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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 16, Iss. 11 — Nov. 1, 1999
  • pp: 2690–2694

Effective-focal-length calculations and measurements for a radial diffraction grating

Jeffrey A. Koch, Steven Cui, and Michael A. McNeill  »View Author Affiliations

JOSA A, Vol. 16, Issue 11, pp. 2690-2694 (1999)

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As the technology for manufacturing radial diffraction gratings improves, there will be increasing interest in using high-groove-density radial gratings as scales for precision interferometer-type rotary encoders. If efficiency in optical designs is to be optimized, the focusing properties of these gratings must be understood. We use analytical geometry to investigate the focusing properties of a radial diffraction grating illuminated by laser light diverging from a pointlike source. We compare the results with experiments that we performed with a state-of-the-art custom radial grating, and we obtain excellent agreement with our calculations, which improve on earlier analytical work.

© 1999 Optical Society of America

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(080.2720) Geometric optics : Mathematical methods (general)
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(230.3120) Optical devices : Integrated optics devices

Original Manuscript: March 4, 1999
Revised Manuscript: June 10, 1999
Manuscript Accepted: June 10, 1999
Published: November 1, 1999

Jeffrey A. Koch, Steven Cui, and Michael A. McNeill, "Effective-focal-length calculations and measurements for a radial diffraction grating," J. Opt. Soc. Am. A 16, 2690-2694 (1999)

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  1. W. H. Stevenson, “Optical frequency shifting by means of a rotating diffraction grating,” Appl. Opt. 9, 649–652 (1970). [CrossRef] [PubMed]
  2. K. Matsumoto, “Method for optical detection and/or measurement of movement of a diffraction grating,” U.S. patent3,726,595 (April17, 1973).
  3. A. Kozlowska, “Fibre optic grating interferometer for in-plane displacement measurement,” in Interferometry ’94: Interferometric Fiber Sensing, E. Udd, R. P. Tatam, eds., Proc. SPIE2341, 124–131 (1994). [CrossRef]
  4. Linear grating interferometers are available from several companies, including Heidenhain Corporation, Schaumburg, Ill.
  5. A. Teimel, “Technology and applications of grating interferometers in high-precision measurement,” Precis. Eng. 14, 147–154 (1992). [CrossRef]
  6. Rotary grating interferometers are available from Canon U.S.A. Inc., Lake Success, N.Y.
  7. Conventional optical rotary encoders are commonly used for many applications and are a continuing focus for research efforts; see, for example, K. Engelhardt, P. Seitz, “Absolute, high-resolution optical position encoder,” Appl. Opt. 35, 201–208 (1996). [CrossRef] [PubMed]
  8. Yu Hong-Lin, “Research on the small-sized high-resolution radial grating,” in Measurement Technology and Intelligent Instruments, L. Zhu, ed., Proc. SPIE2101, 1168–1171 (1993). [CrossRef]
  9. S. V. Gordeev, B. G. Turukhano, “Investigation of the interference field of two spherical waves for holographic recording of precision radial gratings,” Opt. Laser Technol. 28, 225–261 (1996). [CrossRef]
  10. R. Sawada, H. Tanaka, O. Ohguchi, J. Shimada, S. Hara, “Fabrication of active integrated optical micro-encoder,” in Proceedings of the Micro Electro-Mechanical Systems Conference 1991 (IEEE, New York, 1991), pp. 233–238.
  11. R. Sawada, O. Ohguchi, K. Mise, M. Tsubamoto, “Fabrication of advanced integrated optical micro-encoder chip,” in Proceedings of the Micro Electro-Mechanical Systems Conference 1994 (IEEE, New York, 1994), pp. 337–342.
  12. M. Dobosz, “Application of a divergent laser beam in a grating interferometer for high-resolution displacement measurements,” Opt. Eng. 33, 897–901 (1994). [CrossRef]
  13. J. A. Koch, “Compact high-resolution soft x-ray spectrograph design using two matched grazing-incidence gratings,” Appl. Opt. 34, 3693–3701 (1995). [CrossRef] [PubMed]
  14. J. H. Underwood, J. A. Koch, “High resolution tunable spectrograph for x-ray laser line width measurements using a plane varied line spacing grating,” Appl. Opt. 36, 4913–4921 (1997). [CrossRef] [PubMed]
  15. J. Alonso, E. Bernabeu, “Spatial evolution of Gaussian beams diffracted by radial gratings,” Opt. Commun. 98, 323–330 (1993). [CrossRef]
  16. J. Alonso, E. Bernabeu, “Use of effective focal lengths to describe laser-beam evolution after diffraction in radial gratings,” J. Opt. Soc. Am. A 10, 1963–1970 (1993). [CrossRef]
  17. J. F. James, R. S. Sternberg, The Design of Optical Spectrometers (Chapman & Hall, London, 1969).
  18. M. C. Hettrick, S. Bowyer, “Variable line-space gratings: new designs for use in grazing incidence spectrometers,” Appl. Opt. 22, 3921–3924 (1983). [CrossRef] [PubMed]

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