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Applied Optics

Applied Optics


  • Vol. 39, Iss. 19 — Jul. 1, 2000
  • pp: 3232–3242

Imaging with four spherical mirrors

Joseph M. Howard and Bryan D. Stone  »View Author Affiliations

Applied Optics, Vol. 39, Issue 19, pp. 3232-3242 (2000)

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We investigate unobstructed, plane-symmetric imaging systems of four spherical mirrors. Fifteen parameters are necessary to specify the configuration of such a system. Constraints are determined that ensure that any resultant system possesses a given set of first-order properties. These constraints remove four parameters as available degrees of freedom. To illustrate the efficacy of this design approach, we present two example studies: one for a class of systems with the object at infinity and another for finite-conjugate projection systems. For each study a global optimizer is used as the primary search tool. Example systems from these studies are presented.

© 2000 Optical Society of America

OCIS Codes
(080.2740) Geometric optics : Geometric optical design
(080.3620) Geometric optics : Lens system design
(220.2740) Optical design and fabrication : Geometric optical design
(220.3620) Optical design and fabrication : Lens system design
(230.4040) Optical devices : Mirrors

Original Manuscript: October 1, 1999
Published: July 1, 2000

Joseph M. Howard and Bryan D. Stone, "Imaging with four spherical mirrors," Appl. Opt. 39, 3232-3242 (2000)

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  1. D. Korsch, Reflective Optics (Academic, Boston, Mass., 1991), Chaps. 10 and 12.
  2. The original research can be found in K. Schwarzschild, “Untersuchungen zur geometrischen Optik. II. Theorie der Spiegelteleskope,” Abh. Koenigl. Ges. Wiss. Goettingen Math.-Phys. Klasse, Folge 9 IV(2), 1–28 (1905). A modern summary translated into English can be found in R. N. Wilson, “Karl Schwarzschild Lecture of the German Astronomical Society,” Rev. Mod. Astron. 7, 1–29 (1993).
  3. G. D. Wassermann, E. Wolf, “On the theory of aplanatic aspheric system,” Proc. Phys. Soc. B 62, 2–8 (1949). [CrossRef]
  4. D. R. Shafer, “Four-mirror unobscured anastigmatic telescopes with all-spherical surfaces,” Appl. Opt. 17, 1072–1074 (1977). [CrossRef]
  5. J. M. Howard, B. D. Stone, “Imaging with three spherical mirrors,” Appl. Opt. 39, 3216–3231 (2000). [CrossRef]
  6. For the mechanics of differential ray tracing through homogeneous media, see, for example, A. Cox, A System of Optical Design (Focal, London, 1964), pp. 112–121; D. P. Feder, “Differentiation of ray-tracing equations with respect to construction parameters of rotationally symmetric optics,” J. Opt. Soc. Am. 58, 1494–1505 (1968).
  7. Details of this global optimization method can be found in A. E. W. Jones, G. W. Forbes, “An adaptive simulated annealing algorithm for global optimization over continuous variables,” J. Global Optimization 6, 1–34 (1995).
  8. Details of imaging with a single spherical mirror can be found in J. M. Howard, B. D. Stone, “Imaging a point to a line with a single spherical mirror,” Appl. Opt. 37, 1826–1834 (1998).
  9. J. M. Howard, B. D. Stone, “Imaging a point with two spherical mirrors,” J. Opt. Soc. Am. A 15, 3045–3056 (1998). [CrossRef]

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