched surface error with a best-fit sphere. The residual is true surface error which can be used for corrective figuring or for tolerance assessment. The self-calibrated stitching and troubleshooting are demonstrated experimentally.

© 2015 Optical Society of America

## 1. Introduction

Spherical surfaces are usually measured in null test configuration with a standard interferometer by using a transmission sphere (TS). The center of curvature of the test surface coincides with the focus of the TS. The last surface of the TS is the aplanatic surface acting as the reference. In such a cofocal configuration, the aperture and the radius of curvature of the test surface are both smaller than those of the TS. Currently the TS with the smallest f/number available off the shelf is f/0.65 TS for a 4-inch interferometer. Therefore limited by the aperture and f/number of the TS, large convex spheres or small R/number spheres cannot be tested in a single measurement with a standard interferometer. R/number is the ratio of the radius of curvature to the aperture.

If the convex surface belongs to a refractive lens, it can then be tested through the back with certain auxiliary optics [1]. The aperture limit is broken since the test surface is flipped as a concave one. To break the limit of f/number, Creath *et al.* proposed a method by using aplanatic imaging points [2]. The test lens is plano-convex with its convex surface facing the interferometer. After refraction at the convex surface, the spherical test beam converges to the rear flat surface in the lens. Such a skew-symmetric configuration requires special design of the test lens and the transmitted wavefront is polluted by the material inhomogeneity.

Large convex and/or small R/number spheres can find more applications in modern optical systems such as the lithographic lens [1] and large telescopes. In telescope design, spherical and aspherical lenses are usually used as corrector near the focal plane. For example, the Large Synoptic Survey Telescope (LSST) includes a refractive camera design with 3 lenses and a set of filters [3]. One of the lenses has a convex sphere of 1.55m in diameter and 2.824m in radius of curvature.

This paper focuses on interferometric test of a half meter-class convex sphere shown in Fig. 1
. The clear aperture is 506 mm and the R/number is smaller than R/0.61 and the angle between the optical axis and the normal at the edge approaches 55°. It is near hemisphere. To avoid coupling with the material inhomogeneity, we have to reject the transmitted wavefront test method. Subaperture stitching test is employed instead to break the limits of both aperture and f/number by dividing the full aperture into a series of smaller subapertures. This method has been well discussed or applied to test of large flats, spherical surfaces and even large convex aspheres combined with null optics [4] or near-null optics [5,6
]. Stitching algorithm is critical to remove the misalignment induced aberrations from the subaperture measurements. Yan *et al.* [7] proposed a stitching algorithm with distortion corrected before the stitching coefficients are optimized, though the two aspheres measured are not steep. The distortion correction is similar with coordinate transformation between the subaperture and full aperture, and fiducials are required for aspheres. But serious problems still arises in stitching test of such a large steep surface. We present the stitching algorithm and troubleshooting in this paper. As the subaperture is much smaller than the full aperture, a self-calibrated stitching algorithm is required to separate the TS reference error. Because of the strong nonlinearlity of the coordinate mapping, we need to further process the stitching result to get the true surface error rather than directly removing power as we usually do. Although Zhang *et al.* [8] proposed a method for annular subaperture stitching of steep aspheres, self-calibration and projection nonlinearity were not discussed. Their major contribution is effectively extending the dynamic range of measurement by employing partial null lens.

## 2. Experimental setup

The subaperture test is still in a cofocal null test configuration, so the TS radius must be longer than that of the test surface (308mm). The options available are Zygo 4-inch f/7.1 TS and 6-inch f/3.5 TS. The former has a long radius of 681.7 mm and consequently requires more than 300 subapertures to cover the full aperture. We prefer the 6-inch f/3.5 TS with the radius of 475.8 mm. The size of the subaperture is about 88 mm and we need totally 90 off-axis subapertures arranged on 5 rings around the central one, as shown in Fig. 2 . The off-axis angles are 10°, 19°, 29°, 39° and 49° respectively. Subapertures are evenly spaced on each ring. Because of the big off-axis angle at the edge, the footprint of the outmost subaperture on OXY plane is elliptical.

The lens is mounted in an aluminum ring with adjustable radial clearance. Flexible material is filled between the rim-contact interfaces to spread the gravity effect around the lens rim. The aluminum ring with the lens is then mounted on the rotary table. The optical axis of the lens is carefully aligned with the axis of rotation by means of the reflective beam spot in the interferometer. The rotary table is mounted on a yawing table. Both tables are numerically controlled and fixed on a pitch adjustment device. The 6-inch interferometer is placed on a three-axis numerical control machine. The experimental setup is shown in Fig. 3 .

During the subaperture test, the test surface is moved to the subaperture’s nominal position by yawing and rotating, while the three translational axes are responsible for fine aligning and nulling the interferometer. Figure 4 shows the measurement results of the central and off-axis subapertures on different rings centered on X axis. Subaperture error on inner rings is comparable in magnitude with the TS reference error. And the outer subapertures show mostly astigmatism which indicates apparent spherical aberration exists in the full aperture error. All subaperture measurement results are saved with piston, tip-tilt and power removed. In the following sections, we present the stitching algorithm and demonstrate the effect of self-calibration with the reference error separated. We also show that the strong nonlinear coordinate mapping existing in such a small R/number sphere can produce significantly incorrect surface error. A postprocessing method is hence proposed for troubleshooting.

## 3. Self-calibrated stitching algorithm and experimental verification

#### 3.1 Self-calibrated stitching algorithm

Although the test surface is near hemisphere, there is still one-to-one coordinate mapping between the points on the sphere and their projections on OXY plane since the solid angle is smaller than 2π. Therefore we do not need to calculate the overlapping correspondence by virtue of coordinates of latitude and longitude [9].

Denote the subaperture measurements by the triplets *W* = {(*u*,*v*,*φ*)}, where *φ* is the measured height on the imaging pixel (*u*,*v*). The test geometry is shown in Fig. 5
with a local coordinate frame built at the subaperture center. Because the last surface of the TS is aplanatic, the (*x*
_{0},*y*
_{0}) coordinates on the test surface are linear with the height in the collimated space in the interferometer. Hence the lateral coordinates in the object space are related to the pixel (*u*,*v*) as below:

*r*is the radius of curvature of the convex sphere,

*r*

_{ts}and

*d*

_{ts}are the radius of curvature and diameter of the TS, respectively. And

*d*is the diameter of the imaging area in pixels.

_{p}The corresponding points (*x*
_{0},*y*
_{0},*z*
_{0}) on the nominal surface are then determined by the surface equation:

*β*and azimuthal angle

*γ*, the coordinates (

*x*,

*y*,

*z*) in the global frame are related to those in the local frame by rigid body transformations:

When the overlapping point pairs are determined, the height differences Δ*φ* between subaperture measurements are minimized in the sense of least-squares by the stitching algorithm. It can be realized by simply removing piston, tip-tilt, power, lateral shifts, clocking and Zernike polynomials from the subaperture measurements [11]:

*a*,

*b*,

*c*,

*p*,

*s*,

*t*and

*θ*are coefficients of each subaperture.

*Z*is Zernike polynomial terms describing the TS reference error.

_{i}*M*is the number of terms and

*q*is the coefficient.

_{i}The subaperture stitching and localization (SASL) algorithm we developed for general surfaces [10] is also applicable. Since the algorithm uses Cartesian coordinates by optimizing rigid body transformation parameters, we need to calculate coordinates of the measuring points (*x ^{M}*,

*y*,

^{M}*z*) in the global frame by adding signed height in surface normal direction

^{M}**n**as below:

Considering the misalignment during subaperture test, the real coordinates are finally corrected by applying rigid body transformation *g* for each subaperture and the stitching optimization is based on the Cartesian coordinates:

*s*is the number of subapertures, and

*N*

_{o}is the total number of overlapping point pairs. The left superscript

*ik*indicates the overlapping region between subapertures

*k*and

*i*.

**H**= [

*x*,

^{M}*y*,

^{M}*z*,1]

^{M}^{T}are homogeneous coordinates of the measuring points and

**h**= [

*x*

_{0},

*y*

_{0},

*z*

_{0},1]

^{T}are homogeneous coordinates of the corresponding points on the nominal sphere. Readers are referred to [10] for details on the algorithm.

The stitching optimization is finally modeled as a linear least-squares problem based on the overlapping height difference either in the form of Eq. (4) or Eq. (6). Note that as a number of subapertures are treated simultaneously, the sparse technique and blockwise QR decomposition are suggested to save memory and time [12]. Since the linear relationship between the optimization variables and the height change is valid only in a sufficiently small neighborhood of the minima, we can also add some constraints on the magnitude of the variables. The problem is therefore a ball-constrained least-squares problem:

where “s.t.” stands for “subject to”. Ball-constrained least-squares problem is typical by constraining the variables in a ball of certain radius, i.e., the norm of the vector*x*is equal to or less than the magnitude of the ball radius. It can still be solved by singular value decomposition. Additional computation involved is to solve a non-linear equation [13]. For unconstrained problem,

*α*is infinity. In our stitching experience for other spherical or aspheric surfaces, unconstrained stitching sometimes results in obviously incorrect surface error.

#### 3.2 Experimental verification of self-calibration

When we first apply the stitching algorithm to subaperture measurements of this large steep convex sphere, the reference error is not considered and the stitched error map is shown in Fig. 6(a) . It is easy to see subaperture traces almost all over the map. Such remarkable overlapping inconsistency may be ascribed to the reference error as it is comparable with the subaperture error. Therefore we try the self-calibrated stitching and get the stitched error map shown in Fig. 6(b) without visible subaperture traces in contrast. Note for the purpose of comparison, the stitching result is imported into Zygo MetroPro software and piston, tip-tilt, power, astigmatism, coma and primary spherical aberration are all removed from both error maps. The true surface error map will be given in the following section.

The calibrated reference error is described in 37-term Zernike polynomials. It is rotated by about 70° and shown in Fig. 7(a) for comparison. It is approximately consistent with Zygo’s final report of the TS. Figure 7(b) shows the scanned copy of the report including the PVr and Zernike fit result.

Readers may worry about coupling of the systematic error with the surface error. But in our algorithm the surface error is unaffected by introducing additional Zernike terms during stitching, which is indirectly verified by the consistency of self-calibrated TS reference error and Zygo’s report. On the one hand, only those components of the surface error giving identical subaperture errors are not differentiable from systematic errors. For example, the spherical aberration of rotational symmetry gives identical subaperture error for those subapertures lying on the same ring (see Fig. 2). But there is no such component giving identical error for all subapertures with different off-axis distances. Moreover, the identical subaperture error is composed of certain terms with certain proportions. For example, spherical aberration in the full aperture contains typically coma and astigmatism with certain proportions when masked in a local off-axis subaperture. Therefore, the surface error will not be smoothed out by fitting extra Zernike polynomials because we use the same systematic error form for all subapertures. On the other hand, the stitching process does not add extra Zernike terms to the surface error either. Because the QR decomposition used to solve the least-squares problem *Ax* = *b* gives a solution with at most *k* nonzero elements, where *k* is the column-rank of matrix *A*. If *A* has full column-rank, then the solution has the minimal norm. This mathematical background indicates that no extra Zernike terms will be added as systematic error to each subaperture, even if they do not contribute to the overlapping mismatch error.

## 4. Sphericity assessment of the stitching result

#### 4.1 Problem caused by nonlinear mapping

Usually when we get the error map for spherical or aspherical surfaces, we remove the piston, tip-tilt and power to evaluate the true surface error regardless of the uncertainty of the radius of curvature. However a serious problem arises when we do that for this very small R/number sphere. As mentioned above, we can add ball constraints on the stitching optimization variables. Figures 8(a) and 8(b)
show two distinct error maps which are obtained by self-calibrated stitching without ball constraints and with *α* = 0.1 described in Eq. (7), respectively.

Great difference of mostly spherical aberration exists between the two error maps. But the calibrated reference errors in Zernike polynomials are almost identical as we check them both in polynomial coefficients and in surface distribution map. In order to judge which one is incorrect we then pick up the outmost subaperture on X axis from the stitched error map. The subaperture maps shown in Figs. 8(c) and 8(d) have distinct astigmatism and Fig. 8(d) more complies with the original subaperture measurements shown in Fig. 4. However that does not necessarily mean the stitching result with *α* = 0.1 is correct. In fact both stitching results are correct and even identical if we eliminate the effect of nonlinear mapping between the subaperture’s local coordinates and the full aperture’s global coordinates. Due to this strong nonlinearity, we cannot directly remove power from the full aperture error map as we usually do.

As we can see from Eq. (3), the lateral coordinates in subaperture’s local frame is related to those in the full aperture’s global frame as below:

*γ*= 0 for off-axis subapertures on X axis. Equation (8) indicates the compression in X coordinate. As a result, power in the global frame produces astigmatism in the local frame:

For test surfaces of big R/number, the off-axis angle is relatively small and the astigmatism in Eq. (9) can be neglected. For example, the coefficient of astigmatism is only 0.015 for *β* = 10°. However for the test surface here, the angle of the outmost subaperture is *β* = 49° and the coefficient of astigmatism is about 0.285. The strong nonlinearity generates incorrect subaperture astigmatism and finally leads to significantly incorrect spherical aberration in the full aperture map. Postprocession of the stitching result is necessary to get the true surface error rather than directly removing power as we usually do.

Image distortion is a normal situation for high numerical aperture lens but it’s different in this case. As we mentioned above, the lateral coordinates on the test surface are linear with the height (in CCD pixels) in the collimated space in the interferometer because the last surface of the TS is aplanatic. Note it is not the arc length on the test surface (or proportionally the arc length on the reference sphere) that is linear with the CCD pixel height. Hence even for small f/number TS, the linear mapping between the mirror coordinates and the CCD pixel coordinates still exists with negligible image distortion. And we can still remove power from the measurement result as we usually do for spherical surface test. But this is no longer true when we stitch subapertures with big off-axis angles. The problem is special for stitching and arises from the strong nonlinearity of coordinate mapping. It may be avoided if the free compensators for each subaperture are treated with miscellaneous distortion corrected as suggested in [7]. For example, power in the subaperture measurements may incorporate some astigmatism after distortion correction since it is now described in the distorted elliptical subaperture. However this treatment is not preferred because it is not general and furthermore, Zernike polynomials are not applied on the distorted subaperture. In our algorithm, power and Zernike reference removal is processed before the subaperture coordinates are transformed into global coordinates. Both Eqs. (4) and (5)
show that the removal is based on the pixel coordinates (*u*,*v*), not the mirror coordinates (*x*,*y*) instead. In this way, the subaperture measurements are still described in circular region and we do not need to consider distortion correction during stitching. Such a stitching model is general for various surfaces. What we need to do is to eliminate the effect of nonlinear mapping at the stage of postprocession.

#### 4.2 Postprocession with sphericity assessment

The stitch result is a set of triplets {*p _{i}* = (

*x*,

_{i}*y*,

_{i}*δ*)} where (

_{i}*x*,

_{i}*y*) are lateral coordinates of projections of measuring points on OXY plane in the global frame and

_{i}*δ*is the deviation from the nominal sphere of uncertain radius. The deviation possibly contains components caused by uncertainty of the position and the radius of the sphere which cannot be directly removed by removing tilt and power from the error map. We hence propose to assess the sphericity by removing the kinematic error in rigid body transformation.

_{i}Sphericity assessment algorithms are now well developed in the field of computer-aided tolerancing. Mathematically it is to find the optimal transformation *g*∈*Q*, such that the following function is minimized:

*d*(

*g*,

_{i}*p*) is signed distance of point

_{i}*p*to the best-fit sphere and

_{i}*C*is a slack variable dealing with the radius uncertainty. Note the transformation should be acted on Cartesian coordinates of point

*p*, which are calculated by adding the signed deviation

_{i}*δ*in normal direction to the corresponding points on the nominal sphere. The configuration space

_{i}*Q*for sphericity assessment is simply three-dimensional translational group, i.e., we need only search for the optimal transformation in three orthogonal translations. Details of the algorithm for general geometric tolerances can be found in [14].

The same postprocession algorithm is now applied to the two stitching results obtained without ball constraints and with *α* = 0.1, respectively. The residual is the deviation from the best-fit sphere. It is the true surface error which can be used for corrective figuring or for tolerance assessment. The residual maps are shown in Figs. 9(a) and 9(b)
which are almost identical. We can also find that Fig. 8(b) is more close to the residual map. It is because much smaller power is removed and thus much smaller spherical aberration is incorrectly generated than in Fig. 8(a). Due to the nonlinearity of coordinate mapping, the two error maps with power directly removed are both wrong, though the two stitching results are both correct and identical in fact.

## 5. Conclusion

Subaperture stitching test of large steep spheres implies that a lot of subapertures are most likely required to cover the full aperture and strong nonlinearity exists in the mapping from the subaperture’s local frame to the full aperture’s global frame. Self-calibrated stitching algorithm is accordingly suggested to separate the TS reference error as it is comparable in magnitude with the small subaperture measurements. The effect of reference error on stitching can be revealed by the subaperture traces existing in the uncalibrated stitching map. Compliance of the calibrated reference error with the factory final report of the TS indirectly verifies the stitching algorithm.

Another serious problem arises from the nonlinearity to which we have not paid much attention. The nonlinearity increases remarkably with the off-axis angle. Because the nonlinear mapping compresses the coordinate unilaterally, power in the global frame generates incorrect astigmatism in the local frame. Consequently great error of mostly spherical aberration is generated as we remove power directly from the stitched error map. Postprocession of sphericity assessment by removing the kinematic error in rigid body transformation is proposed instead to get the true surface error map. The residual after procession is the deviation from the best-fit sphere which can be used for corrective figuring or for tolerance assessment.

## Acknowledgments

This project is supported by National Natural Science Foundation of China (Grant No. 51375488) and National Basic Research Program of China (Grant No. 2011CB013200).

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