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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References

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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).
    [CrossRef]
  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 (1)

1998 (1)

1996 (1)

1995 (1)

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 (3)

1992 (4)

1975 (1)

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

1972 (1)

1970 (1)

1969 (1)

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).
[CrossRef]

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. (2)

J. Global Optim. (1)

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. (1)

J. Opt. Soc. Am. A (8)

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]

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]

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

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).

Opt. Eng. (1)

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

Sky Telesc. (2)

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 (19)

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.

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.

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 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).

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).

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).
[CrossRef]

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.

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

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Figures (6)

Fig. 1
Fig. 1

Schematic representation of the cross section of the plane of symmetry of the systems investigated here. The base ray through the system is shown, along with tilted object and image planes.

Fig. 2
Fig. 2

Rendering of a five-dimensional merit-function space for a three-spherical-mirror system with the object at infinity. The color indicates the RMS spot radius of the basal image point. White areas are systems with obstruction, a virtual image, or a rms spot radius greater than 10.0-RMS spot radius. The plot with the asterisk is illustrated in Fig. 3. The systems labeled by letters are illustrated in Figs. 4 and 5.

Fig. 3
Fig. 3

Two-dimensional portion of the plotted space in Fig. 2. Values of the fixed, unconstrained parameters are {c 1 = -0.003, d 1 = 70.0, θ1 = 10.0°}. The systems of interest in the plot (areas with color) naturally divide into quadrants and are identified by the sign of the tilt angles for that region. For example, the upper right portion of the plot marked +-+ is the area where θ1 is positive, θ2 is negative and, θ3 is positive. The hatched areas represent systems that have a virtual image. Systems labeled with letters are illustrated in Fig. 4.

Fig. 4
Fig. 4

Systems A–F correspond to the points labeled A–F in Figs. 2 and 3, which were found by a systematic search. The numbers next to the images represent the RMS spot radii of the basal image points. Systems B is illustrated at half the scale of the others. The sequence of + and - after each label denotes the sign of each of the corresponding mirror tilt angles. Note that, because the first mirror always has a positive tilt angle, each sequence begins with a +.

Fig. 5
Fig. 5

Systems G–L correspond to the points labeled G–L in Fig. 2, which were found with a global optimizer (ASA). The numbers next to the images represent the RMS spot radii of the basal image points. All systems, except system H, are illustrated at the same scale. The sequence of + and - after each system label corresponds to the sign of the tilt angle for each of the three mirrors.

Fig. 6
Fig. 6

Systems M–R were found by use of ASA. The numbers next to the images represent the maximum RMS spot radii of the 15 field points evaluated. In systems M and O the first two mirrors are nearly concentric, and their centers of curvature are labeled with × and Δ, respectively. System P is nearly an off-axis section of a rotationally symmetric system, and an approximate axis is included in the figure. A scale is included with each system. The sequence of + and - after each system label corresponds to the sign of the tilt angle for each of the three mirrors.

Tables (3)

Tables Icon

Table 1 Parameters for Systems in Fig. 4

Tables Icon

Table 2 Parameters for Systems in Fig. 5

Tables Icon

Table 3 Parameters for Systems in Fig. 6a

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

W2y, p=δ+y sin θobj+½F11y2+F22z2+M11ypy+sin θim+M22zpz+½B11py+sin θim2+B22pz2+O3,
y=-W2y, ppy,  z=-W2y, ppz.
y=-M11y+B11py+sin θim+O2,
z=-M22z+B22pz+O2.
M11=-m1,  M22=-m2,
B11=0,  B22=0.
Tp, p=δ+½F11py2+F22pz2+M11pypy+sin θim+M22pzpz+½B11py+sin θim2+B22pz2+O3.
y=-Tp, ppy=-M11py+B11py+sin θim+O2,
z=-Tp, ppz=-M22pz+B22pz+O2.
M11=-f1,  M22=-f2,
B11=0,  B22=0,
M11=cos θobj cos θ1 cos θ2 cos θ3 sec θimcos θ1 cos θ2q1+2c3d0+d1+2c2q1d0+d1cos θ1+2c1d0q1+2c3d1cos θ2+2c2d1q1,
M22=1q2+2d0+d1c2q2 cos θ2+c3 cos θ3+2c1d0 cos θ1q2+2d1c2q2 cos θ2+c3 cos θ3,
B11=(M11 sec θ1 sec θ2d0+d1cos θ1 cos θ2+d2cos θ2+2c2d0+d1cos θ1+2c1d02c2d1d2+d0+d1cos θ2-d3)sec2 θim,
B22=M22q3+d01+2c2d2 cos θ2+2c1q3 cos θ1-d3,
q1=2c2d3+cos θ3,
q2=1+2c2d3 cos θ3,
q3=d1+d2+2c2d1d2 cos θ2.
d0=-q4mˆ1q5 sin2 θ3+m2q3 cos2 θ1 cos θ2 sin2 θ3-mˆ1 sin2 θ12c2d12cos2 θ2-cos2 θ3+d1 cos θ2+d2cos θ2+2c2d11+2c2d1 cos θ2sin2 θ3,
d3=q4mˆ1m2-mˆ1q3 cos θ2 cos2 θ3+m2q5sin2 θ1+-2c2d22cos2 θ1-cos2 θ2+d1 cos θ2+d2cos θ2+2c2d11+cos2 θ2+4c22d1d22 cos θ2sin2 θ1sin2 θ3,
c1=-2c2d2mˆ1+m2 cos2 θ2-mˆ1m2 sin2 θ2sin2 θ3+mˆ1+m2mˆ1 cos2 θ3+m2-sin2 θ3cos θ2cos θ1/2mˆ1q5 sin2 θ3+m2q3 cos2 θ1 cos θ2 sin2 θ3-mˆ1 sin2 θ1d1cos θ2 sin3 θ2+2c2d1cos2 θ2-cos2 θ3+cos θ2+2c2d11+2c2d1 cos θ2d2 sin θ32,
c3=-cos θ3cos θ2mˆ1+m2mˆ1+m2 cos2 θ1-mˆ1m2 sin2 θ1cos θ2mˆ1+m2+2c2d1mˆ1 cos2 θ2+m2-sin2 θ2/2mˆ1m22c2d2cos2 θ2-cos2 θ1d2+d11+2c2d2 cos θ2sin2 θ1sin2 θ3+sin2 θ1-m2q5-q3 cos θ2mˆ1 cos2 θ3-sin2 θ3,
mˆ1=m1 sec θobj cos θim,
q4=(cos θ2mˆ1+m2mˆ1 cos2 θ3+m2 cos2 θ1+2c2mˆ1m2d1cos2 θ3-cos2 θ2sin2 θ1+d2cos2 θ1-cos θ3+2c2d1 sin2 θ1sin2 θ3)-1,
q5=2c2d1d2+d1+d2cos θ2.
M11=-cos θobj cos θ1 cos θ2 cos θ3 sec θim/2c3q6×cos θ2+cos θ3+2c3d2c1 cos θ2 +c2q6,
M22=1/2c1q2 cos θ1+1+2c1d1 cos θ1×c2q2 cos θ2+c3 cos θ3,
B11=(M11 sec θ1 sec θ2cos θ1cos θ2+2c2d2+2c12c2d1d2+cos θ2d1+d2-d3)sec2 θim,
B22=-M221+2c2d2 cos θ2+2c1q3 cos θ1-d3,
q6=2c1d1+cos θ1.
d2=q6fˆ1 cos2 θ3+q7f2 cos θ1 cos2 θ22c1d1fˆ1+f2 cos2 θ1+cos θ1fˆ1+f2/q8 cos θ1 sin2 θ3,
d3=fˆ1f22c1q6fˆ1 cos2 θ3+q7f2 cos θ1q6 cos θ1-q7 cos2 θ2+q6q7 cos θ1 sin2 θ2 sin2 θ3/q8×cos θ1 sin2 θ3,
c2=-cos θ2cos θ1 sin2 θ3+2c1fˆ1 cos2 θ3+f2 cos2 θ1/2q6fˆ1 cos2 θ3+2q7f2 cos θ1 cos2 θ2,
c3=-q82 cos θ1 cos θ3 sin2 θ3/2fˆ1f2q6fˆ1 cos2 θ3+q7f2 cos θ1 cos2 θ22c1q6fˆ1 cos2 θ3+q7f2 cos θ1q6 cos θ1-q7 cos2 θ2+q6q7 cos θ1 sin2 θ2 sin2 θ3,
fˆ1=f1 sec θim,
q7=1+2c1d1 cos θ1,
q8=q6fˆ1+f2q7 cos θ1 cos2 θ2+2c1fˆ1q6 cos θ1-q7 cos2 θ2.
yp=abcdyp+O2,
abcd=BM-1F-MBM-1-M-1F-M-1,
abcd=-BM-1BM-1F-MM-1-M-1F.

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