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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Franco Gori
  • Vol. 28, Iss. 4 — Apr. 1, 2011
  • pp: 667–674

Differential geometry of the ruled surfaces optically generated by mirror-scanning devices. I. Intrinsic and extrinsic properties of the scan field

Yajun Li  »View Author Affiliations


JOSA A, Vol. 28, Issue 4, pp. 667-674 (2011)
http://dx.doi.org/10.1364/JOSAA.28.000667


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Abstract

Rectilinear propagation of light rays in homogeneous isotropic media makes it possible for optical generation of ruled surfaces as the ray is deflected by a rotatable mirror. Scan patterns on a plane or curved surface are merely curves on the ruled surface. Based on this understanding, structures of the scan fields produced by mirror- scanning devices of different configurations are investigated in terms of differential geometry. Expressions of the first and second fundamental coefficients and the first and second Gauss differential forms are given for an investigation of the intrinsic properties of the optically generated ruled surfaces. The Plücker ruled conoid is then generalized for mathematical modeling of the scan fields produced by single-mirror scanning devices of different configurations. Part II will be devoted to a study of multi-mirror scanning systems for optical generation of well-known ruled surfaces such as helicoids and hyperbolic paraboloids.

© 2011 Optical Society of America

OCIS Codes
(080.0080) Geometric optics : Geometric optics
(080.2720) Geometric optics : Mathematical methods (general)
(120.5800) Instrumentation, measurement, and metrology : Scanners
(220.2740) Optical design and fabrication : Geometric optical design

History
Original Manuscript: November 23, 2010
Manuscript Accepted: January 12, 2011
Published: March 30, 2011

Citation
Yajun Li, "Differential geometry of the ruled surfaces optically generated by mirror-scanning devices. I. Intrinsic and extrinsic properties of the scan field," J. Opt. Soc. Am. A 28, 667-674 (2011)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-28-4-667


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References

  1. D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, (Chelsea, 1956), pp. 15–341.
  2. A. Gray, E. Abbena, and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed. (Chapman & Hall/CRC, 2006), Chap. 14, “Ruled Surfaces.”
  3. V. I. Smirnov, A Course of Higher Mathemantics II: Advanced Calculus, trans. by D. E. Brown, trans. ed. I. N. Sneddon (Pergamon, 1964), Chap. V, “Foundations of Differential Geometry.”
  4. Y. Li, “Laser beam scanning by rotary mirrors. II. Conic-section scan patterns,” Appl. Opt. 34, 6417–6430 (1995). [CrossRef] [PubMed]
  5. Y. Li and J. Katz, “Laser beam scanning by rotary mirrors. I. Modeling mirror scanning devices,” Appl. Opt. 34, 6403–6416 (1995). [CrossRef] [PubMed]
  6. Y. Li and J. Katz, “Asymmetric distribution of the scanned field of a rotating reflective polygon,” Appl. Opt. 36, 342–352 (1997). [CrossRef] [PubMed]
  7. Y. Li, “Single-mirror beam steering system: analysis and synthesis of high-order conic-section scan patterns,” Appl. Opt. 47, 386–397 (2008). [CrossRef] [PubMed]
  8. Y. Li, “Beam deflection and scanning by two-mirror and two-axis systems of different architectures: a unified approach,” Appl. Opt. 47, 5976–5984 (2008). [CrossRef] [PubMed]
  9. R. J. Sherman, “Polygon scanners: applications, performance and design,” in Optical Scanning, G.F.Marshall, ed. (Dekker, 1991), pp. 351–406.
  10. W. L. Edge, The Theory of Ruled Surfaces, 1st ed. (Cambridge University, 1931), Sec. 3.2.2.

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