Abstract

Unobstructed, plane-symmetric systems of three spherical mirrors are investigated. Twelve parameters are necessary to specify the configuration of such a system. Constraints are determined to eliminate four of these parameters as independent degrees of freedom. These constraints ensure appropriate first-order behavior and are used to aid in two example design studies—one for a class of systems with the object at infinity and another for a class of finite conjugate projection systems. For the first study, a portion of the associated merit-function space is systematically evaluated and plotted, and the results are compared with those obtained when a global optimizer is used. For the second study, a global optimizer is employed as the primary search tool. Example systems from both studies are presented.

© 2000 Optical Society of America

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  1. D. Korsch, Reflective Optics (Academic, Boston, Mass., 1991), Chaps. 9, 12, and 13.
  2. A. Kutter, Der Schiefspiegler (Fritz Weichhard, Biberach an der Riss, Germany, 1953). An edited and translated summary of this work can be found in A. Kutter, “The Schiefspiegler (oblique telescope),” Sky Telesc. Bull. A (1958).
  3. B. Tatian, “A first look at computer design of optical systems without any symmetry,” in Recent Trends in Optical Systems Design: Computer Lens Design Workshop, C. Londono, R. E. Fischer, eds., Proc. SPIE766, 38–47 (1985).
  4. R. A. Buchroeder, “Tilted-component telescopes. I. Theory,” Appl. Opt. 9, 2169–2171 (1970). Example systems can be found in R. A. Buchroeder, “Design examples of tilted-component telescopes (TCT’s) (a class of unobscured reflectors), (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1971).
  5. K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1980).
  6. J. R. Rogers, “Vector aberration theory and the design of off-axis systems,” in International Lens Design Conference, W. H. Taylor, D. T. Moore, eds., Proc. SPIE554, 76–81 (1985).
  7. P. Sands, “Aberration coefficients of plane symmetric systems,” J. Opt. Soc. Am. 62, 1211–1220 (1972).
    [CrossRef]
  8. J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33, 2045–2061 (1994).
    [CrossRef]
  9. B. D. Stone, G. W. Forbes, “Foundations of first-order layout for asymmetric systems: an applications of Hamilton’s methods,” J. Opt. Soc. Am. A 9, 96–110 (1992).
    [CrossRef]
  10. B. D. Stone, G. W. Forbes, “Foundations of second-order layout for asymmetric systems,” J. Opt. Soc. Am. A 9, 2067–2082 (1992).
    [CrossRef]
  11. The original works of Hamilton can be found in A. W. Conway, J. L. Synge, The Mathematical Papers of Sir William Rowan Hamilton (Cambridge U. Press, Cambridge, 1931).
  12. B. D. Stone, G. W. Forbes, “Second-order design methods for definitive studies of plane-symmetric, two-mirror systems,” J. Opt. Soc. Am. A 11, 3292–3307 (1994).
    [CrossRef]
  13. B. D. Stone, G. W. Forbes, “Illustration of second-order design methods: global merit function plots for a class of projection systems,” J. Opt. Soc. Am. A 11, 3308–3321 (1994).
    [CrossRef]
  14. J. M. Howard, B. D. Stone, “Imaging a point with two spherical mirrors,” J. Opt. Soc. Am. A 15, 3045–3056 (1998).
    [CrossRef]
  15. In a separate study, Stone and Forbes also consider the first-order layout of asymmetric systems composed of three spherical mirrors: B. D. Stone, G. W. Forbes, “First-order layout of asymmetric systems composed of three spherical mirrors,” J. Opt. Soc. Am. A 9, 110–120 (1992). However, in that paper they are concerned only with systems that are pseudosymmetric—defined as systems whose first-order imaging properties are equivalent to those of some axially symmetric system. Such constraints are overly restrictive for the study presented here; we are interested in plane-symmetric configurations of three spherical mirrors that form sharp images of only a single object plane.
  16. For a description of Hamilton’s characteristic functions see, for example, H. A. Buchdahl, Introduction to Hamiltonian Optics (Dover, New York, 1993), Chap. 2.
  17. For a description of the imaging properties that correspond to the terms of degree two in the Taylor expansion of Hamilton’s point-angle mixed characteristic see, for example, B. D. Stone, G. W. Forbes, “Characterization of first order optical properties for asymmetric systems,” J. Opt. Soc. Am. A 9, 478–489 (1992), and references therein.
  18. The geometrical interpretation associated with Hamilton’s characteristic functions can be found in Chap. 2 of the reference cited in Note 16 or alternatively in J. L. Synge, Geometrical Optics, An Introduction to Hamilton’s Method (Cambridge U. Press, Cambridge, 1937), Sec. 6.
  19. A detailed discussion of the geometric interpretation associated with the coefficients of degree two in the Taylor expansion of a characteristic function can be found in Sec. 2 of the paper cited in Note 17. Note that there is a subtle notational difference between that paper and this. In that paper the reference planes are taken to be perpendicular to the base ray segments in object and image space. When quantities associated with tilted object and image planes are discussed, they are denoted by the addition of a tilde (˜). Here we tilt the reference planes along with the object and the image planes; therefore a tilde is unnecessary and is not used.
  20. See Sec. 6 of the book by Buchdahl cited in Note 16.
  21. A discussion of the consequences of object and image tilts in the context of first-order optics can be found in Sec. 4 of the paper cited in Note 17.
  22. A discussion of the consequences of object and image tilts in the context of second-order optics can be found in Sec. 3 of Ref. 10.
  23. A detailed discussion of these conclusions can be found in Sec. 2.C.1 of Ref. 12.
  24. This method for global optimization is described in A. E. W. Jones, G. W. Forbes, “An adaptive simulated annealing algorithm for global optimization over continuous variables,” J. Global Optim. 6, 1–34 (1995).
  25. J. M. Howard, B. D. Stone, “Imaging with four spherical mirrors,” Appl. Opt. 39, 3232–3242 (2000).
    [CrossRef]
  26. M. Brunn, “Unobstructed all-reflecting telescopes of the Schiefspiegler type,” U.S. patent5,142,417 (25August1992).
  27. C. T. Cotton, “Design of an all-spherical, three-mirror, off-axis telescope objective,” in International Optical Design Conference, G. W. Forbes, ed., Vol. 22 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 349–351.
  28. R. A. Buchroeder, “A new three-mirror off-axis amateur telescope,” Sky Telesc. 38, 418–423 (1969).
  29. A. Kutter, “A new three-mirror unobstructed reflector,” Sky Telesc. 49, 46–49 (1975).
  30. A. Offner, “Unit power imaging catoptric anastigmat,” U.S. patent3,748,015 (24July1973).
  31. L. H. J. F. Beckmann, D. Ehrlichmann, “Three-mirror off-axis systems for laser applications,” in International Optical Design Conference, G. W. Forbes, ed., Vol. 22 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 340–348.
  32. Specific examples of employing computer algebra in aberration theory can be found in A. Walther, “Eikonal theory and computer algebra,” J. Opt. Soc. Am. A 13, 523–531 (1996), and in the follow-on paper, A. Walther, “Eikonal theory and computer algebra. II,” J. Opt. Soc. Am. A 13, 1763–1765 (1996).
  33. An introduction to matrix methods for symmetric systems can be found in Sec. 6.2.1 of E. Hecht, Optics, 3rd ed. (Addison-Wesley, Reading Mass., 1998). A generalization of matrix optics for asymmetric systems is discussed in Sec. 4 of B. D. Stone, “Determination of initial ray configurations for asymmetric systems,” J. Opt. Soc. Am. A 14, 3415–3429 (1997).
  34. A detailed derivation of Eq. (17) can be found in Sec. 10 of the book cited in Note 16 or in Sec. 2 of the paper referenced in Note 17. Equation (18) can be determined by a similar procedure.

2000

1998

1996

1995

This method for global optimization is described in A. E. W. Jones, G. W. Forbes, “An adaptive simulated annealing algorithm for global optimization over continuous variables,” J. Global Optim. 6, 1–34 (1995).

1994

1992

1975

A. Kutter, “A new three-mirror unobstructed reflector,” Sky Telesc. 49, 46–49 (1975).

1972

1970

1969

R. A. Buchroeder, “A new three-mirror off-axis amateur telescope,” Sky Telesc. 38, 418–423 (1969).

Beckmann, L. H. J. F.

L. H. J. F. Beckmann, D. Ehrlichmann, “Three-mirror off-axis systems for laser applications,” in International Optical Design Conference, G. W. Forbes, ed., Vol. 22 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 340–348.

Brunn, M.

M. Brunn, “Unobstructed all-reflecting telescopes of the Schiefspiegler type,” U.S. patent5,142,417 (25August1992).

Buchdahl, H. A.

For a description of Hamilton’s characteristic functions see, for example, H. A. Buchdahl, Introduction to Hamiltonian Optics (Dover, New York, 1993), Chap. 2.

Buchroeder, R. A.

Conway, A. W.

The original works of Hamilton can be found in A. W. Conway, J. L. Synge, The Mathematical Papers of Sir William Rowan Hamilton (Cambridge U. Press, Cambridge, 1931).

Cotton, C. T.

C. T. Cotton, “Design of an all-spherical, three-mirror, off-axis telescope objective,” in International Optical Design Conference, G. W. Forbes, ed., Vol. 22 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 349–351.

Ehrlichmann, D.

L. H. J. F. Beckmann, D. Ehrlichmann, “Three-mirror off-axis systems for laser applications,” in International Optical Design Conference, G. W. Forbes, ed., Vol. 22 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 340–348.

Forbes, G. W.

This method for global optimization is described in A. E. W. Jones, G. W. Forbes, “An adaptive simulated annealing algorithm for global optimization over continuous variables,” J. Global Optim. 6, 1–34 (1995).

B. D. Stone, G. W. Forbes, “Second-order design methods for definitive studies of plane-symmetric, two-mirror systems,” J. Opt. Soc. Am. A 11, 3292–3307 (1994).
[CrossRef]

B. D. Stone, G. W. Forbes, “Illustration of second-order design methods: global merit function plots for a class of projection systems,” J. Opt. Soc. Am. A 11, 3308–3321 (1994).
[CrossRef]

B. D. Stone, G. W. Forbes, “Foundations of first-order layout for asymmetric systems: an applications of Hamilton’s methods,” J. Opt. Soc. Am. A 9, 96–110 (1992).
[CrossRef]

B. D. Stone, G. W. Forbes, “Foundations of second-order layout for asymmetric systems,” J. Opt. Soc. Am. A 9, 2067–2082 (1992).
[CrossRef]

In a separate study, Stone and Forbes also consider the first-order layout of asymmetric systems composed of three spherical mirrors: B. D. Stone, G. W. Forbes, “First-order layout of asymmetric systems composed of three spherical mirrors,” J. Opt. Soc. Am. A 9, 110–120 (1992). However, in that paper they are concerned only with systems that are pseudosymmetric—defined as systems whose first-order imaging properties are equivalent to those of some axially symmetric system. Such constraints are overly restrictive for the study presented here; we are interested in plane-symmetric configurations of three spherical mirrors that form sharp images of only a single object plane.

For a description of the imaging properties that correspond to the terms of degree two in the Taylor expansion of Hamilton’s point-angle mixed characteristic see, for example, B. D. Stone, G. W. Forbes, “Characterization of first order optical properties for asymmetric systems,” J. Opt. Soc. Am. A 9, 478–489 (1992), and references therein.

Hecht, E.

An introduction to matrix methods for symmetric systems can be found in Sec. 6.2.1 of E. Hecht, Optics, 3rd ed. (Addison-Wesley, Reading Mass., 1998). A generalization of matrix optics for asymmetric systems is discussed in Sec. 4 of B. D. Stone, “Determination of initial ray configurations for asymmetric systems,” J. Opt. Soc. Am. A 14, 3415–3429 (1997).

Howard, J. M.

Jones, A. E. W.

This method for global optimization is described in A. E. W. Jones, G. W. Forbes, “An adaptive simulated annealing algorithm for global optimization over continuous variables,” J. Global Optim. 6, 1–34 (1995).

Korsch, D.

D. Korsch, Reflective Optics (Academic, Boston, Mass., 1991), Chaps. 9, 12, and 13.

Kutter, A.

A. Kutter, “A new three-mirror unobstructed reflector,” Sky Telesc. 49, 46–49 (1975).

A. Kutter, Der Schiefspiegler (Fritz Weichhard, Biberach an der Riss, Germany, 1953). An edited and translated summary of this work can be found in A. Kutter, “The Schiefspiegler (oblique telescope),” Sky Telesc. Bull. A (1958).

Offner, A.

A. Offner, “Unit power imaging catoptric anastigmat,” U.S. patent3,748,015 (24July1973).

Rogers, J. R.

J. R. Rogers, “Vector aberration theory and the design of off-axis systems,” in International Lens Design Conference, W. H. Taylor, D. T. Moore, eds., Proc. SPIE554, 76–81 (1985).

Sands, P.

Sasian, J. M.

J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33, 2045–2061 (1994).
[CrossRef]

Stone, B. D.

J. M. Howard, B. D. Stone, “Imaging with four spherical mirrors,” Appl. Opt. 39, 3232–3242 (2000).
[CrossRef]

J. M. Howard, B. D. Stone, “Imaging a point with two spherical mirrors,” J. Opt. Soc. Am. A 15, 3045–3056 (1998).
[CrossRef]

B. D. Stone, G. W. Forbes, “Illustration of second-order design methods: global merit function plots for a class of projection systems,” J. Opt. Soc. Am. A 11, 3308–3321 (1994).
[CrossRef]

B. D. Stone, G. W. Forbes, “Second-order design methods for definitive studies of plane-symmetric, two-mirror systems,” J. Opt. Soc. Am. A 11, 3292–3307 (1994).
[CrossRef]

B. D. Stone, G. W. Forbes, “Foundations of first-order layout for asymmetric systems: an applications of Hamilton’s methods,” J. Opt. Soc. Am. A 9, 96–110 (1992).
[CrossRef]

B. D. Stone, G. W. Forbes, “Foundations of second-order layout for asymmetric systems,” J. Opt. Soc. Am. A 9, 2067–2082 (1992).
[CrossRef]

In a separate study, Stone and Forbes also consider the first-order layout of asymmetric systems composed of three spherical mirrors: B. D. Stone, G. W. Forbes, “First-order layout of asymmetric systems composed of three spherical mirrors,” J. Opt. Soc. Am. A 9, 110–120 (1992). However, in that paper they are concerned only with systems that are pseudosymmetric—defined as systems whose first-order imaging properties are equivalent to those of some axially symmetric system. Such constraints are overly restrictive for the study presented here; we are interested in plane-symmetric configurations of three spherical mirrors that form sharp images of only a single object plane.

For a description of the imaging properties that correspond to the terms of degree two in the Taylor expansion of Hamilton’s point-angle mixed characteristic see, for example, B. D. Stone, G. W. Forbes, “Characterization of first order optical properties for asymmetric systems,” J. Opt. Soc. Am. A 9, 478–489 (1992), and references therein.

Synge, J. L.

The original works of Hamilton can be found in A. W. Conway, J. L. Synge, The Mathematical Papers of Sir William Rowan Hamilton (Cambridge U. Press, Cambridge, 1931).

The geometrical interpretation associated with Hamilton’s characteristic functions can be found in Chap. 2 of the reference cited in Note 16 or alternatively in J. L. Synge, Geometrical Optics, An Introduction to Hamilton’s Method (Cambridge U. Press, Cambridge, 1937), Sec. 6.

Tatian, B.

B. Tatian, “A first look at computer design of optical systems without any symmetry,” in Recent Trends in Optical Systems Design: Computer Lens Design Workshop, C. Londono, R. E. Fischer, eds., Proc. SPIE766, 38–47 (1985).

Thompson, K. P.

K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1980).

Walther, A.

Appl. Opt.

J. Global Optim.

This method for global optimization is described in A. E. W. Jones, G. W. Forbes, “An adaptive simulated annealing algorithm for global optimization over continuous variables,” J. Global Optim. 6, 1–34 (1995).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

B. D. Stone, G. W. Forbes, “Foundations of second-order layout for asymmetric systems,” J. Opt. Soc. Am. A 9, 2067–2082 (1992).
[CrossRef]

J. M. Howard, B. D. Stone, “Imaging a point with two spherical mirrors,” J. Opt. Soc. Am. A 15, 3045–3056 (1998).
[CrossRef]

Specific examples of employing computer algebra in aberration theory can be found in A. Walther, “Eikonal theory and computer algebra,” J. Opt. Soc. Am. A 13, 523–531 (1996), and in the follow-on paper, A. Walther, “Eikonal theory and computer algebra. II,” J. Opt. Soc. Am. A 13, 1763–1765 (1996).

B. D. Stone, G. W. Forbes, “Second-order design methods for definitive studies of plane-symmetric, two-mirror systems,” J. Opt. Soc. Am. A 11, 3292–3307 (1994).
[CrossRef]

B. D. Stone, G. W. Forbes, “Illustration of second-order design methods: global merit function plots for a class of projection systems,” J. Opt. Soc. Am. A 11, 3308–3321 (1994).
[CrossRef]

In a separate study, Stone and Forbes also consider the first-order layout of asymmetric systems composed of three spherical mirrors: B. D. Stone, G. W. Forbes, “First-order layout of asymmetric systems composed of three spherical mirrors,” J. Opt. Soc. Am. A 9, 110–120 (1992). However, in that paper they are concerned only with systems that are pseudosymmetric—defined as systems whose first-order imaging properties are equivalent to those of some axially symmetric system. Such constraints are overly restrictive for the study presented here; we are interested in plane-symmetric configurations of three spherical mirrors that form sharp images of only a single object plane.

B. D. Stone, G. W. Forbes, “Foundations of first-order layout for asymmetric systems: an applications of Hamilton’s methods,” J. Opt. Soc. Am. A 9, 96–110 (1992).
[CrossRef]

For a description of the imaging properties that correspond to the terms of degree two in the Taylor expansion of Hamilton’s point-angle mixed characteristic see, for example, B. D. Stone, G. W. Forbes, “Characterization of first order optical properties for asymmetric systems,” J. Opt. Soc. Am. A 9, 478–489 (1992), and references therein.

Opt. Eng.

J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33, 2045–2061 (1994).
[CrossRef]

Sky Telesc.

R. A. Buchroeder, “A new three-mirror off-axis amateur telescope,” Sky Telesc. 38, 418–423 (1969).

A. Kutter, “A new three-mirror unobstructed reflector,” Sky Telesc. 49, 46–49 (1975).

Other

A. Offner, “Unit power imaging catoptric anastigmat,” U.S. patent3,748,015 (24July1973).

L. H. J. F. Beckmann, D. Ehrlichmann, “Three-mirror off-axis systems for laser applications,” in International Optical Design Conference, G. W. Forbes, ed., Vol. 22 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 340–348.

For a description of Hamilton’s characteristic functions see, for example, H. A. Buchdahl, Introduction to Hamiltonian Optics (Dover, New York, 1993), Chap. 2.

An introduction to matrix methods for symmetric systems can be found in Sec. 6.2.1 of E. Hecht, Optics, 3rd ed. (Addison-Wesley, Reading Mass., 1998). A generalization of matrix optics for asymmetric systems is discussed in Sec. 4 of B. D. Stone, “Determination of initial ray configurations for asymmetric systems,” J. Opt. Soc. Am. A 14, 3415–3429 (1997).

A detailed derivation of Eq. (17) can be found in Sec. 10 of the book cited in Note 16 or in Sec. 2 of the paper referenced in Note 17. Equation (18) can be determined by a similar procedure.

The original works of Hamilton can be found in A. W. Conway, J. L. Synge, The Mathematical Papers of Sir William Rowan Hamilton (Cambridge U. Press, Cambridge, 1931).

M. Brunn, “Unobstructed all-reflecting telescopes of the Schiefspiegler type,” U.S. patent5,142,417 (25August1992).

C. T. Cotton, “Design of an all-spherical, three-mirror, off-axis telescope objective,” in International Optical Design Conference, G. W. Forbes, ed., Vol. 22 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 349–351.

The geometrical interpretation associated with Hamilton’s characteristic functions can be found in Chap. 2 of the reference cited in Note 16 or alternatively in J. L. Synge, Geometrical Optics, An Introduction to Hamilton’s Method (Cambridge U. Press, Cambridge, 1937), Sec. 6.

A detailed discussion of the geometric interpretation associated with the coefficients of degree two in the Taylor expansion of a characteristic function can be found in Sec. 2 of the paper cited in Note 17. Note that there is a subtle notational difference between that paper and this. In that paper the reference planes are taken to be perpendicular to the base ray segments in object and image space. When quantities associated with tilted object and image planes are discussed, they are denoted by the addition of a tilde (˜). Here we tilt the reference planes along with the object and the image planes; therefore a tilde is unnecessary and is not used.

See Sec. 6 of the book by Buchdahl cited in Note 16.

A discussion of the consequences of object and image tilts in the context of first-order optics can be found in Sec. 4 of the paper cited in Note 17.

A discussion of the consequences of object and image tilts in the context of second-order optics can be found in Sec. 3 of Ref. 10.

A detailed discussion of these conclusions can be found in Sec. 2.C.1 of Ref. 12.

D. Korsch, Reflective Optics (Academic, Boston, Mass., 1991), Chaps. 9, 12, and 13.

A. Kutter, Der Schiefspiegler (Fritz Weichhard, Biberach an der Riss, Germany, 1953). An edited and translated summary of this work can be found in A. Kutter, “The Schiefspiegler (oblique telescope),” Sky Telesc. Bull. A (1958).

B. Tatian, “A first look at computer design of optical systems without any symmetry,” in Recent Trends in Optical Systems Design: Computer Lens Design Workshop, C. Londono, R. E. Fischer, eds., Proc. SPIE766, 38–47 (1985).

K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1980).

J. R. Rogers, “Vector aberration theory and the design of off-axis systems,” in International Lens Design Conference, W. H. Taylor, D. T. Moore, eds., Proc. SPIE554, 76–81 (1985).

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