Synthetic phase-shifting for optical testing: Point-diffraction interferometry without null optics or phase shifters |
Optics Express, Vol. 21, Issue 22, pp. 26398-26417 (2013)
http://dx.doi.org/10.1364/OE.21.026398
Acrobat PDF (4370 KB)
Abstract
An innovative iterative search method called the synthetic phase-shifting (SPS) algorithm is proposed. This search algorithm is used for maximum-likelihood (ML) estimation of a wavefront that is described by a finite set of Zernike Fringe polynomials. In this paper, we estimate the coefficient, or parameter, values of the wavefront using a single interferogram obtained from a point-diffraction interferometer (PDI). In order to find the estimates, we first calculate the squared-difference between the measured and simulated interferograms. Under certain assumptions, this squared-difference image can be treated as an interferogram showing the phase difference between the true wavefront deviation and simulated wavefront deviation. The wavefront deviation is the difference between the reference and the test wavefronts. We calculate the phase difference using a traditional phase-shifting technique without physical phase-shifters. We present a detailed forward model for the PDI interferogram, including the effect of the finite size of a detector pixel. The algorithm was validated with computational studies and its performance and constraints are discussed. A prototype PDI was built and the algorithm was also experimentally validated. A large wavefront deviation was successfully estimated without using null optics or physical phase-shifters. The experimental result shows that the proposed algorithm has great potential to provide an accurate tool for non-null testing.
© 2013 Optical Society of America
1. Introduction
1. V. Genberg, G. Michels, and K. B. Doyle, “Orthogonality of Zernike polynomials,” Proc. SPIE 4771, 276–286 (2002). [CrossRef]
3. J. E. Greivenkamp and R. O. Gappinger, “Design of a nonnull interferometer for aspheric wave fronts,” Appl. Opt. 43(27), 5143–5151 (2004). [CrossRef] [PubMed]
3. J. E. Greivenkamp and R. O. Gappinger, “Design of a nonnull interferometer for aspheric wave fronts,” Appl. Opt. 43(27), 5143–5151 (2004). [CrossRef] [PubMed]
5. J. E. Greivenkamp, A. E. Lowman, and R. J. Palum, “Sub‐Nyquist interferometry: implementation and measurement capability,” Opt. Eng. 35(10), 2962–2969 (1996). [CrossRef]
7. H. H. Barrett, C. Dainty, and D. Lara, “Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions,” J. Opt. Soc. Am. A 24(2), 391–414 (2007). [CrossRef] [PubMed]
8. J. A. Sakamoto and H. H. Barrett, “Maximum-likelihood estimation of parameterized wavefronts from multifocal data,” Opt. Express 20(14), 15928–15944 (2012). [CrossRef] [PubMed]
8. J. A. Sakamoto and H. H. Barrett, “Maximum-likelihood estimation of parameterized wavefronts from multifocal data,” Opt. Express 20(14), 15928–15944 (2012). [CrossRef] [PubMed]
9. J. A. Sakamoto, H. H. Barrett, and A. V. Goncharov, “Inverse optical design of the human eye using likelihood methods and wavefront sensing,” Opt. Express 16(1), 304–314 (2008). [CrossRef] [PubMed]
7. H. H. Barrett, C. Dainty, and D. Lara, “Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions,” J. Opt. Soc. Am. A 24(2), 391–414 (2007). [CrossRef] [PubMed]
2. Theory
2.1 Maximum-likelihood estimation
7. H. H. Barrett, C. Dainty, and D. Lara, “Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions,” J. Opt. Soc. Am. A 24(2), 391–414 (2007). [CrossRef] [PubMed]
2.2 The synthetic phase-shifting algorithm
12. R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two‐dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988). [CrossRef]
2.3 The iterative process of the synthetic phase-shifting algorithm
3. Numerical studies
3.1 An interferogram with a spatial frequency lower than the Nyquist condition
3.2 An interferogram with a spatial frequency higher than the Nyquist condition
3.3 The convergence study
4. Experimental study
4.1 General design and layout
4.2 Point-diffraction interferometer plate
18. J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift point-diffraction interferometer,” Proc. SPIE 5531, 264–272 (2004). [CrossRef]
4.3 Application of the synthetic phase-shifting algorithm
5. Discussion
6. Conclusion
7. Appendix
7.1 Forward model for the point-diffraction interferometry
7.2 The derivation of the squared difference between the measured and simulated CCD outputs
Acknowledgments
References and links
1. | V. Genberg, G. Michels, and K. B. Doyle, “Orthogonality of Zernike polynomials,” Proc. SPIE 4771, 276–286 (2002). [CrossRef] |
2. | D. Malacara, Optical Shop Testing (Wiley, 1978). |
3. | J. E. Greivenkamp and R. O. Gappinger, “Design of a nonnull interferometer for aspheric wave fronts,” Appl. Opt. 43(27), 5143–5151 (2004). [CrossRef] [PubMed] |
4. | J. E. Greivenkamp, “Sub-Nyquist interferometry,” Appl. Opt. 26(24), 5245–5258 (1987). [CrossRef] [PubMed] |
5. | J. E. Greivenkamp, A. E. Lowman, and R. J. Palum, “Sub‐Nyquist interferometry: implementation and measurement capability,” Opt. Eng. 35(10), 2962–2969 (1996). [CrossRef] |
6. | H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004). |
7. | H. H. Barrett, C. Dainty, and D. Lara, “Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions,” J. Opt. Soc. Am. A 24(2), 391–414 (2007). [CrossRef] [PubMed] |
8. | J. A. Sakamoto and H. H. Barrett, “Maximum-likelihood estimation of parameterized wavefronts from multifocal data,” Opt. Express 20(14), 15928–15944 (2012). [CrossRef] [PubMed] |
9. | J. A. Sakamoto, H. H. Barrett, and A. V. Goncharov, “Inverse optical design of the human eye using likelihood methods and wavefront sensing,” Opt. Express 16(1), 304–314 (2008). [CrossRef] [PubMed] |
10. | W. P. Linnik, “A simple interferometer for the investigation of optical systems,” C. R. Acad. Sci. URSS 5, 208–210 (1933). |
11. | R. N. Smartt and W. H. Steel, “Theory and Application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1975). |
12. | R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two‐dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988). [CrossRef] |
13. | D. Ruijters, B. M. ter Harr Romeny, and P. Suetens, "Efficient GPU-accelerated elastic image registration," in Proceedings Sixth IASTED international conference on biomedical engineering (BioMed), pp. 419-424 (2008). |
14. | G. E. Sommargren, "Phase shifting diffraction interferometry for measuring extreme ultraviolet optics," No. UCRL-JC--123549, CONF-9604150--1, Lawrence Livermore National Lab., CA (1996). |
15. | C. R. Mercer and K. Creath, “Liquid-crystal point-diffraction interferometer for wave-front measurements,” Appl. Opt. 35(10), 1633–1642 (1996). [CrossRef] [PubMed] |
16. | R. M. Neal and J. C. Wyant, “Polarization phase-shifting point-diffraction interferometer,” Appl. Opt. 45(15), 3463–3476 (2006). [CrossRef] [PubMed] |
17. | M. Paturzo, F. Pignatiello, S. Grilli, S. De Nicola, and P. Ferraro, “Phase-shifting point-diffraction interferometer developed by using the electro-optic effect in ferroelectric crystals,” Opt. Lett. 31(24), 3597–3599 (2006). [CrossRef] [PubMed] |
18. | J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift point-diffraction interferometer,” Proc. SPIE 5531, 264–272 (2004). [CrossRef] |
19. | P. Su, J. Burge, R. A. Sprowl, and J. Sasian,"Maximum likelihood estimation as a general method of combining subaperture data for interferometric testing," Proc. SPIE 6342, 1X-1X-6 (2006). |
OCIS Codes
(100.2650) Image processing : Fringe analysis
(120.5050) Instrumentation, measurement, and metrology : Phase measurement
(110.3175) Imaging systems : Interferometric imaging
ToC Category:
Instrumentation, Measurement, and Metrology
History
Original Manuscript: July 1, 2013
Revised Manuscript: October 13, 2013
Manuscript Accepted: October 21, 2013
Published: October 28, 2013
Citation
Ryeojin Park, Dae Wook Kim, and Harrison H. Barrett, "Synthetic phase-shifting for optical testing: Point-diffraction interferometry without null optics or phase shifters," Opt. Express 21, 26398-26417 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-22-26398
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References
- V. Genberg, G. Michels, and K. B. Doyle, “Orthogonality of Zernike polynomials,” Proc. SPIE4771, 276–286 (2002). [CrossRef]
- D. Malacara, Optical Shop Testing (Wiley, 1978).
- J. E. Greivenkamp and R. O. Gappinger, “Design of a nonnull interferometer for aspheric wave fronts,” Appl. Opt.43(27), 5143–5151 (2004). [CrossRef] [PubMed]
- J. E. Greivenkamp, “Sub-Nyquist interferometry,” Appl. Opt.26(24), 5245–5258 (1987). [CrossRef] [PubMed]
- J. E. Greivenkamp, A. E. Lowman, and R. J. Palum, “Sub‐Nyquist interferometry: implementation and measurement capability,” Opt. Eng.35(10), 2962–2969 (1996). [CrossRef]
- H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).
- H. H. Barrett, C. Dainty, and D. Lara, “Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions,” J. Opt. Soc. Am. A24(2), 391–414 (2007). [CrossRef] [PubMed]
- J. A. Sakamoto and H. H. Barrett, “Maximum-likelihood estimation of parameterized wavefronts from multifocal data,” Opt. Express20(14), 15928–15944 (2012). [CrossRef] [PubMed]
- J. A. Sakamoto, H. H. Barrett, and A. V. Goncharov, “Inverse optical design of the human eye using likelihood methods and wavefront sensing,” Opt. Express16(1), 304–314 (2008). [CrossRef] [PubMed]
- W. P. Linnik, “A simple interferometer for the investigation of optical systems,” C. R. Acad. Sci. URSS5, 208–210 (1933).
- R. N. Smartt and W. H. Steel, “Theory and Application of point-diffraction interferometers,” Jpn. J. Appl. Phys.14, 351–356 (1975).
- R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two‐dimensional phase unwrapping,” Radio Sci.23(4), 713–720 (1988). [CrossRef]
- D. Ruijters, B. M. ter Harr Romeny, and P. Suetens, "Efficient GPU-accelerated elastic image registration," in Proceedings Sixth IASTED international conference on biomedical engineering (BioMed), pp. 419-424 (2008).
- G. E. Sommargren, "Phase shifting diffraction interferometry for measuring extreme ultraviolet optics," No. UCRL-JC--123549, CONF-9604150--1, Lawrence Livermore National Lab., CA (1996).
- C. R. Mercer and K. Creath, “Liquid-crystal point-diffraction interferometer for wave-front measurements,” Appl. Opt.35(10), 1633–1642 (1996). [CrossRef] [PubMed]
- R. M. Neal and J. C. Wyant, “Polarization phase-shifting point-diffraction interferometer,” Appl. Opt.45(15), 3463–3476 (2006). [CrossRef] [PubMed]
- M. Paturzo, F. Pignatiello, S. Grilli, S. De Nicola, and P. Ferraro, “Phase-shifting point-diffraction interferometer developed by using the electro-optic effect in ferroelectric crystals,” Opt. Lett.31(24), 3597–3599 (2006). [CrossRef] [PubMed]
- J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift point-diffraction interferometer,” Proc. SPIE5531, 264–272 (2004). [CrossRef]
- P. Su, J. Burge, R. A. Sprowl, and J. Sasian,"Maximum likelihood estimation as a general method of combining subaperture data for interferometric testing," Proc. SPIE 6342, 1X-1X-6 (2006).
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