## Practical methods for retrace error correction in nonnull aspheric testing

Optics Express, Vol. 17, Issue 9, pp. 7025-7035 (2009)

http://dx.doi.org/10.1364/OE.17.007025

Acrobat PDF (582 KB)

### Abstract

Nonnull test is often adopted for aspheric testing. But due to its
violation of null condition, the testing rays will follow different paths from the reference and aberrations from the interferometer will not cancel out, leading to widely difference between the obtained surface figure and that of the real, which is called the Retrace-error accordingly. In this paper, retrace error of nonnull aspheric testing is analyzed in detail with conclusions that retrace error has much to do with the aperture, *F* number and surface shape error of the aspheric under test. Correcting methods are proposed according to the manner of the retrace errors. Both computer simulation and experimental results show that the proposed methods can correct the retrace error effectively. The analysis and proposed correction methods bring much to the application of nonnull aspheric testing.

© 2009 Optical Society of America

## 1. Introduction

3. F. Y. Pan, J. Burge, D. Anderson, and A. Poleshchuk, “Efficient Testing of Segmented Aspherical Mirrors by Use of a Reference Plate and Computer-Generated Holograms. II. Case Study, Error Analysis, and Experimental Validation,” Appl. Opt . **43**, 5313–5322 (2004). [PubMed]

4. D. Liu, Y. Yang, L. Wang, and Y. Zhuo, “Real time diagnosis of transient pulse laser with high repetition by radial shearing interferometer,” Appl. Opt . **46**, 8305–8314 (2007). [PubMed]

*RBT*(Random ball test) technique in calibrating a figure measuring interferometer. Blümel et al [12] proposed a method to test an asphere in a spherical test setup by an interferometer, but the method is applicable only in rotational symmetrical aspherics with small deviations from a sphere or flat. Murphy et. al [13] applied aberration theory to interpret and predict imaging errors for aspheric surface testing. But for a real system, ray tracing is easier, faster and more practical.

## 2. Retrace error analysis of nonnull aspheric testing

*Asp*is the aspheric under test, whose function is

*z*=

*f*(

*x*),

*S*is the reference sphere,

_{p}*L*

_{1},

*L*

_{2}are the aplanat and imaging lens, respectively. For convenience, we presume they are ideal lenses, with focal lengths

*f*

_{1}and

*f*

_{2}, and a distance d between them. The focus point of

*L*

_{1}superposes on the center of the reference sphere at

*C*

_{0}, and

*C*is the intersection of the ray reflected from the aspheric with the optical axis.

*F*

_{2}is the focus point of the imaging lens

*L*

_{2}, and the detect plane

*η*is ∆

*L*away from the focal plane of

*L*

_{2}.

*P*is the intersection of the testing ray with the detect plane

*η*. The marginal ray of the plane wave intersects with

*L*

_{1}at point

*I*, and intersects with the aspheric at point

*A*[

*x,f*(

*x*)] after focused by

*L*

_{1}. Because of the deviation between the aspheric and the reference sphere, longitude aberration will be yielded. The ray off the aspheric will intersect with optical axis at point

*C*which is different from

*C*

_{0}, and pass through

*L*

_{1}, then

*L*

_{2}, and finally intersect with the detect plane at point

*P*.

*IA*intersects with reference sphere at point

*S*, and consequently the deviation between the aspheric and the reference sphere is

*R*

_{Sp}is the radius of the reference sphere.

*I*(

*h*from optical axis on the aplanat) to point

*P*on the detect plane

*η*, and the

*OPL*(Optical Path Length) from point

*I*to point

*P*on the detect plane can be easily obtained.

*OPD*(Optical Path Difference) between

*OPL*and the

_{h}*OPL*of the ray along the optical axis

*OPL*is

_{0}*W*recorded by the detector.

*W*is the wavefront detected by the wavephase sensor, and

*N*

_{0}is the deviation between the aspheric under test and the reference sphere. We can call Eq. (3) the

*Coherenttesting-principle*(

*CTP*for short). But it is a pity that the

*CTP*is not always appropriate in nonnull aspheric testing [15]. Figure 2 shows the testing rays reflected from the aspheric cannot follow their original paths due to the large deviation between the aspheric under test and the reference sphere. In Fig. 2,

*AS*is the deviation between the aspheric and the reference sphere, which is

*N*

_{0}in Eq. (3). It is plain to see that the wavefront

*W*, which would be recorded by the detector, is not the simple twice of the deviation

*N*

_{0}.

## 2.1 Retrace phase error

*Err*the absolute testing error when the

*CTP*is adopted in nonnull aspheric testing, then

*CTP*is adopted to test an

*f*/2 paraboloid with the aspheric placed in the so-called

*vertex matched position*. “

*Vertex matched position*” is a situation in which the aspheric has been positioned that the focal point of the aplanat and the center of curvature of the vertex of the aspheric are coincident. Figure 3(a) shows half of the PV (Peak-to-Valley) value of the wavefront recorded by the detector (

*W*/2) and the deviation between the aspheric under test and the reference sphere (

*N*

_{0}). Figures 3(b) and 3(c) are the absolute and the relative testing error respectively when

*CTP*is adopted in nonnull aspheric testing. As is shown in Figs. 3(b) and 3(c), for paraboloids with identical

*F*number, the testing error rapidly increases as the aperture of the aspheric under test grows.

*CTP*is adopted to test a 550mm paraboloid with the aspheric placed in the vertex matched position. Figure 4(a) shows half of the PV value of the wavefront detected by the detector and the deviation between the aspheric under test and the reference sphere. Figures 4(b) and 4(c) are the absolute and the relative testing error respectively. From Fig. 4, it is also obvious that for paraboloids with identical aperture, the testing error increases as the

*F*number of the aspheric under test decreases.

*CTP*is adopted, and only when the aperture and relative aperture are both very small can the

*CTP*be used in nonnull aspheric testing.

## 2.2 Retrace coordinate error

*CTP*would result in another error besides retrace phase error, called retrace coordinate error.

*P*on the detected plane can be obtained using geometric calculation when the aspheric is perfect. Figure 6 shows the normalized coordinates of the testing rays at the aspheric surface and the detect plane, respectively (The aspheric under test is an

*f*/2, 160mm parabloid, positioned in the vertex match position and the detector is at the focal plane of the imaging lens). Figure 6(a) is the regularized coordinates of the testing rays at the aspheric surface, sampled at equally spaced zones and Fig. 6(b) is the corresponding normalized coordinates of the testing rays at the detect plane( The same marker means the same zone rays). Comparing Fig. 6(b) with Fig. 6(a), the equally spaced zone rays from the aspheric reach the detect plane at different spaced zone. That is mainly due to the transverse-ray aberration on the detected plane. For a unique asphere, the imaging lens can be designed to project the test part to the detected plane without distortion [16

16. Z. Malacara and D. Malacara, “Design of lenses to project the image of a pupil in optical testing interferometers,” Appl. Opt . **34**, 739–742 (1995) [PubMed]

*retrace coordinate error*(

*RCE*for short). If the test optics has a small local error at the 0.3 zone, due to the RCE, it will be present at the 0.1 zone in the testing result. Furthermore, lots of simulations have proved that, the change effect also alters with the aspheric under test and the optical system. Consequently the testing result in nonnull test can hardly give the real figure of the surface under test.

*CTP*can only be adopted when the aperture and relative aperture of the aspheric under test are both very small, otherwise the testing data will be doubtful. When the aperture or relative aperture is relatively large, which means the deviation between the aspheric under test and the reference wave is not small, the recorded wavefront by the detector does contain the information of the figure error of the aspheric and also the retrace errors. Measurements should be taken to distinguish the figure of the aspheric from the retrace errors.

## 2.3 Retrace error for aspherics with figure error

*f*/1.8 , 150mm parabloid, which has a figure error and is fixed in the vertex-sphere situation. Figure 7(a) is the deviation between the theoretic value of the parabloid and its vertex sphere (

*N*

_{A-V}). Figure 7(b) shows the simulated figure error of the aspheric (

*W*). Figure 7(c) is half of the wavefront recorded by the detector (

_{Asp}*W*/2). Figure 7(d) is the figure error (

_{CCD}*W*) reconstructed from the wavefront detected by the detector following the

_{Test}*CTP*. Comparing Fig. 7(d) with Fig. 7(b), the reconstructed figure error deviates greatly from the actual figure error. And Table 1 is the numerical results of the simulations.

*RPE, RCE*and the figure error of the aspheric, the testing results deviate much from the real figure error when the

*CTP*is adopted in nonnull aspheric testing. The nonnull aspheric testing is widely used to direct production in industry. However, the nonnull testing would not give convincing guidance if the aspheric under test differs greatly from the reference wave.

## 3. Retrace error corrections for nonnull aspheric testing

*W*

_{Test}contains not only the surface figure of the test aspheric, but also the retrace phase error, retrace coordinate error and the error induced by the nominal figure of the aspheric. So the wavefront on the detector

*W*

_{Test}can be expressed as

*Z*is the retrace phase error,

*R*is the retrace coordinate error,

*E*is the error induced by the surface figure of the aspheric and ∑ is the surface figure of the test aspheric. Notice that, the “⊕” in Eq. (6) denotes all the variables are not simply added up. In fact, the tested wavefront on the detector is the coactions of the

*RPE, RCE*and the figure of the aspheric.

*W*on the detector plane could be obtained simply by ray tracing. Removing

_{ideal}*W*from the tested wavefront by the detector, the retrace phase error Z would be reduced to deviations from nominal behavior.

_{ideal}## 4. Experimental validations for retrace error correction in nonnull testing

*RPE*and the

*RCE*plot, correct the measured data using the methods proposed above and compare it with the testing result by ZYGO GPI interferometer. Figure 11(a) is the result by ZYGO GPI interferometer and Fig. 11(b) the nonnull testing result after retrace-error correction. Table 2 is the numerical results of the experiment, where

*W*is the result obtained in nonnull interferometer without error correction,

_{NN}*W*the phase with

_{PCNN}*RPE*corrected from

*W*,

_{NN}*W*the phase with

_{PCNN}*RPE*and

*RPE*both corrected from

*W*, and

_{NN}*W*surface figure obtained in ZYGO interferometer. We can find that combined with the retrace error correcting method, the nonnull interferometer shows satisfying performance with high accuracy.

_{ZYGO}## 5. Error considerations and system optimization

*null fringe*even if the surface under test is perfect. The

*reference wavefront*here can be any wavefront which makes the wavefront off the aspheric resolved by the detector. Aplanats can be adopted to generate reference sphere for mild aspherics, but for deep ones, the most important thing is to find the appropriate reference wave which can reduce the wavefront slope on the detector [17, 18].

*λ*/400 can be obtained. Thus, modeling error of the system and retrace error from the figure of test part remain the main error sources of the correcting method leaves.

*μm*, and the locating precision of the transfer 0.5

*μm*, computer simulation shows the accuracy of the correcting method proposed in this paper can be better than PV

*λ*/15 . Notice that, the correcting method proposed here is more suitable for home-built testing system than commercial interferometer, because the designs of the commercial system are usually proprietary. And for a home-built interferometer, as long as the optical elements been modeled precisely, the only thing left in every testing is to determine the distance between the aspheric and the interferometer. Actually, most of the aspheric testing methods should model the testing system, such as sub-Nyquist interferometer [19

19. J. E. Greivenkamp and R. O. Gappinger, “Design of a Nonnull Interferometer for Aspheric Wave Fronts,” Appl. Opt . **43**, 5143–5151 (2004). [PubMed]

## 6. Conclusions

## References and links

1. | D. Malacara, Editor, |

2. | A. Offner, “A Null Corrector for Paraboloidal Mirrors,” Appl. Opt . |

3. | F. Y. Pan, J. Burge, D. Anderson, and A. Poleshchuk, “Efficient Testing of Segmented Aspherical Mirrors by Use of a Reference Plate and Computer-Generated Holograms. II. Case Study, Error Analysis, and Experimental Validation,” Appl. Opt . |

4. | D. Liu, Y. Yang, L. Wang, and Y. Zhuo, “Real time diagnosis of transient pulse laser with high repetition by radial shearing interferometer,” Appl. Opt . |

5. | D. Liu, Y. Yang, J. Weng, X. Zhang, B. Chen, and X. Qin, “Measurement of transient near-infrared laser pulse wavefront with high precision by radial shearing interferometer,” Opt. Commun . |

6. | O. Kown, J. C. Wyant, and C. R. Hayslett, “Rough surface interferometry at 10.6 microns,” Appl. Opt . |

7. | J. C. Wyant and K. Creath, “Two-wavelength phase-Shifting interferometer and method,” U.S. Patent No.4,832,489 (1989). |

8. | T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial shearing interferometer for in-process measurement of diamond turning,” Opt. Eng . |

9. | D. Liu, Y. Yang, Y. Shen, J. Weng, and Y. Zhuo, “System optimization of radial shearing interferometer for aspheric testing,” Proc. SPIE |

10. | J. M. Geary, M. Yoo, and G. Si, “Retrace error: a case study,” Proc. SPIE . |

11. | N. Gardner and A. Davies, “Retrace error evaluation on a figure-measuring interferometer,” Proc. SPIE |

12. | B. Thomas and B. Markus, “Interferometric asphere testing in a spherical test setup,” Proc. SPIE |

13. | P. E. Murphy, T. G. Brown, and D. T. Moore, “Interference Imaging for Aspheric Surface Testing,” Appl. Opt . |

14. | T.-C. Poon and T. Kim. “Geometrical Optics” in |

15. | J. Weng, Y. Yang, D. Liu, Y. Shen, and Y. Zhuo, “The wavefront aberration analysis and testing accuracy evaluation for the large aberration aspheric system based on the best fit sphere,” Proc. SPIE |

16. | Z. Malacara and D. Malacara, “Design of lenses to project the image of a pupil in optical testing interferometers,” Appl. Opt . |

17. | Y. Yang, D. Liu, Y. Shen, J. Weng, and Y. Zhuo, “Study on testing larger asphericity in non-null interferometer,” Proc. SPIE |

18. | J. J. Sullivan and J. E. Greivenkamp, “Design of partial nulls for testing of fast aspheric surfaces,” Proc. SPIE |

19. | J. E. Greivenkamp and R. O. Gappinger, “Design of a Nonnull Interferometer for Aspheric Wave Fronts,” Appl. Opt . |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.4640) Instrumentation, measurement, and metrology : Optical instruments

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: March 5, 2009

Revised Manuscript: April 1, 2009

Manuscript Accepted: April 2, 2009

Published: April 13, 2009

**Citation**

Dong Liu, Yongying Yang, Chao Tian, Yongjie Luo, and Lin Wang, "Practical methods for retrace error correction in nonnull aspheric testing," Opt. Express **17**, 7025-7035 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7025

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### References

- D. Malacara, Editor, Optical shop testing, 3rd Ed. (John Wiley and Sons, Inc., New Jersey 2007).
- A. Offner, "A Null Corrector for Paraboloidal Mirrors," Appl. Opt. 2, 153-156 (1963).
- F. Y. Pan, J. Burge, D. Anderson, and A. Poleshchuk, "Efficient Testing of Segmented Aspherical Mirrors by Use of a Reference Plate and Computer-Generated Holograms. II. Case Study, Error Analysis, and Experimental Validation," Appl. Opt. 43, 5313-5322 (2004). [PubMed]
- D. Liu, Y. Yang, L. Wang, and Y. Zhuo, "Real time diagnosis of transient pulse laser with high repetition by radial shearing interferometer," Appl. Opt. 46, 8305-8314 (2007). [PubMed]
- D. Liu, Y. Yang, J. Weng, X. Zhang, B. Chen, and X. Qin, "Measurement of transient near-infrared laser pulse wavefront with high precision by radial shearing interferometer, " Opt. Commun. 275, 173-178 (2007).
- O. Kown, J. C. Wyant, and C. R. Hayslett, "Rough surface interferometry at 10.6 microns," Appl. Opt. 19, 1862-1869 (1980).
- J. C. Wyant and K. Creath, "Two-wavelength phase-Shifting interferometer and method," U.S. Patent No.4,832,489 (1989).
- T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, "Radial shearing interferometer for in-process measurement of diamond turning," Opt. Eng. 39, 2696-2699 (2000).
- D. Liu, Y. Yang, Y. Shen, J. Weng, and Y. Zhuo, "System optimization of radial shearing interferometer for aspheric testing," Proc. SPIE 6834, 68340U (2007).
- J. M. Geary, M. Yoo, and G. Si, "Retrace error: a case study, " Proc. SPIE. 1776, 98-105 (1992).
- N. Gardner and A. Davies, "Retrace error evaluation on a figure-measuring interferometer," Proc. SPIE 5869, 58690V.1-8 (2005).
- B. Thomas and B. Markus, "Interferometric asphere testing in a spherical test setup," Proc. SPIE 5965, 596514.1-8 (2005).
- P. E. Murphy, T. G. Brown, and D. T. Moore, "Interference Imaging for Aspheric Surface Testing," Appl. Opt. 39, 2122-2129 (2000).
- T.-C. Poon and T. Kim. "Geometrical Optics" in Engineering Optics with Matlab, Ting-Chung Poon ed., World Scientific Publishing Co.Pte.Ltd (2006).
- J. Weng, Y. Yang, D. Liu, Y. Shen, and Y. Zhuo, "The wavefront aberration analysis and testing accuracy evaluation for the large aberration aspheric system based on the best fit sphere, " Proc. SPIE 6834, 68342V (2007).
- Z. Malacara and D. Malacara, "Design of lenses to project the image of a pupil in optical testing interferometers," Appl. Opt. 34, 739-742 (1995) [PubMed]
- Y. Yang, D. Liu, Y. Shen, J. Weng, and Y. Zhuo, "Study on testing larger asphericity in non-null interferometer, " Proc. SPIE 6834, 68340T (2007).
- J. J. Sullivan and J. E. Greivenkamp, "Design of partial nulls for testing of fast aspheric surfaces, " Proc. SPIE 6671, 66710W (2007).
- J. E. Greivenkamp and R. O. Gappinger, "Design of a Nonnull Interferometer for Aspheric Wave Fronts," Appl. Opt. 43, 5143-5151 (2004). [PubMed]

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