## Azimuthal position error correction algorithm for absolute test of large optical surfaces

Optics Express, Vol. 14, Issue 20, pp. 9169-9177 (2006)

http://dx.doi.org/10.1364/OE.14.009169

Acrobat PDF (296 KB)

### Abstract

Absolute test needs test part rotation to separate errors of the interferometer itself from errors due to the test surfaces. At this time, previous absolute test algorithms assume no azimuthal position error during part rotation. For large optics whose diameters are 0.6 m and over, however, exact rotations are physically difficult. Motivated by this, we propose a new algorithm that adopts least squares technique to determine the true azimuthal positions of part rotation and consequently eliminates testing errors caused by rotation inaccuracy.

© 2006 Optical Society of America

## 1. Introduction

16. Ulf Griesmann, “Three-flat test solutions based on simple mirror symmetry” Appl. Opt. **45**, 5856–5865 (2006). [CrossRef] [PubMed]

17. C. Evans, R. Hocken, and W. Estler, “Self-calibration: reversal, redundancy, error separation, and absolute testing,” Annals of the CIRP **45**, 617–634 (1996). [CrossRef]

7. C. Ai and J. C. Wyant, “Absolute testing of flats using even and odd functions,” Appl. Opt. **32**, 4698–4703 (1993). [CrossRef] [PubMed]

8. C. Evans and R. Kestner, “Test optics error removal,” Appl. Opt. **35**, 1015–1021 (1996). [CrossRef] [PubMed]

10. W. T. Estler, C. J. Evans, and L. Z. Shao, “Uncertainty estimation for multi-position form error metrology,” Prec. Eng. **21**, 72–82 (1997). [CrossRef]

11. R. E. Parks, L. Shao, and C. J. Evans, “Pixel-based absolute topography test for three flats,” Appl. Opt. **37**, 5951–5956 (1998). [CrossRef]

16. Ulf Griesmann, “Three-flat test solutions based on simple mirror symmetry” Appl. Opt. **45**, 5856–5865 (2006). [CrossRef] [PubMed]

18. S. Kim, H. S. Yang, Y. W. Lee, and S. W. Kim, “Merit Function regression method for efficient alignment control of two-mirror optical system,” Opt. Express (to be published). [PubMed]

## 2. Basic theory

*W*from any interferometric optical testing is composed of two partial wavefront components of

*T*is the systematic error of the instrument including the reference surface, while

*P*is the surface of the part to be measured. The above simple linear superposition of the two wavefronts is not strictly true but generally valid if they are of small orders as usual in most cases of optical shop testing. To separate the two wavefronts from each other,

*P*is rotated about the optical axis by some physical means while

*T*remains stationary. Then, let

*W*

_{j}be the wavefront of

*W*sampled when

*P*is stationed at an azimuthal position of

*α*

_{j}. The subscript

*j*indicates the rotation index ranging from 0 to

*N*-1, where

*α*

_{0}=0 and

*N*is the total number of part rotations.

*T*undergoing no changes vanishes in the difference wavefront

*D*

_{j}, which is intermediately defined as the subtraction of

*P*

_{0}from

*D*

_{j}, the expression of Zernike polynomials is adopted to decompose

*P*

_{0}such as

*r*and

*θ*are the normalized radial and angular coordinates;

*(r)*the radial polynomials;

*c*

_{lk}and

*d*

_{lk}the coefficients of the angular terms. In line with

*P*

_{0}expressed in Eq. (3), the rotated wavefront

*P*

_{j}is also described as

*c*

_{lk}and

*d*

_{lk}comply with the well-known transformation rule of vector rotation when they are regarded as the two orthogonal magnitude components of a two-dimensional vector. Then, substituting both the Zernike expressions of

*P*

_{0}and

*P*

_{j}into Eq. (2) allows the coefficients of the difference wavefront Dj to be obtained such as

*Δc*

_{lk}

*≡c*

_{lk}

*-c’*

_{lk}and

*Δd*

_{lk}

*≡d*

_{lk}

*-d’ lk*. Therefore, once

*D*

_{j}have actually been sampled and fitted to solve for

*Δc*

_{lk}and

*Δd*

_{lk}, the coefficients of the original part wavefront

*P*

_{0}are readily determined as

8. C. Evans and R. Kestner, “Test optics error removal,” Appl. Opt. **35**, 1015–1021 (1996). [CrossRef] [PubMed]

11. R. E. Parks, L. Shao, and C. J. Evans, “Pixel-based absolute topography test for three flats,” Appl. Opt. **37**, 5951–5956 (1998). [CrossRef]

16. Ulf Griesmann, “Three-flat test solutions based on simple mirror symmetry” Appl. Opt. **45**, 5856–5865 (2006). [CrossRef] [PubMed]

*α*

_{j}

*=2πj/N*, with the intention of making the most of the invariant properties of the sine and cosine harmonic functions of

8. C. Evans and R. Kestner, “Test optics error removal,” Appl. Opt. **35**, 1015–1021 (1996). [CrossRef] [PubMed]

## 3. Least square algorithm

*α*

_{j}are treated as additional unknowns together with the coefficients

*c*

_{lk}and

*d*

_{lk}. Then their actual values are determined from the measured wavefronts

*D*

_{j}using least squares technique. Our new algorithm beings with decomposing the part wavefront in terms of the angular order in

*k*. Letting

*L*(

*k*) be the maximum radial order to be considered for each

*k*, the partial sum is made up in detail such as

*D*

_{j}is also expressed as

*D*

_{j}(

*r, θ*)=

*r, θ*) in which

*r,θ*) such as

*k*allows

*α*

_{j}are not correctly estimated, the computed values of

*k*such as

*α*

_{j}. On the other hand, the latter

*and*ξ 0 l k , ξ ˜

^{k}_{0l}*α*

_{j}. The necessary conditions are derived as

*k*, an initial guess is made for the azimuthal positions

*α*

_{j}so that

*are computed from Eq. (15). Then by using computed values of*ξ ˜

^{k}_{0l}*, the azimuthal positions*ξ ˜

^{k}_{0l}*α*

_{j}are upgraded from Eq. (16). Next step is go back to Eq. (15) with the new values of

*α*

_{j}and repeat the computation of

*, and*ξ ˜

^{k}_{0l}*α*

_{j}is adjusted again. The iterative computation between Eqs. (15) and (16) continues until the change of

*α*

_{j}converges into a predefined small value. Total computation time is influenced by the number of terms of Zernike polynomials in consideration, the number of rotation

*N*, and the initially guess of

*α*

_{j}. Finally, with the converged values of

*P*

_{0}is reconstructed.

## 4. Simulation and experiment

*k*=5. Comparison reveals that the least squares algorithm effectively removes almost all the instrument error although part rotations are not accurately induced as intended. On the other hand, the averaging algorithm is limited in restoring the part wavefront especially around the circumference of the measured area.

*k*=6 to 8 as shown in (a). If the number of part rotation is taken as 6, the averaging algorithm fails to remove the 6th angular harmonic error components as illustrated in (b). On the other hand, the least squares algorithm suppresses the higher order instrument errors even with the same number of part rotations, demonstrating that higher order surface irregularities of the part are examined accurately. Detailed numerical data for comparison are listed in the Fig. 2.

*k*=1-8 It has 1.763 µm in P-V value and 0.11 λ in RMS as show in Fig. 2(a). The diameter was 480 pixels. 2) Using the original wavefront, six rotated wavefronts were generated. The rotated angles were 0°, 60°, 120°, 180°, 240°, and 300°. 3) Using C++ code, six different random noise sets were automatically generated within the wavefront area. The P-V value of the noise was 0.01736 µm (1 % magnitude of the original wavefront) 4) Each random noise set (Step 3) was added on each rotated wavefront (Step 2) pixel by pixel. 5) We applied both algorithms, and calculated the residual errors due to the random noise. 6) We repeated the simulation procedure with different noise magnitude (2%, 3%, 4%, 5%, and 10% of the original wavefront.) The result is shown in Fig. 3. Our algorithm can reduce the random noise effect by 94.4 %.

*α*

_{j}as 0°, 92.17°, 180.98°, 271.20°. Table 1 shows typical aberration coefficients.

## 5. Conclusions

19. H. S. Yang, Y. W. Lee, J. B. Song, and I. W. Lee, “Null Hartmann test for fabrication of large aspheric surfaces,” Opt. Express **6**, 1839–1847 (2005). [CrossRef]

## References and links

1. | D. Malacara, “Phase shifting interferometry,” in |

2. | A. E. Jensen, “Absolute calibration method for laser Twyman-Green wave-front testing interferometers,” J. Opt. Soc. Am. |

3. | K. L. Shu, “Ray-trace analysis and data reduction methods for the Ritchey-Common test,” Appl. Opt. |

4. | B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. |

5. | K.-E. Elssner, R. Burow, J. Grzanna, and R. Spolaczyk, “Absolute sphericity measurement,” Appl. Opt. |

6. | G. Shulz and J. Grzanna, “Absolute flatness testing by the rotation method with optimal measuring error compensation,” Appl. Opt. |

7. | C. Ai and J. C. Wyant, “Absolute testing of flats using even and odd functions,” Appl. Opt. |

8. | C. Evans and R. Kestner, “Test optics error removal,” Appl. Opt. |

9. | P. Hariharan, “Interferometric testing of optical surfaces: absolute measurement of flatness,” Opt. Eng. |

10. | W. T. Estler, C. J. Evans, and L. Z. Shao, “Uncertainty estimation for multi-position form error metrology,” Prec. Eng. |

11. | R. E. Parks, L. Shao, and C. J. Evans, “Pixel-based absolute topography test for three flats,” Appl. Opt. |

12. | V. Greco, R. Tronconi, C. D. Vecchio, M. Trivi, and G. Molesini, “Absolute measurement of planarity with Fritz’s method: uncertainty evaluation,” Appl. Opt. |

13. | P. E. Murphy, T. G. Brown, and D. T. Moore, “Interference imaging for aspheric surface testing,” Appl. Opt. |

14. | K. R. Freischlad, “Absolute interferomtric testing based on reconstruction of rotational shear,” Appl. Opt. |

15. | S. Reichelt, C. Pruss, and H. J. Tiziani, “Absolute interferometric test of aspheres by use of twin computer-generated holograms,” Appl. Opt. |

16. | Ulf Griesmann, “Three-flat test solutions based on simple mirror symmetry” Appl. Opt. |

17. | C. Evans, R. Hocken, and W. Estler, “Self-calibration: reversal, redundancy, error separation, and absolute testing,” Annals of the CIRP |

18. | S. Kim, H. S. Yang, Y. W. Lee, and S. W. Kim, “Merit Function regression method for efficient alignment control of two-mirror optical system,” Opt. Express (to be published). [PubMed] |

19. | H. S. Yang, Y. W. Lee, J. B. Song, and I. W. Lee, “Null Hartmann test for fabrication of large aspheric surfaces,” Opt. Express |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.3940) Instrumentation, measurement, and metrology : Metrology

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

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: August 22, 2006

Revised Manuscript: September 14, 2006

Manuscript Accepted: September 21, 2006

Published: October 2, 2006

**Citation**

Hyug-Gyo Rhee, Yun-Woo Lee, and Seung-Woo Kim, "Azimuthal position error correction algorithm for absolute test of large optical surfaces," Opt. Express **14**, 9169-9177 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-9169

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### References

- D. Malacara, "Phase shifting interferometry," in Optical Shop Testing, 2nd ed., (Wiley, New York, 1992), Chap.14.
- A. E. Jensen, "Absolute calibration method for laser Twyman-Green wave-front testing interferometers," J. Opt. Soc. Am. 63, 1313A (1973).
- K. L. Shu, "Ray-trace analysis and data reduction methods for the Ritchey-Common test," Appl. Opt. 22, 1879-1886 (1983). [CrossRef] [PubMed]
- B. S. Fritz, "Absolute calibration of an optical flat," Opt. Eng. 23, 379-383 (1984).
- K.-E. Elssner, R. Burow, J. Grzanna, and R. Spolaczyk, "Absolute sphericity measurement," Appl. Opt. 28, 4649-4661 (1989). [CrossRef] [PubMed]
- G. Shulz and J. Grzanna, "Absolute flatness testing by the rotation method with optimal measuring error compensation," Appl. Opt. 31, 3767-3780 (1992). [CrossRef]
- C. Ai and J. C. Wyant, "Absolute testing of flats using even and odd functions," Appl. Opt. 32, 4698-4703 (1993). [CrossRef] [PubMed]
- C. Evans, R. Kestner, "Test optics error removal," Appl. Opt. 35, 1015-1021 (1996). [CrossRef] [PubMed]
- P. Hariharan, "Interferometric testing of optical surfaces: absolute measurement of flatness," Opt. Eng. 36, 2478-2481 (1997). [CrossRef]
- W. T. Estler, C. J. Evans, and L. Z. Shao, "Uncertainty estimation for multi-position form error metrology," Prec. Eng. 21, 72-82 (1997). [CrossRef]
- R. E. Parks, L. Shao, and C. J. Evans, "Pixel-based absolute topography test for three flats," Appl. Opt. 37, 5951-5956 (1998). [CrossRef]
- V. Greco, R. Tronconi, C. D. Vecchio, M. Trivi, and G. Molesini, "Absolute measurement of planarity with Fritz’s method: uncertainty evaluation," Appl. Opt. 38, 2018-2027 (1999). [CrossRef]
- P. E. Murphy, T. G. Brown, and D. T. Moore, "Interference imaging for aspheric surface testing," Appl. Opt. 39, 2122-2129 (2000). [CrossRef]
- K. R. Freischlad, "Absolute interferomtric testing based on reconstruction of rotational shear," Appl. Opt. 401637-1648 (2001). [CrossRef]
- S. Reichelt, C. Pruss, and H. J. Tiziani, "Absolute interferometric test of aspheres by use of twin computer-generated holograms," Appl. Opt. 42, 4468-4479 (2003). [CrossRef] [PubMed]
- Ulf Griesmann, "Three-flat test solutions based on simple mirror symmetry" Appl. Opt. 45, 5856-5865 (2006). [CrossRef] [PubMed]
- C. Evans, R. Hocken, and W. Estler, "Self-calibration: reversal, redundancy, error separation, and absolute testing," Annals of the CIRP 45, 617-634 (1996). [CrossRef]
- S. Kim, H. S. Yang, Y. W. Lee, and S. W. Kim, "Merit Function regression method for efficient alignment control of two-mirror optical system," Opt. Express (to be published). [PubMed]
- H. S. Yang, Y. W. Lee, J. B. Song, and I. W. Lee, "Null Hartmann test for fabrication of large aspheric surfaces," Opt. Express 6, 1839-1847 (2005). [CrossRef]

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