## Experimental study on subaperture testing with iterative triangulation algorithm |

Optics Express, Vol. 21, Issue 19, pp. 22628-22644 (2013)

http://dx.doi.org/10.1364/OE.21.022628

Acrobat PDF (3287 KB)

### Abstract

Applying the iterative triangulation stitching algorithm, we provide an experimental demonstration by testing a Φ120mm flat mirror, a Φ1450mm off-axis parabolic mirror and a convex hyperboloid mirror. By comparing the stitching results with the self-examine subaperture, it shows that the reconstruction results are in consistent with that of the subaperture testing. As all the experiments are conducted with a 5-dof adjustment platform with big adjustment errors, it proves that using the above mentioned algorithm, the subaperture stitching can be easily performed without a precise positioning system. In addition, with the algorithm, we accomplish the coordinate unification between the testing and processing that makes it possible to guide the processing by the stitching result.

© 2013 OSA

## 1. Introduction

1. J. G. Thunen and O. Y. Kwon, “Full aperture testing with subaperture test optics,” Proc. SPIE **351**, 19–27 (1983). [CrossRef]

2. W. W. Chow and G. N. Lawrence, “Method for subaperture testing interferogram reduction,” Opt. Lett. **8**(9), 468–470 (1983). [CrossRef] [PubMed]

3. T. W. Stuhlinger, “Subaperture optical testing: experimental verification,” Proc. SPIE **656**, 118–127 (1986). [CrossRef]

4. A. Kulawiec, P. Murphy, and M. D. Marco, “Measurement of high-departure aspheres using subaperture stitching with the Variable Optical Null (VON^{TM}),” Proc. SPIE **7655**, 765512-1–765512-4 (2010). [CrossRef]

5. C. Supranowitz, C. M. Fee, and P. Murphy, “Asphere metrology using variable optical null technology,” Proc. SPIE **8416**, 841604-1– 841604-5 (2012). [CrossRef]

6. P. Su, J. H. Burge, and R. E. Parks, “Application of maximum likelihood reconstruction of subaperture data for measurement of large flat mirrors,” Appl. Opt. **49**(1), 21–31 (2010). [CrossRef] [PubMed]

## 2. Theory

### 2.1. The iterative triangulation algorithm

8. D. Liu, Y. Yang, C. Tian, Y. Luo, and L. Wang, “Practical methods for retrace error correction in nonnull aspheric testing,” Opt. Express **17**(9), 7025–7035 (2009). [CrossRef] [PubMed]

#### 2.1.1 Delaunay triangulation interpolation

10. P. F. Zhang, H. Zhao, X. Zhou, and J. J. Li, “Sub-aperture stitching interferometry using stereovision positioning technique,” Opt. Express **18**(14), 15216–15222 (2010). [CrossRef] [PubMed]

*X*-

*Y*plane. Then, all of the points in the

*X*-

*Y*plane. For the sake of conciseness, only the grid points falling in the overlapping area are shown as dot points in Fig. 2(a). The coordinates of the dot points in the z direction need to be calculated respectively in each subaperture. Consider the grid points in the overlapping area between the

*a*,

*b*and

*d*are coefficients of the equation,which is used to describe the plane equation of the triangle (the coefficient of z won’t be zero).

*Z*direction can be interpolated from the plane function according to the data from the

*Z*direction according to the testing data from the

#### 2.1.2. Calculation of stitching coefficients

*N*subaperture measurements and the coordinates of grid points in the

*z*direction have been calculated in each subaperture. For convenience, the

*L*is the number of terms to be fit. When we perform the stitching to a plane mirror, usually the fitting functions are limited to tip/tilt and piston and the function

*P*,

*Q*and

*R*are defined as follows,

*P*is a vector in length of

*j*represents the sequence number of the subaperture while

*k*is the sequence number of predefined functions. That means,and

*Q*is a matrix in size of

*Q*

_{((}

_{j-}_{1)⋅}

_{k}_{)((}

_{l-}_{1)⋅}

_{k’}_{)}is used to represent the element in the row

*l*also represents the sequence number of the subaperture while

*Q*

_{((}

_{j-}_{1)⋅}

_{k}_{)((}

_{l-}_{1)⋅}

_{k’}_{)}can be written as,

*R*is a vector in length of

#### 2.1.3. Position calculation of each subaperture with two-dimensional cross-correlation

#### 2.1.4. Stitching factor in the overlapping area

*O*

_{1}and

*O*

_{2}are the centers of each subaperture and

*P*is the point to be calculated in the overlapping area between the subaperture 1 and 2.

*P*can be written as,where

*P*in the subaperture 1 and 2 respectively.

*P*,

*P*,

*P*is in the overlapping area of

*N*subapertures, in a similar way to Eq. (13), the phase date of point

*P*can be written as,where

*P*in the

### 2.2. Alignment between processing and testing coordinates

11. M. Novak, C. Zhao, and J. H. Burge, “Distortion mapping correction in aspheric null testing,” Proc. SPIE **7063**, 1–8 (2008). [CrossRef]

12. C. Zhao, R. A. Sprowl, M. Bray, and J. H. Burge, “Figure measurement of a large optical flat with a Fizeau interferometer and stitching technique,” Proc. SPIE **6293**, 1–9 (2006). [CrossRef]

*θ*is the relative rotation degree between them.

*s*is the magnification to each pixel in the testing results. After correction of distortion to each phase map,

*s*should be the same to every point.

*N*is the number of marked points, matrix T can be solved and the relationship between the processing and testing coordinates can be obtained.

## 3. Experimental verification

### 3.1. *Φ120mm flat mirror*

*X*,

*Y*and

*Z*) and rotational tables (yaw and pitch) are adjusted manually. As a verification experiment, four subapertures are measured with the interferometer (up, down, left and right subapertures). The SAT results are shown in Fig. 6. At the same time, the full aperture testing to the flat mirror can be accomplished with the same interferometer. Figure 7 gives the full aperture testing result. In the experiment, the testing accuracy is within 2/1000λ. The alignments between subapertures in both

*X*and

*Y*directions are accomplished with the marked point. Figures 8 and 9 give the stitching results with traditional stitching algorithm [9] and the iterative triangulation algorithm respectively. In the experiment, the stopping criterion of the stitching is that the RMS of the residual map between every two adjacent subapertures is less than 1.5nm. After 3 circles of iteration, the relative stitching result is calculated.

### 3.2. Φ1450mm off-axis parabolic mirror

*X*,

*Y*and

*Z*) and rotational tables (yaw and pitch) are adjusted manually. The alignments between subapertures in both

*X*and

*Y*directions are accomplished with targets. In the experiment, the testing accuracy is within 3/1000λ. In total, 3 subapertures (up, middle and down) are tested and the measured results are shown in Fig. 13.

13. S. Y. Chen, S. Y. Li, Y. F. Dai, L. Y. Ding, and S. Y. Zeng, “Experimental study on subaperture testing with iterative stitching algorithm,” Opt. Express **16**(7), 4760–4765 (2008). [CrossRef] [PubMed]

### 3.3. *Φ130mm convex hyperboloid mirror*

*R*is 1227.65mm. A simple 5-dof platform is built for the SAT of the mirror as shown in Fig. 20 and the alignments between subapertures in both

*X*and

*Y*directions are accomplished with targets. In the experiment, the testing accuracy is within 3/1000λ. A Φ150mm aperture interferometer and a standard transmission sphere with

*F*= 11 are chosen to accomplish the SAT.

*r*is about:

8. D. Liu, Y. Yang, C. Tian, Y. Luo, and L. Wang, “Practical methods for retrace error correction in nonnull aspheric testing,” Opt. Express **17**(9), 7025–7035 (2009). [CrossRef] [PubMed]

## 4. Conclusion

## Acknowledgments

## References and links

1. | J. G. Thunen and O. Y. Kwon, “Full aperture testing with subaperture test optics,” Proc. SPIE |

2. | W. W. Chow and G. N. Lawrence, “Method for subaperture testing interferogram reduction,” Opt. Lett. |

3. | T. W. Stuhlinger, “Subaperture optical testing: experimental verification,” Proc. SPIE |

4. | A. Kulawiec, P. Murphy, and M. D. Marco, “Measurement of high-departure aspheres using subaperture stitching with the Variable Optical Null (VON |

5. | C. Supranowitz, C. M. Fee, and P. Murphy, “Asphere metrology using variable optical null technology,” Proc. SPIE |

6. | P. Su, J. H. Burge, and R. E. Parks, “Application of maximum likelihood reconstruction of subaperture data for measurement of large flat mirrors,” Appl. Opt. |

7. | P. Su, |

8. | D. Liu, Y. Yang, C. Tian, Y. Luo, and L. Wang, “Practical methods for retrace error correction in nonnull aspheric testing,” Opt. Express |

9. | M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurement of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. |

10. | P. F. Zhang, H. Zhao, X. Zhou, and J. J. Li, “Sub-aperture stitching interferometry using stereovision positioning technique,” Opt. Express |

11. | M. Novak, C. Zhao, and J. H. Burge, “Distortion mapping correction in aspheric null testing,” Proc. SPIE |

12. | C. Zhao, R. A. Sprowl, M. Bray, and J. H. Burge, “Figure measurement of a large optical flat with a Fizeau interferometer and stitching technique,” Proc. SPIE |

13. | S. Y. Chen, S. Y. Li, Y. F. Dai, L. Y. Ding, and S. Y. Zeng, “Experimental study on subaperture testing with iterative stitching algorithm,” Opt. Express |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.6650) Instrumentation, measurement, and metrology : Surface measurements, figure

(220.4610) Optical design and fabrication : Optical fabrication

(220.4840) Optical design and fabrication : Testing

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: May 30, 2013

Revised Manuscript: August 29, 2013

Manuscript Accepted: September 3, 2013

Published: September 18, 2013

**Citation**

Lisong Yan, Xiaokun Wang, Ligong Zheng, Xuefeng Zeng, Haixiang Hu, and Xuejun Zhang, "Experimental study on subaperture testing with iterative triangulation algorithm," Opt. Express **21**, 22628-22644 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-19-22628

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### References

- J. G. Thunen, O. Y. Kwon, “Full aperture testing with subaperture test optics,” Proc. SPIE 351, 19–27 (1983). [CrossRef]
- W. W. Chow, G. N. Lawrence, “Method for subaperture testing interferogram reduction,” Opt. Lett. 8(9), 468–470 (1983). [CrossRef] [PubMed]
- T. W. Stuhlinger, “Subaperture optical testing: experimental verification,” Proc. SPIE 656, 118–127 (1986). [CrossRef]
- A. Kulawiec, P. Murphy, M. D. Marco, “Measurement of high-departure aspheres using subaperture stitching with the Variable Optical Null (VONTM),” Proc. SPIE 7655, 765512-1–765512-4 (2010). [CrossRef]
- C. Supranowitz, C. M. Fee, P. Murphy, “Asphere metrology using variable optical null technology,” Proc. SPIE 8416, 841604-1– 841604-5 (2012). [CrossRef]
- P. Su, J. H. Burge, R. E. Parks, “Application of maximum likelihood reconstruction of subaperture data for measurement of large flat mirrors,” Appl. Opt. 49(1), 21–31 (2010). [CrossRef] [PubMed]
- P. Su, Absolute Measurements of Large Mirrors (The University Of Arizona 2008).
- D. Liu, Y. Yang, C. Tian, Y. Luo, L. Wang, “Practical methods for retrace error correction in nonnull aspheric testing,” Opt. Express 17(9), 7025–7035 (2009). [CrossRef] [PubMed]
- M. Otsubo, K. Okada, J. Tsujiuchi, “Measurement of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33(2), 608–613 (1994).
- P. F. Zhang, H. Zhao, X. Zhou, J. J. Li, “Sub-aperture stitching interferometry using stereovision positioning technique,” Opt. Express 18(14), 15216–15222 (2010). [CrossRef] [PubMed]
- M. Novak, C. Zhao, J. H. Burge, “Distortion mapping correction in aspheric null testing,” Proc. SPIE 7063, 1–8 (2008). [CrossRef]
- C. Zhao, R. A. Sprowl, M. Bray, J. H. Burge, “Figure measurement of a large optical flat with a Fizeau interferometer and stitching technique,” Proc. SPIE 6293, 1–9 (2006). [CrossRef]
- S. Y. Chen, S. Y. Li, Y. F. Dai, L. Y. Ding, S. Y. Zeng, “Experimental study on subaperture testing with iterative stitching algorithm,” Opt. Express 16(7), 4760–4765 (2008). [CrossRef] [PubMed]

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