OSA's Digital Library

Applied Optics

Applied Optics

APPLICATIONS-CENTERED RESEARCH IN OPTICS

  • Vol. 38, Iss. 23 — Aug. 10, 1999
  • pp: 5024–5031

Solution to the shearing problem

Clemens Elster and Ingolf Weingärtner  »View Author Affiliations


Applied Optics, Vol. 38, Issue 23, pp. 5024-5031 (1999)
http://dx.doi.org/10.1364/AO.38.005024


View Full Text Article

Enhanced HTML    Acrobat PDF (128 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

Lateral shearing interferometry is a promising reference-free measurement technique for optical wave-front reconstruction. The wave front under study is coherently superposed by a laterally sheared copy of itself, and from the interferogram difference measurements of the wave front are obtained. From these difference measurements the wave front is then reconstructed. Recently, several new and efficient algorithms for evaluating lateral shearing interferograms have been suggested. So far, however, all evaluation methods are somewhat restricted, e.g., assume a priori knowledge of the wave front under study, or assume small shears, and so on. Here a new, to our knowledge, approach for the evaluation of lateral shearing interferograms is presented, which is based on an extension of the difference measurements. This so-called natural extension allows for reconstruction of that part of the underlying wave front whose information is contained in the given difference measurements. The method is not restricted to small shears and allows for high lateral resolution to be achieved. Since the method uses discrete Fourier analysis, the reconstructions can be efficiently calculated. Furthermore, it is shown that, by application of the method to the analysis of two shearing interferograms with suitably chosen shears, exact reconstruction of the underlying wave front at all evaluation points is obtained up to an arbitrary constant. The influence of noise on the results obtained by this reconstruction procedure is investigated in detail, and its stability is shown. Finally, applications to simulated measurements are presented. The results demonstrate high-quality reconstructions for single shearing interferograms and exact reconstructions for two shearing interferograms.

© 1999 Optical Society of America

OCIS Codes
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.4630) Instrumentation, measurement, and metrology : Optical inspection
(120.5050) Instrumentation, measurement, and metrology : Phase measurement

History
Original Manuscript: February 3, 1999
Revised Manuscript: May 11, 1999
Published: August 10, 1999

Citation
Clemens Elster and Ingolf Weingärtner, "Solution to the shearing problem," Appl. Opt. 38, 5024-5031 (1999)
http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-38-23-5024


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. V. Ronchi, “Le frangie di combinazione nello studio delle superficie e dei sistemi ottici,” Riv. Ottica Mecc. Precis. 2, 9–35 (1923).
  2. W. J. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc. London 59, 940–952 (1947). [CrossRef]
  3. G. Schulz, “Ein Interferenzverfahren zur absoluten Ebenheitsprüfung längs beliebiger Zentralschnitte,” Opt. Acta 14, 375–388 (1967). [CrossRef]
  4. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1989).
  5. V. Ronchi, “Forty years of history of a grating interferometer,” Appl. Opt. 3, 437–450 (1964). [CrossRef]
  6. S. Bäumer, “Quantitative Mikro-Messtechnik mit einem Lateral Shearing Interferometer,” Ph.D. dissertation (Optics Institute of Berlin Technical University, Berlin, 1995).
  7. H. von Brug, “Zernike polynomials as a basis for wave-front fitting in lateral shearing interferometry,” Appl. Opt. 36, 2788–2790 (1997). [CrossRef]
  8. K. R. Freischlad, C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A 3, 1852–1861 (1986). [CrossRef]
  9. D. L. Fried, “Least-squares fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977). [CrossRef]
  10. R. L. Frost, C. K. Rushforth, B. S. Baxter, “Fast FFT-based algorithm for phase estimation in speckle imaging,” Appl. Opt. 18, 2056–2061 (1979). [CrossRef] [PubMed]
  11. D. C. Ghiglia, L. A. Romero, “Direct phase estimation from phase differences using elliptic partial differential equation solvers,” Opt. Lett. 14, 1107–1109 (1989). [CrossRef] [PubMed]
  12. F. Guse, “Auswertung von Messungen mit Optimierungsverfahren—demonstriert an Interferometrie und Ellipsometrie,” Ph.D. dissertation (Optics Institute of Berlin Technical University, Berlin, 1996).
  13. G. Harbers, P. J. Kunst, G. W. R. Leibbrandt, “Analysis of lateral shearing interferograms by use of Zernike polynomials,” Appl. Opt. 35, 6162–6172 (1996). [CrossRef] [PubMed]
  14. R. H. Hudgin, “Wavefront reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977). [CrossRef]
  15. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979). [CrossRef]
  16. G. W. R. Leibbrandt, G. Harbers, P. J. Kunst, “Wave-front analysis with high accuracy by use of a double-grating lateral shearing interferometer,” Appl. Opt. 35, 6151–6161 (1996). [CrossRef] [PubMed]
  17. S. Loheide, I. Weingärtner, “New procedure for wavefront reconstruction,” Optik 108, 53–62 (1998).
  18. R. J. Noll, “Phase estimates from slope-type wavefront sensors,” J. Opt. Soc. Am. 68, 139–140 (1978). [CrossRef]
  19. M. P. Rimmer, “Method for evaluating lateral shearing interferometer,” Appl. Opt. 13, 623–629 (1974). [CrossRef] [PubMed]
  20. M. P. Rimmer, J. C. Wyant, “Evaluation of large aberrations using a lateral-shear interferometer having variable shear,” Appl. Opt. 14, 142–150 (1975). [CrossRef] [PubMed]
  21. F. Roddier, C. Roddier, “Wavefront reconstruction using iterative Fourier transforms,” Appl. Opt. 30, 1325–1327 (1991). [CrossRef] [PubMed]
  22. H. Schreiber, J. Schwider, “Lateral shearing interferometer based on two Ronchi gratings in series,” Appl. Opt. 36, 5321–5324 (1997). [CrossRef] [PubMed]
  23. M. Servin, D. Malacara, J. L. Marroquin, “Wave-front recovery from two orthogonal sheared interferograms,” Appl. Opt. 35, 4343–4348 (1996). [CrossRef] [PubMed]
  24. W. H. Southwell, “Wavefront estimation from wavefront slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980). [CrossRef]
  25. H. Takajo, T. Takahashi, “Least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 416–425 (1988). [CrossRef]
  26. X. Tian, M. Itoh, T. Yatagai, “Simple algorithm for large-grid phase reconstruction of lateral-shearing interferometry,” Appl. Opt. 34, 7213–7220 (1995). [CrossRef] [PubMed]
  27. K. Freischlad, “Sensitivity of heterodyne shearing interferometers,” Appl. Opt. 26, 4053–4054 (1987). [CrossRef] [PubMed]
  28. S. Loheide, I. Weingärtner, “Verfahren zur Ermittlung einer optischen, mechanischen, elektrischen oder anderen Messgrösse,” German patent197 05 609.1 (15May1997).
  29. I. Weingärtner, H. Stenger, “A simple shear-tilt interferometer for the measurement of wavefront aberration,” Optik 70, 124–126 (1985).
  30. J. B. Saunders, “Measurement of wavefronts without a reference standard. Part 1. The wavefront shearing interferometer,” J. Res. Natl. Bur. Stand. Sect. B 65B, 239–244 (1961). [CrossRef]
  31. A. Fricke, I. Weingärtner, “Verfahren zur Ermittlung einer Winkelteilung aus der Winkelteilungsdifferenz, bestimmt aus der Differenz zweier Winkelpositionen mit jeweils konstantem Winkelabstand,” German patent197 20 968.8 (patent pending).
  32. C. Elster, S. Loheide, I. Weingärtner, “Verfahren zur Ermittlung einer optischen, mechanischen, elektrischen oder anderen Messgrösse,” German patent198 33 269.6 (patent pending).
  33. C. Elster, I. Weingärtner, “Method for exact wave-front reconstruction from two lateral shearing interferograms,” J. Opt. Soc. Am. A (in press).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited