OSA's Digital Library

Applied Optics

Applied Optics

APPLICATIONS-CENTERED RESEARCH IN OPTICS

  • Editor: James C. Wyant
  • Vol. 47, Iss. 9 — Mar. 20, 2008
  • pp: 1327–1335

Tilt sensitivity of the two-grating interferometer

Christopher N. Anderson and Patrick P. Naulleau  »View Author Affiliations


Applied Optics, Vol. 47, Issue 9, pp. 1327-1335 (2008)
http://dx.doi.org/10.1364/AO.47.001327


View Full Text Article

Enhanced HTML    Acrobat PDF (1448 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

Fringe formation in the two-grating interferometer is analyzed in the presence of a small parallelism error between the diffraction gratings assumed in the direction of grating shear. Our analysis shows that with partially coherent illumination, fringe contrast in the interference plane is reduced in the presence of nonzero grating tilt with the effect proportional to the grating tilt angle and the grating spatial frequencies. Our analysis also shows that for a given angle between the gratings there is an angle between the final grating and the interference plane that optimizes fringe contrast across the field.

© 2008 Optical Society of America

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(050.0050) Diffraction and gratings : Diffraction and gratings
(070.0070) Fourier optics and signal processing : Fourier optics and signal processing
(110.3175) Imaging systems : Interferometric imaging

ToC Category:
Imaging Systems

History
Original Manuscript: October 2, 2007
Revised Manuscript: January 4, 2008
Manuscript Accepted: January 9, 2008
Published: March 18, 2008

Citation
Christopher N. Anderson and Patrick P. Naulleau, "Tilt sensitivity of the two-grating interferometer," Appl. Opt. 47, 1327-1335 (2008)
http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-47-9-1327


Sort:  Year  |  Journal  |  Reset  

References

  1. F. J. Weinberg and N. B. Wood, “Interferometer based on four diffraction gratings,” J. Sci. Instrum. 36, 227-230 (1959). [CrossRef]
  2. E. N. Leith and B. J. Chang, “Space-invariant holography with quasicoherent light,” Appl. Opt. 12, 1957-1963 (1973). [CrossRef] [PubMed]
  3. E. N. Leith and B. J. Chang, “Image formation with an achromatic interferometer,” Opt. Commun. 23, 217-219 (1977). [CrossRef]
  4. B. J. Chang, R. C. Alferness, and E. N. Leith, “Space-invariant achromatic grating interferometers: theory,” Appl. Opt. 14, 1592-1600 (1975). [CrossRef] [PubMed]
  5. B. J. Chang, “Grating-based interferometers,” Ph.D. dissertation, (University Michigan, 1974), University Microfilm 74-23-170.
  6. Y. S. Cheng, “Fringe formation in incoherent light with a two-grating interferometer,” Appl. Opt. 23, 3057-3059 (1984). [CrossRef] [PubMed]
  7. Y. S. Cheng, “Temporal coherence requirement in a symmetric-path grating interferometer,” Appl. Opt. 36, 800-804 (1997). [CrossRef] [PubMed]
  8. K. Patorski, “Talbot interferometry with increased shear,” Appl. Opt. 24, 4448-4453 (1985). [CrossRef] [PubMed]
  9. K. Patorski, “Talbot interferometry with increased shear: further considerations,” Appl. Opt. 25, 1111-1116 (1986). [CrossRef] [PubMed]
  10. Q. Liu and R. Ohba, “Effects of unparallel grating planes in Talbot interferometry,” Appl. Opt. 38, 4111-4116 (1999). [CrossRef]
  11. Q. Liu and R. Ohba, “Effects of unparallel grating planes in Talbot interferometry II,” Appl. Opt. 39, 2084-2090 (2000). [CrossRef]
  12. M. Wei, E. Gullikson, J. H. Underwood, T. K. Gustafson, and D. T. Attwood, “White-light spatial frequency multiplication using soft x-rays,” Proc. of SPIE 2516, 233-239 (1995). [CrossRef]
  13. H. Meiling, H. Meijer, V. Banine, R. Moors, R. Groeneveld, H.-J. Voorma, and U. MicKan, “First performance results of the ASML alpha demo tool,” Proc. SPIE 6151, 615108 (2006). [CrossRef]
  14. M. Miura, K. Murakami, K. Suzuki, Y. Kohama, Y. Ohkubo, and T. Asami, “Nikon EUVL development progress summary,” Proc. SPIE 6151, 615105 (2006). [CrossRef]
  15. S. Heinbuch, M. Grisham, D. Martz, and J. J. Rocca, “Demonstration of a desk-top size high repetition rate soft x-ray laser,” Opt. Express 13, 4050-4055, (2005). [CrossRef] [PubMed]
  16. S. F. Horne, M. M. Besen, D. K. Smith, P. A. Blackborow, and R. DAgostino, “Application of a high-brightness electrodeless Z-pinch EUV source for metrology, inspection, and resist development,” Proc. SPIE 6151, 61510P (2006). [CrossRef]
  17. J. Goodman, Introduction to Fourier Optics, second ed. (McGraw-Hill, 1968), Eq. 3-74. We note that we have written the propagation phase in terms of angle and wavelength rather than in terms of spatial frequency as done by Goodman. It is for this reason we do not call call the propagation phase a transfer function. We also note that this phase is exact; the Fresnel approximation has not yet been made.
  18. J. Goodman, Introduction to Fourier Optics, second ed. (McGraw-Hill, 1968), Sec. 4.5.2.
  19. E. Hecht, Optics, third ed. (Addison-Wesley Longman, 1998), Eq. 10.61.
  20. Because the DOF dephasing term is odd in theta we need to use the full NA (2Δθ) to get the full dephasing for the DOF term. The tilt dephasing term is even in theta so only the half NA is required here.
  21. Here the small correction term does not contain a small parameter (g, γ, d) so we go to second order to maintain reasonable accuracy in the expansion.

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 Fig. 5
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited