## Abstract

Based on our previously reported annular subaperture reconstruction algorithm with Zernike annular polynomials and matrix method, we provide an experimental demonstration by testing a parabolic mirror with the complementary annular subaperture interferometric method. By comparing the results of annular subaperture method with that of the classical auto-collimation method, it is shown that the reconstruction results are in good agreement with the auto-collimation measurement. In addition, we discuss some limitations of characterizing annular subaperture measurement data with finite Zernike coefficients in our algorithm, and also show the possibility of characterizing higher spatial frequency information with adequate Zernike coefficients. It is believable that the reported method can be extended to test the surface shape of some large concave aspheric mirrors with acceptable accuracy.

©2007 Optical Society of America

## 1. Introduction

The subaperture interferometric method [1–16] has been developed to overcome the aperture-size and slope sampling limitations of a conventional interferometer, in which the primary goal is to obtain a full-aperture map from many subaperture measurements without having to measure the entire part at one time. The subaperture method has been discussed for more than 20 years. As is shown in Fig. 1(a), this method [1–5] was originally introduced to avoid the high cost of fabricating large reference optics for testing the large parabolic mirrors. Subsequently, most efforts were made to test large planar optics [7, 8]. In fact, the applications of testing aspheric mirror with the subaperture method would be faced with more challenges. In 2003, QED Technologies has developed a general-purpose stitching interferometer workstation (SSI®) [10, 11] that can automatically carry out high-quality subaperture stitching of flat, spherical, and mild-departure aspheric surfaces up to 200 mm in diameter. Recently, the QED’s SSI-A^{TM} [12] adds asphere metrology to the SSI’s capabilities, in which the subaperture configuration is shown in Fig. 1(b). Except for the above circle subaperture method, the annular subaperture interferometric method [14–16] provides an attractive alternative to testing aspheric surfaces with any special null-forming elements (null lenses or computer-generated holograms).

The annular subaperture method for testing large aspheric surfaces is promising, but the full-aperture reconstruction accuracy is a major concern. The earliest reduction method [14] based on Zernike circle polynomials [17] was proposed by Liu *et al*. in 1988, which can calculate full-aperture Zernike coefficients from the subaperture Zernike coefficients. Liu *et al*. indicated that the overlapping areas or gaps may exist in the subaperture configuration, and overlapping will contribute to the reconstruction accuracy. As is shown in Fig. 1(c), another reduction method with successive overlapping phase maps was presented by Melozzi *et al*. [15] in 1993. Starting from the inner phase map and subtracting the misalignment errors from its adjacent annular map, the latter is brought to coincidence to the former. This process is then repeated until the requested diameter, obtaining the total phase map of aspheric surfaces under test. F.Granados-Agustín *et al*. [16] improved the method by simultaneously fitting of misalignment errors among multiple overlapping subapertures.

However, when applying the circle subaperture method with overlapping subaperture configuration [10–13] to large aspheric surface, more subapertures are needed to cover the full aperture relative to the complementary annular subaperture configuration. Hence the computational efficiency of subaperture stitching algorithm and construction of high stability and high precision hardware platform will become a big problem, and some strategies will be necessary [13]. Note that, the complementary annular subapertures shown in Fig. 1(d) can exactly cover full aperture, but the non-overlapping circle subapertures can not cover the full aperture. The complementary annular subapertures can be regarded as a special example of non-overlapping subaperture configuration. The strategy of solution for non-overlapping subaperture reconstruction and overlapping subaperture stitching is different. One attempts to achieve the full-aperture surface reconstruction by fitting polynomials to non-overlapping subapertures [1–6, 14], and the other [7–13, 15, 16] attempts to achieve high-spatial-resolution stitching result of the full aperture by minimizing the mismatch among the overlapping areas of all subapertures. Obviously, in the complementary annular subaperture configuration, the edge of subaperture data can be well determined, and the solution for the corresponding relation of overlapping areas of the adjacent annular subapertures data is not necessary to get the misalignment coefficients. Moreover, the number of complementary subapertures is relatively fewer, and computational efficiency is relatively higher.

In a previous work [18], the theoretical and experimental analysis results show that the annular wavefront expressed with the orthogonal Zernike annular polynomials [19] is more accurate and meaningful for the decomposition of aberrations and the removal of misalignments in interferometry. Recently, we reported a more accurate and efficient reconstruction algorithm based on Zernike annular polynomials and provided some numerical simulations [20]. In this paper, we focus on experimental demonstration of the high-accuracy surface shape measurement of aspheric mirror with complementary annular subaperture method. We have applied our reconstruction algorithm to real annular subaperture measurement data for an approximate 130-mm diameter and *f*/2 parabolic mirror, and we compare the reconstruction results with that of the classical auto-collimation null measurement and full-aperture non-null measurement. This paper is organized as follows. In Section 2, the basic theory of annular subaperture interferometric method is described. In Section 3, we describe the experimental setup and results. In Section 4, we discuss some limitations of characterizing annular subaperture measurement data with finite Zernike coefficients, and also show the possibility of characterizing higher spatial frequency information with adequate Zernike coefficients. The conclusion is given in Sec.5.

## 2. Theory

Let us consider the optical system shown in Fig. 2, where *f* is the focal distance of the transmission sphere, *R*
_{0} is the vertex curvature radius of aspheric surface, and *d* is the moving
distance relative to the vertex curvature measurement. When *d* = 0, the aspheric surface will be tested at its vertex center of curvature. When *d* > 0 , the aspheric surface will be tested with some defocus from its vertex center of curvature.

Increasing gradually with the distance *d*, a set of simulated interferograms are shown in Fig. 3. Note that, every over-sampled interferogram can not accurately represent the difference between the aspheric surface and the reference spherical wavefront owing to the higher fringe density involved. If a series of interferograms are taken at different longitudinal position of the aspheric surface, the resolvable subaperture interferograms can be extracted from those over-sampled interferograms containing the full-aperture surface information with the “mask” method, which is used to define the interesting sub-areas within the test part. The next critical step is to combine all subaperture data together with a suitable algorithm to get a complete surface map.

As indicated above, there are mainly different piston, tilts and defocus contributions in each subaperture measurement, owing to the misalignments introduced during the relative moving the aspheric mirror from transmission sphere of interferometer. The full-aperture wavefront with misalignments, *W*(P,Θ,ε_{0}) , can be segmented into the global surface information and the local subaperture misalignment information in terms of the Zernike annular polynomials,

where (*ρ _{k}*, θ) is the normalized local pixel coordinate system for the

*k*th subaperture, and (P, Θ) is the normalized global coordinate system for the full-aperture.

*K*is the number of subapertures,

*L*is the number of Zernike annular polynomials,

*b*is the misalignment coefficient for the

_{ki}*k*th subaperture and

*i*th Zernike mode, and

*B*is the full-aperture coefficient for the

_{i}*i*th Zernike mode. The obscuration ratio of full-aperture and the

*k*th subaperture is

*ε*

_{0}and ε

*, respectively. The*

_{k}*Z*(

_{ki}*ρ*,

_{k}*θ*,

*ε*) is the

_{k}*i*th Zernike annular polynomial of the

*k*th subaperture, and the

*Z*(P, Θ,

_{i}*ε*

_{0}) is the

*i*th Zernike annular polynomial of the full-aperture. The Zernike polynomials are ordered according to Ref. 17 and are also adopted in the commercial interferogram reduction software MetroPro [21] developed by the Zygo Corp.

In a similar way to piecewise-polynomial functions, the full-aperture wavefront can be also described as,

where *a _{ki}* is the subaperture coefficient for the

*k*th subaperture and

*i*th Zernike mode. But since the aspheric surface itself has not changed, Eq. (1) must equal Eq. (2), giving

Based on some matrix representation and transform to Eq. (3), the full-aperture Zernike coefficients *B _{i}* can be calculated, which was described in detail in a published paper [20].

## 3. Experimental results

As is shown in Fig. 4, the experimental setup for testing aspheric mirror with annular subaperture method is very simple, which consists of a Fizeau interferometer (ZYGO GPI XP 4’) and a 5-axis adjustor. The *f*/2 and 130 mm parabolic mirror under test is mounted with a 5-axis adjustor. The aspheric departure from the best-fit sphere is about four microns. The annular subaperture data can be accurately extracted by the fiducial mark and “Mask” function of MetroPro software [21]. In this experiment, two complementary subapertures were required to cover the full-aperture surface. The experimental results of inner and outer subaperture are shown in Fig. 5. From the graph, we can see that the local fringe density of every subaperture is relatively low, and can be well resolved by CCD of interferometer. The surface map of inner and outer subaperture after masking is shown in Fig. 5(b) and Fig. 5(d), respectively.

Theoretically, if an aberration-free parabolic mirror is tested at its vertex curvature by a spherical wavefront, the surface departure is purely rotationally symmetric, and has mainly the primary spherical aberration contribution. The nominal aspheric prescription will be subtracted from the reconstruction result to obtain the interesting surface form error. In addition, the tilt misalignments will introduce some coma error. Generally, the piston, tilts, defocus and coma will be removed in the null measurement of parabolic mirror. The reconstruction result with misalignments and nominal aspheric prescription removed is shown in Fig. 6, in which the first 36-term annular Zernike fitting is used.

The results of auto-collimation measurement removed piston, tilts, defocus and coma are shown in Figs. 7(a) and 7(b), and the corresponding 36-term Zernike fitting results are shown in Figs. 7(c) and 7(d). From the Fig. 7, we can see that the maps based on the phase data remain higher resolution information, and the PV (peak-to-valley) and rms (root mean square) value is approximate 0.323 *λ* and 0.061 *λ*, respectively. The maps based on the first 36-term Zernike fitting results are relatively smooth, and mainly represent the surface shape. By comparison the Figs. 6(a) and 6(b) with Figs. 7(c) and 7(d), the surface map is highly similar, and the difference of PV and rms value is approximate 0.02 *λ* and 0.001 *λ*, respectively. In summary, the reconstruction results with annular subaperture method have good agreement with the classical auto-collimation measurement.

In this experiment, the tested mild parabolic mirror may admit of an efficient full-aperture measurement with a spherical reference wavefront. The non-null measurement results only removed misalignments for full-aperture and annular subaperture method are shown in Figs. 8(a) and 8(b), respectively. Similarly, it is shown that the reconstruction result with annular subaperture method has good agreement with the full-aperture non-null measurement, and the surface map consists mainly of the primary spherical aberration mentioned above.

## 4. Discussion

The above experimental results are satisfactory, but there are some noticeable problems:(1) the adjacent annular zones are required to be complementary, and the complementarity of subapertures can be ensured by so-called characteristic fiducials in this experiment; (2) the inner and outer radius and center of each annular zone is required to determined accurately in the reconstruction algorithm, and the edge determination error of annular zone will lead to decrease the measurement precision; (3) the fitting residual rms error of inner and outer subaperture is 0.0397 and 0.0265, respectively, and the Zernike fitting error will affect on the full-aperture reconstruction accuracy; (4) the relationship between the Zernike coefficients and tilt, defocus, decentering misalignments needs further research.

Here, we discuss the existed problems characterizing the subaperture measurement data with finite Zernike coefficients. The original phase map of the inner subaperture data removed piston, tilts and defocus is shown in Fig. 9(a), and the corresponding profile and 3-D plot is shown in Figs. 9(b) and 9(c), respectively. It is noticeable that there is a great local irregularity in the center area. The surface maps represented with the first 36, 100 and 230-term Zernike coefficients are shown in Figs. 9(d), 9(e) and 9(f).Table 1 lists the residual rms value of Zernike fitting with different term number. Note that the value of the residual rms gradually decreases by increasing the terms of the polynomials. Dutton *et al*. [22] and Rimmer and Wyant [23] suggested fitting data to successively higher orders and stopping when there is no further significant reduction in the rms residual error. In this experiment, the residual rms difference between the first 36 and 45-term Zernike fitting is very small, and the fitting with the first 36-term Zernike polynomials [Fig. 9(d)] has provided an acceptable accuracy for low-order surface shape. The relatively large residual rms value (0.0397 waves) was mainly affected by the local irregularity. When the first 100 and 230-term Zernike fitting was selected, the corresponding residual rms value is 0.0222 and 0.0089 wave, respectively. As are shown in Figs. 9(e) and 9(f), the local irregular information is maintained. Figure 10 shows the residual surface maps of the first 36-term Zernike fitting for the inner and outer subaperture. Furthermore, we can clearly see that local irregularity and high-spatial-frequency information. Nevertheless, the influence of residual error in the surface shape reconstruction will become acceptable by selecting the optimal term number of Zernike fitting. Actually, the above experiment results provide a good demonstrated example.

## 5. Conclusion

We have provided an experimental demonstration of testing aspheric surface shape with complementary annular subaperture interferometric method and discussed some limitations of characterizing annular subaperture measurement data with finite Zernike annular coefficients in our algorithm. The above experimental results prove that the reported method can obtain the reconstructed full-aperture surface shape with satisfactory accuracy, and also show the concerned ability of testing the surface shape of aspheric mirrors without special additional elements. Further theoretical and experimental development of the reported method will allow characterization of some interesting higher-spatial-frequency information in the full aperture reconstruction.

## Acknowledgments

This work was supported with innovation funds from the Institute of Optics and Electronics, Chinese Academy of Sciences.

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