Diffraction effects for interferometric measurements due to imaging aberrations |
Optics Express, Vol. 20, Issue 4, pp. 4403-4418 (2012)
http://dx.doi.org/10.1364/OE.20.004403
Acrobat PDF (1898 KB)
Abstract
Aspheric surfaces are often measured using interferometers with null correctors, either refractive or diffractive. The use of null correctors allows high accuracy in the measurement, but also introduces imaging aberrations, such as mapping distortion and field curvature. These imaging aberrations couple with diffraction effects and limit the accuracy of the measurements, causing high frequency features in the surface under test to be filtered out and creating artifacts near boundaries, especially at edges. We provide a concise methodology for analyzing these effects using the astigmatic field curves to define the aberration, and showing how this couples with diffraction as represented by the Talbot effect and Fresnel edge diffraction. The resulting relationships are validated with both computer simulations and direct measurements from an interferometer with CGH null corrector.
© 2012 OSA
1. Introduction
3. J. H. Burge, “Applications of computer-generated holograms for interferometric measurement of large aspheric optics,” Proc. SPIE 2576, 258–269 (1995). [CrossRef]
4. C. Zhao and J. H. Burge, “Imaging aberrations from null correctors,” Proc. SPIE 6723, 67230L(2007). [CrossRef]
5. M. Novak, C. Zhao, and J. H. Burge, “Distortion mapping correction in aspheric null testing,” Proc. SPIE 7063, 706313, 706313-8 (2008). [CrossRef]
6. P. Zhou and J. H. Burge, “Analysis of wavefront propagation using the Talbot effect,” Appl. Opt. 49(28), 5351–5359 (2010). [CrossRef] [PubMed]
9. J. H. Burge, C. Zhao, and P. Zhou, “Imaging issues for interferometry with CGH null correctors,” Proc. SPIE 7739, 77390T (2010). [CrossRef]
2. Imaging aberrations in interferometry
10. E. Novak, C. Ai, and J. C. Wyant, “Transfer function characterization of laser Fizeau interferometer for high spatial frequency phase measurements,” Proc. SPIE 3134, 114–121 (1997). [CrossRef]
4. C. Zhao and J. H. Burge, “Imaging aberrations from null correctors,” Proc. SPIE 6723, 67230L(2007). [CrossRef]
- • The surface defects appear shifted. This can be mitigated by remapping the data using fiducial marks, which are placed on the optic under test to find the mapping relation;
- • Lower order alignment errors appear as higher order wavefront errors. This has been discussed in detail by Selberg and Murphy [11, 12].
12. P. E. Murphy, T. G. Brown, and D. T. Moore, “Measurement and calibration of interferometric imaging aberrations,” Appl. Opt. 39(34), 6421–6429 (2000). [CrossRef] [PubMed]
- • High frequency data is filtered out. This phase smoothing can be treated using a small-phase approximation to the well-known Talbot imaging relations [6].
6. P. Zhou and J. H. Burge, “Analysis of wavefront propagation using the Talbot effect,” Appl. Opt. 49(28), 5351–5359 (2010). [CrossRef] [PubMed]
- • Diffraction ripples from the edges introduce measurement artifacts. This effect can be approximated using Fresnel integrals for a knife edge [13].
3. Diffraction effects in interferometry
3.1 Phase smoothing analysis using Talbot model
6. P. Zhou and J. H. Burge, “Analysis of wavefront propagation using the Talbot effect,” Appl. Opt. 49(28), 5351–5359 (2010). [CrossRef] [PubMed]
6. P. Zhou and J. H. Burge, “Analysis of wavefront propagation using the Talbot effect,” Appl. Opt. 49(28), 5351–5359 (2010). [CrossRef] [PubMed]
3.2 Edge effects
4. Coupling of quadratic imaging aberration with diffraction effects
4.1 Smoothing effect due to field curves and mapping distortion
14. C. Zhao and J. H. Burge, “Generalization of the Coddington equations to include hybrid diffractive surfaces,” Proc. SPIE 7652, 76522U (2010). [CrossRef]
- z_{t}(x, y) and z_{s}(x, y) give the focus error at a field point (x, y) given by the t and s field curves
- 2a is the diameter of the surface under test, and
- m_{t}(x, y) and m_{s}(x, y) are the local magnification for the image in the t and s directions.
4.2 Edge effect due to field curves and mapping distortion
4.3 Diffraction effects for features with general orientation
14. C. Zhao and J. H. Burge, “Generalization of the Coddington equations to include hybrid diffractive surfaces,” Proc. SPIE 7652, 76522U (2010). [CrossRef]
4.4 Validation for diffraction effects for features with general orientation
15. Zemax, “Design tools,” http://www.zemax.com/.
5. Physical insight of the imaging aberrations
6. P. Zhou and J. H. Burge, “Analysis of wavefront propagation using the Talbot effect,” Appl. Opt. 49(28), 5351–5359 (2010). [CrossRef] [PubMed]
6. Experimental verification
6.1 Phase smoothing
6. P. Zhou and J. H. Burge, “Analysis of wavefront propagation using the Talbot effect,” Appl. Opt. 49(28), 5351–5359 (2010). [CrossRef] [PubMed]
6.2 Edge effect
7. Conclusion
References and links
1. | D. Malacara, Optical Shop Testing, 3rd ed. (Wiley 2007). |
2. | J. M. Sasian, “Design of null lens correctors for the testing of astronomical optics,” Opt. Eng. 27, 1051 (1988). |
3. | J. H. Burge, “Applications of computer-generated holograms for interferometric measurement of large aspheric optics,” Proc. SPIE 2576, 258–269 (1995). [CrossRef] |
4. | C. Zhao and J. H. Burge, “Imaging aberrations from null correctors,” Proc. SPIE 6723, 67230L(2007). [CrossRef] |
5. | M. Novak, C. Zhao, and J. H. Burge, “Distortion mapping correction in aspheric null testing,” Proc. SPIE 7063, 706313, 706313-8 (2008). [CrossRef] |
6. | P. Zhou and J. H. Burge, “Analysis of wavefront propagation using the Talbot effect,” Appl. Opt. 49(28), 5351–5359 (2010). [CrossRef] [PubMed] |
7. | P. Zhou and J. H. Burge, “Diffraction effects in interferometry,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OMA3. |
8. | P. Zhou, J. H. Burge, and C. Zhao, “Imaging issues for interferometric measurement of aspheric surfaces using CGH null correctors,” Proc. SPIE 7790, 77900L (2010). |
9. | J. H. Burge, C. Zhao, and P. Zhou, “Imaging issues for interferometry with CGH null correctors,” Proc. SPIE 7739, 77390T (2010). [CrossRef] |
10. | E. Novak, C. Ai, and J. C. Wyant, “Transfer function characterization of laser Fizeau interferometer for high spatial frequency phase measurements,” Proc. SPIE 3134, 114–121 (1997). [CrossRef] |
11. | L. A. Selberg, “Interferometer accuracy and precision,” Proc. SPIE 1400, 24–32 (1990). |
12. | P. E. Murphy, T. G. Brown, and D. T. Moore, “Measurement and calibration of interferometric imaging aberrations,” Appl. Opt. 39(34), 6421–6429 (2000). [CrossRef] [PubMed] |
13. | J. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005), pp. 88–91. |
14. | C. Zhao and J. H. Burge, “Generalization of the Coddington equations to include hybrid diffractive surfaces,” Proc. SPIE 7652, 76522U (2010). [CrossRef] |
15. | Zemax, “Design tools,” http://www.zemax.com/. |
OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(120.3180) Instrumentation, measurement, and metrology : Interferometry
ToC Category:
Instrumentation, Measurement, and Metrology
History
Original Manuscript: November 22, 2011
Revised Manuscript: January 20, 2012
Manuscript Accepted: January 24, 2012
Published: February 8, 2012
Citation
Ping Zhou, Yong Shu, Chunyu Zhao, and Jim H. Burge, "Diffraction effects for interferometric measurements due to imaging aberrations," Opt. Express 20, 4403-4418 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-4403
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References
- D. Malacara, Optical Shop Testing, 3rd ed. (Wiley 2007).
- J. M. Sasian, “Design of null lens correctors for the testing of astronomical optics,” Opt. Eng.27, 1051 (1988).
- J. H. Burge, “Applications of computer-generated holograms for interferometric measurement of large aspheric optics,” Proc. SPIE2576, 258–269 (1995). [CrossRef]
- C. Zhao and J. H. Burge, “Imaging aberrations from null correctors,” Proc. SPIE6723, 67230L(2007). [CrossRef]
- M. Novak, C. Zhao, and J. H. Burge, “Distortion mapping correction in aspheric null testing,” Proc. SPIE7063, 706313, 706313-8 (2008). [CrossRef]
- P. Zhou and J. H. Burge, “Analysis of wavefront propagation using the Talbot effect,” Appl. Opt.49(28), 5351–5359 (2010). [CrossRef] [PubMed]
- P. Zhou and J. H. Burge, “Diffraction effects in interferometry,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OMA3.
- P. Zhou, J. H. Burge, and C. Zhao, “Imaging issues for interferometric measurement of aspheric surfaces using CGH null correctors,” Proc. SPIE7790, 77900L (2010).
- J. H. Burge, C. Zhao, and P. Zhou, “Imaging issues for interferometry with CGH null correctors,” Proc. SPIE7739, 77390T (2010). [CrossRef]
- E. Novak, C. Ai, and J. C. Wyant, “Transfer function characterization of laser Fizeau interferometer for high spatial frequency phase measurements,” Proc. SPIE3134, 114–121 (1997). [CrossRef]
- L. A. Selberg, “Interferometer accuracy and precision,” Proc. SPIE1400, 24–32 (1990).
- P. E. Murphy, T. G. Brown, and D. T. Moore, “Measurement and calibration of interferometric imaging aberrations,” Appl. Opt.39(34), 6421–6429 (2000). [CrossRef] [PubMed]
- J. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005), pp. 88–91.
- C. Zhao and J. H. Burge, “Generalization of the Coddington equations to include hybrid diffractive surfaces,” Proc. SPIE7652, 76522U (2010). [CrossRef]
- Zemax, “Design tools,” http://www.zemax.com/ .
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