Abstract

Spaceborne visible-light images for observing the large angular extent of the solar corona require 0.1% differential broadband photometry over ∼1° sky bins. When we are using a CCD camera, this specification requires spreading unresolved images over many pixels. Large images ease correction for aberration or field curvature. Permitting large images allows simple and lightweight very-wide-angle designs employing spherical and toroidal mirrors and thick lenses that can view almost the entire sky. We present formulas and graphic results relating sky angle to focal-plane position and determining the tangential and sagittal focal surfaces governing image size at the CCD. Laboratory measurements with two prototype configurations confirm the calculations.

© 1998 Optical Society of America

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References

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  1. S. B. Mende, R. H. Eather, E. K. Aamodt, “Instrument for the monochromatic observation of all sky auroral images,” Appl. Opt. 16, 1691–1700 (1977).
    [CrossRef] [PubMed]
  2. S. B. Mende, “Monochromatic imaging of the 6300 angstrom emissions from South Pole station,” Antarct. J. 19, 235–236 (1984).
  3. J. E. Shields, R. W. Johnson, T. L. Koehler, “Automated whole sky imaging systems for cloud field assessment,” in Proceedings of the Fourth Symposium of Global Change Studies (American Meteorological Society, Boston, 1993), pp. 228–231.
  4. H. P. Brueggemann, Conic Mirrors (Focal Press, London, 1968), pp. 35–39.
  5. B. V. Jackson, A. Buffington, P. L. Hick, S. W. Kahler, R. C. Altrock, R. E. Gold, D. F. Webb, “The solar mass ejection imager,” in Solar Wind Eight, D. Winterhalter, J. T. Gosling, S. R. Habbal, W. S. Kurth, M. Neugebauer, eds., AIP Conf. Proc.382, 536–534 (1996).
  6. B. V. Jackson, A. Buffington, P. P. Hick, R. Harrison, N. R. Waltham, F. M. Ipavich, A. B. Galvin, J. A. Slavin, M. H. Acuña, R. A. Howard, M. Koomin, C. M. Korendyke, D. J. Michels, “PERSPECTIVE—a deep-space mission to investigate the Sun–Earth connection,” a proposal in response to NASA’s SMEX announcement of flight opportunity (June1997).
  7. C. L. Korendyke, D. K. Prinz, D. G. Socker, R. A. Howard, J. W. Cook, A. Buffington, P. Hick, B. V. Jackson, “All-sky and high resolution coronagraphs for ‘FIRE,’” a proposal in response to NASA’s Advanced Instrument Concepts for a Near-Sun Fly-by Mission announcement of development opportunity (January1996).
  8. B. V. Jackson, C. Leinnert, “HELIOS images of solar mass ejections,” J. Geophys. Res. 90, 10,759–10,769 (1985).
    [CrossRef]
  9. B. V. Jackson, “Imaging of coronal mass ejections,” Sol. Phys. 100, 563–575 (1985).
    [CrossRef]
  10. B. V. Jackson, R. A. Howard, N. R. Sheeley, D. J. Michels, M. J. Koomen, R. M. E. Illing, “HELIOS spacecraft and Earth perspective observations of three looplike solar mass ejection transients,” J. Geophys. Res. 90, 5075–5081 (1985).
    [CrossRef]
  11. D. F. Webb, B. V. Jackson, “The identification and characteristics of solar mass ejections observed in the heliosphere with the HELIOS-2 photometers,” J. Geophys. Res. 95, 20641–20661 (1990).
    [CrossRef]
  12. B. V. Jackson, “HELIOS spacecraft photometer observation of elongated corotating structures in the interplanetary medium,” J. Geophys. Res. 96, 11307–11318 (1991).
    [CrossRef]
  13. B. V. Jackson, A. Buffington, P. L. Hick, S. W. Kahler, D. F. Webb, “A spaceborne near-Earth asteroid detection system,” Astron. Astrophys. Suppl. Ser. 108, 279–285 (1994).
  14. A. Buffington, B. V. Jackson, C. M. Korendyke, “Wide-angle stray-light reduction for a spaceborne optical hemispherical imager,” Appl. Opt. 35, 6669–6673 (1996).
    [CrossRef] [PubMed]
  15. M. V. Berry, “Reflections on a Christmas-tree bauble,” Phys. Ed. 7, 1–6 (1972).
    [CrossRef]
  16. R. S. Longhurst, Geometrical and Physical Optics, 2nd ed. (Wiley, New York, 1967), p. 35.
  17. M. G. J. Minnaert, The Nature of Light and Colour in the Open Air, translated by H. M. Kremer, revised by K. E. Brian Jay (Dover, New York, 1954), Section I-11.
  18. See, for example, the garden globe in Mole’s garden that “reflected everything all wrong and had a very pleasing effect,” in K. Grahame, The Wind in the Willows (Scribner’s, New York1954), Chap. 5 (Dulce Domum).
  19. See, for example, E. Hall, The Arnolfini Betrothal (U. California Press, Berkeley, Calif., 1994), describing the portrait and its wide-angle convex mirror behind, a painting from 1434 by Jan van Eyck.
  20. J. L. Locher, ed., Escher (Thames and Hudson, London, 1982), presenting five different drawings of spherical reflectors by M. C. Escher.
  21. P. S. Theocaris, “Properties of caustics from conic reflectors. 1: Meridional rays,” Appl. Opt. 16, 1705–1716 (1977).
    [CrossRef] [PubMed]
  22. A. Buffington, H. S. Hudson, C. H. Booth, “A laboratory measurement of CCD photometric and dimensional stability,” Publ. Astron. Soc. Pac. 102, 688–697 (1990).
    [CrossRef]
  23. A. Buffington, C. H. Booth, H. S. Hudson, “Using image area to control CCD systematic errors in spaceborne photometric and astrometric time-series measurements,” Publ. Astron. Soc. Pac. 103, 685–693 (1991).
    [CrossRef]
  24. W. J. Smith, Modern Optical Engineering, the Design of Optical Systems (McGraw-Hill, New York, 1966), Sect. 3.2.3, or see also the introductory discussion in Ref. 4.
  25. M. Born, E. Wolf, Principles of Optics, 6th (corrected) ed. (Pergamon, New York, 1989), Sect. 5.3.
  26. W. J. Smith, Modern Lens Design: a Resource Manual (McGraw-Hill, New York, 1992), p. 165.
  27. See, for example, Ref. 24, Chap. 10.
  28. Spectra-Tech Inc., 2 Research Drive, P.O. Box 869, Shelton, Conn., made the toroidal mirrors.

1996 (1)

1994 (1)

B. V. Jackson, A. Buffington, P. L. Hick, S. W. Kahler, D. F. Webb, “A spaceborne near-Earth asteroid detection system,” Astron. Astrophys. Suppl. Ser. 108, 279–285 (1994).

1991 (2)

A. Buffington, C. H. Booth, H. S. Hudson, “Using image area to control CCD systematic errors in spaceborne photometric and astrometric time-series measurements,” Publ. Astron. Soc. Pac. 103, 685–693 (1991).
[CrossRef]

B. V. Jackson, “HELIOS spacecraft photometer observation of elongated corotating structures in the interplanetary medium,” J. Geophys. Res. 96, 11307–11318 (1991).
[CrossRef]

1990 (2)

D. F. Webb, B. V. Jackson, “The identification and characteristics of solar mass ejections observed in the heliosphere with the HELIOS-2 photometers,” J. Geophys. Res. 95, 20641–20661 (1990).
[CrossRef]

A. Buffington, H. S. Hudson, C. H. Booth, “A laboratory measurement of CCD photometric and dimensional stability,” Publ. Astron. Soc. Pac. 102, 688–697 (1990).
[CrossRef]

1985 (3)

B. V. Jackson, C. Leinnert, “HELIOS images of solar mass ejections,” J. Geophys. Res. 90, 10,759–10,769 (1985).
[CrossRef]

B. V. Jackson, “Imaging of coronal mass ejections,” Sol. Phys. 100, 563–575 (1985).
[CrossRef]

B. V. Jackson, R. A. Howard, N. R. Sheeley, D. J. Michels, M. J. Koomen, R. M. E. Illing, “HELIOS spacecraft and Earth perspective observations of three looplike solar mass ejection transients,” J. Geophys. Res. 90, 5075–5081 (1985).
[CrossRef]

1984 (1)

S. B. Mende, “Monochromatic imaging of the 6300 angstrom emissions from South Pole station,” Antarct. J. 19, 235–236 (1984).

1977 (2)

1972 (1)

M. V. Berry, “Reflections on a Christmas-tree bauble,” Phys. Ed. 7, 1–6 (1972).
[CrossRef]

Aamodt, E. K.

Acuña, M. H.

B. V. Jackson, A. Buffington, P. P. Hick, R. Harrison, N. R. Waltham, F. M. Ipavich, A. B. Galvin, J. A. Slavin, M. H. Acuña, R. A. Howard, M. Koomin, C. M. Korendyke, D. J. Michels, “PERSPECTIVE—a deep-space mission to investigate the Sun–Earth connection,” a proposal in response to NASA’s SMEX announcement of flight opportunity (June1997).

Altrock, R. C.

B. V. Jackson, A. Buffington, P. L. Hick, S. W. Kahler, R. C. Altrock, R. E. Gold, D. F. Webb, “The solar mass ejection imager,” in Solar Wind Eight, D. Winterhalter, J. T. Gosling, S. R. Habbal, W. S. Kurth, M. Neugebauer, eds., AIP Conf. Proc.382, 536–534 (1996).

Berry, M. V.

M. V. Berry, “Reflections on a Christmas-tree bauble,” Phys. Ed. 7, 1–6 (1972).
[CrossRef]

Booth, C. H.

A. Buffington, C. H. Booth, H. S. Hudson, “Using image area to control CCD systematic errors in spaceborne photometric and astrometric time-series measurements,” Publ. Astron. Soc. Pac. 103, 685–693 (1991).
[CrossRef]

A. Buffington, H. S. Hudson, C. H. Booth, “A laboratory measurement of CCD photometric and dimensional stability,” Publ. Astron. Soc. Pac. 102, 688–697 (1990).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th (corrected) ed. (Pergamon, New York, 1989), Sect. 5.3.

Brueggemann, H. P.

H. P. Brueggemann, Conic Mirrors (Focal Press, London, 1968), pp. 35–39.

Buffington, A.

A. Buffington, B. V. Jackson, C. M. Korendyke, “Wide-angle stray-light reduction for a spaceborne optical hemispherical imager,” Appl. Opt. 35, 6669–6673 (1996).
[CrossRef] [PubMed]

B. V. Jackson, A. Buffington, P. L. Hick, S. W. Kahler, D. F. Webb, “A spaceborne near-Earth asteroid detection system,” Astron. Astrophys. Suppl. Ser. 108, 279–285 (1994).

A. Buffington, C. H. Booth, H. S. Hudson, “Using image area to control CCD systematic errors in spaceborne photometric and astrometric time-series measurements,” Publ. Astron. Soc. Pac. 103, 685–693 (1991).
[CrossRef]

A. Buffington, H. S. Hudson, C. H. Booth, “A laboratory measurement of CCD photometric and dimensional stability,” Publ. Astron. Soc. Pac. 102, 688–697 (1990).
[CrossRef]

B. V. Jackson, A. Buffington, P. L. Hick, S. W. Kahler, R. C. Altrock, R. E. Gold, D. F. Webb, “The solar mass ejection imager,” in Solar Wind Eight, D. Winterhalter, J. T. Gosling, S. R. Habbal, W. S. Kurth, M. Neugebauer, eds., AIP Conf. Proc.382, 536–534 (1996).

B. V. Jackson, A. Buffington, P. P. Hick, R. Harrison, N. R. Waltham, F. M. Ipavich, A. B. Galvin, J. A. Slavin, M. H. Acuña, R. A. Howard, M. Koomin, C. M. Korendyke, D. J. Michels, “PERSPECTIVE—a deep-space mission to investigate the Sun–Earth connection,” a proposal in response to NASA’s SMEX announcement of flight opportunity (June1997).

C. L. Korendyke, D. K. Prinz, D. G. Socker, R. A. Howard, J. W. Cook, A. Buffington, P. Hick, B. V. Jackson, “All-sky and high resolution coronagraphs for ‘FIRE,’” a proposal in response to NASA’s Advanced Instrument Concepts for a Near-Sun Fly-by Mission announcement of development opportunity (January1996).

Cook, J. W.

C. L. Korendyke, D. K. Prinz, D. G. Socker, R. A. Howard, J. W. Cook, A. Buffington, P. Hick, B. V. Jackson, “All-sky and high resolution coronagraphs for ‘FIRE,’” a proposal in response to NASA’s Advanced Instrument Concepts for a Near-Sun Fly-by Mission announcement of development opportunity (January1996).

Eather, R. H.

Galvin, A. B.

B. V. Jackson, A. Buffington, P. P. Hick, R. Harrison, N. R. Waltham, F. M. Ipavich, A. B. Galvin, J. A. Slavin, M. H. Acuña, R. A. Howard, M. Koomin, C. M. Korendyke, D. J. Michels, “PERSPECTIVE—a deep-space mission to investigate the Sun–Earth connection,” a proposal in response to NASA’s SMEX announcement of flight opportunity (June1997).

Gold, R. E.

B. V. Jackson, A. Buffington, P. L. Hick, S. W. Kahler, R. C. Altrock, R. E. Gold, D. F. Webb, “The solar mass ejection imager,” in Solar Wind Eight, D. Winterhalter, J. T. Gosling, S. R. Habbal, W. S. Kurth, M. Neugebauer, eds., AIP Conf. Proc.382, 536–534 (1996).

Grahame, K.

See, for example, the garden globe in Mole’s garden that “reflected everything all wrong and had a very pleasing effect,” in K. Grahame, The Wind in the Willows (Scribner’s, New York1954), Chap. 5 (Dulce Domum).

Hall, E.

See, for example, E. Hall, The Arnolfini Betrothal (U. California Press, Berkeley, Calif., 1994), describing the portrait and its wide-angle convex mirror behind, a painting from 1434 by Jan van Eyck.

Harrison, R.

B. V. Jackson, A. Buffington, P. P. Hick, R. Harrison, N. R. Waltham, F. M. Ipavich, A. B. Galvin, J. A. Slavin, M. H. Acuña, R. A. Howard, M. Koomin, C. M. Korendyke, D. J. Michels, “PERSPECTIVE—a deep-space mission to investigate the Sun–Earth connection,” a proposal in response to NASA’s SMEX announcement of flight opportunity (June1997).

Hick, P.

C. L. Korendyke, D. K. Prinz, D. G. Socker, R. A. Howard, J. W. Cook, A. Buffington, P. Hick, B. V. Jackson, “All-sky and high resolution coronagraphs for ‘FIRE,’” a proposal in response to NASA’s Advanced Instrument Concepts for a Near-Sun Fly-by Mission announcement of development opportunity (January1996).

Hick, P. L.

B. V. Jackson, A. Buffington, P. L. Hick, S. W. Kahler, D. F. Webb, “A spaceborne near-Earth asteroid detection system,” Astron. Astrophys. Suppl. Ser. 108, 279–285 (1994).

B. V. Jackson, A. Buffington, P. L. Hick, S. W. Kahler, R. C. Altrock, R. E. Gold, D. F. Webb, “The solar mass ejection imager,” in Solar Wind Eight, D. Winterhalter, J. T. Gosling, S. R. Habbal, W. S. Kurth, M. Neugebauer, eds., AIP Conf. Proc.382, 536–534 (1996).

Hick, P. P.

B. V. Jackson, A. Buffington, P. P. Hick, R. Harrison, N. R. Waltham, F. M. Ipavich, A. B. Galvin, J. A. Slavin, M. H. Acuña, R. A. Howard, M. Koomin, C. M. Korendyke, D. J. Michels, “PERSPECTIVE—a deep-space mission to investigate the Sun–Earth connection,” a proposal in response to NASA’s SMEX announcement of flight opportunity (June1997).

Howard, R. A.

B. V. Jackson, R. A. Howard, N. R. Sheeley, D. J. Michels, M. J. Koomen, R. M. E. Illing, “HELIOS spacecraft and Earth perspective observations of three looplike solar mass ejection transients,” J. Geophys. Res. 90, 5075–5081 (1985).
[CrossRef]

C. L. Korendyke, D. K. Prinz, D. G. Socker, R. A. Howard, J. W. Cook, A. Buffington, P. Hick, B. V. Jackson, “All-sky and high resolution coronagraphs for ‘FIRE,’” a proposal in response to NASA’s Advanced Instrument Concepts for a Near-Sun Fly-by Mission announcement of development opportunity (January1996).

B. V. Jackson, A. Buffington, P. P. Hick, R. Harrison, N. R. Waltham, F. M. Ipavich, A. B. Galvin, J. A. Slavin, M. H. Acuña, R. A. Howard, M. Koomin, C. M. Korendyke, D. J. Michels, “PERSPECTIVE—a deep-space mission to investigate the Sun–Earth connection,” a proposal in response to NASA’s SMEX announcement of flight opportunity (June1997).

Hudson, H. S.

A. Buffington, C. H. Booth, H. S. Hudson, “Using image area to control CCD systematic errors in spaceborne photometric and astrometric time-series measurements,” Publ. Astron. Soc. Pac. 103, 685–693 (1991).
[CrossRef]

A. Buffington, H. S. Hudson, C. H. Booth, “A laboratory measurement of CCD photometric and dimensional stability,” Publ. Astron. Soc. Pac. 102, 688–697 (1990).
[CrossRef]

Illing, R. M. E.

B. V. Jackson, R. A. Howard, N. R. Sheeley, D. J. Michels, M. J. Koomen, R. M. E. Illing, “HELIOS spacecraft and Earth perspective observations of three looplike solar mass ejection transients,” J. Geophys. Res. 90, 5075–5081 (1985).
[CrossRef]

Ipavich, F. M.

B. V. Jackson, A. Buffington, P. P. Hick, R. Harrison, N. R. Waltham, F. M. Ipavich, A. B. Galvin, J. A. Slavin, M. H. Acuña, R. A. Howard, M. Koomin, C. M. Korendyke, D. J. Michels, “PERSPECTIVE—a deep-space mission to investigate the Sun–Earth connection,” a proposal in response to NASA’s SMEX announcement of flight opportunity (June1997).

Jackson, B. V.

A. Buffington, B. V. Jackson, C. M. Korendyke, “Wide-angle stray-light reduction for a spaceborne optical hemispherical imager,” Appl. Opt. 35, 6669–6673 (1996).
[CrossRef] [PubMed]

B. V. Jackson, A. Buffington, P. L. Hick, S. W. Kahler, D. F. Webb, “A spaceborne near-Earth asteroid detection system,” Astron. Astrophys. Suppl. Ser. 108, 279–285 (1994).

B. V. Jackson, “HELIOS spacecraft photometer observation of elongated corotating structures in the interplanetary medium,” J. Geophys. Res. 96, 11307–11318 (1991).
[CrossRef]

D. F. Webb, B. V. Jackson, “The identification and characteristics of solar mass ejections observed in the heliosphere with the HELIOS-2 photometers,” J. Geophys. Res. 95, 20641–20661 (1990).
[CrossRef]

B. V. Jackson, “Imaging of coronal mass ejections,” Sol. Phys. 100, 563–575 (1985).
[CrossRef]

B. V. Jackson, R. A. Howard, N. R. Sheeley, D. J. Michels, M. J. Koomen, R. M. E. Illing, “HELIOS spacecraft and Earth perspective observations of three looplike solar mass ejection transients,” J. Geophys. Res. 90, 5075–5081 (1985).
[CrossRef]

B. V. Jackson, C. Leinnert, “HELIOS images of solar mass ejections,” J. Geophys. Res. 90, 10,759–10,769 (1985).
[CrossRef]

C. L. Korendyke, D. K. Prinz, D. G. Socker, R. A. Howard, J. W. Cook, A. Buffington, P. Hick, B. V. Jackson, “All-sky and high resolution coronagraphs for ‘FIRE,’” a proposal in response to NASA’s Advanced Instrument Concepts for a Near-Sun Fly-by Mission announcement of development opportunity (January1996).

B. V. Jackson, A. Buffington, P. P. Hick, R. Harrison, N. R. Waltham, F. M. Ipavich, A. B. Galvin, J. A. Slavin, M. H. Acuña, R. A. Howard, M. Koomin, C. M. Korendyke, D. J. Michels, “PERSPECTIVE—a deep-space mission to investigate the Sun–Earth connection,” a proposal in response to NASA’s SMEX announcement of flight opportunity (June1997).

B. V. Jackson, A. Buffington, P. L. Hick, S. W. Kahler, R. C. Altrock, R. E. Gold, D. F. Webb, “The solar mass ejection imager,” in Solar Wind Eight, D. Winterhalter, J. T. Gosling, S. R. Habbal, W. S. Kurth, M. Neugebauer, eds., AIP Conf. Proc.382, 536–534 (1996).

Johnson, R. W.

J. E. Shields, R. W. Johnson, T. L. Koehler, “Automated whole sky imaging systems for cloud field assessment,” in Proceedings of the Fourth Symposium of Global Change Studies (American Meteorological Society, Boston, 1993), pp. 228–231.

Kahler, S. W.

B. V. Jackson, A. Buffington, P. L. Hick, S. W. Kahler, D. F. Webb, “A spaceborne near-Earth asteroid detection system,” Astron. Astrophys. Suppl. Ser. 108, 279–285 (1994).

B. V. Jackson, A. Buffington, P. L. Hick, S. W. Kahler, R. C. Altrock, R. E. Gold, D. F. Webb, “The solar mass ejection imager,” in Solar Wind Eight, D. Winterhalter, J. T. Gosling, S. R. Habbal, W. S. Kurth, M. Neugebauer, eds., AIP Conf. Proc.382, 536–534 (1996).

Koehler, T. L.

J. E. Shields, R. W. Johnson, T. L. Koehler, “Automated whole sky imaging systems for cloud field assessment,” in Proceedings of the Fourth Symposium of Global Change Studies (American Meteorological Society, Boston, 1993), pp. 228–231.

Koomen, M. J.

B. V. Jackson, R. A. Howard, N. R. Sheeley, D. J. Michels, M. J. Koomen, R. M. E. Illing, “HELIOS spacecraft and Earth perspective observations of three looplike solar mass ejection transients,” J. Geophys. Res. 90, 5075–5081 (1985).
[CrossRef]

Koomin, M.

B. V. Jackson, A. Buffington, P. P. Hick, R. Harrison, N. R. Waltham, F. M. Ipavich, A. B. Galvin, J. A. Slavin, M. H. Acuña, R. A. Howard, M. Koomin, C. M. Korendyke, D. J. Michels, “PERSPECTIVE—a deep-space mission to investigate the Sun–Earth connection,” a proposal in response to NASA’s SMEX announcement of flight opportunity (June1997).

Korendyke, C. L.

C. L. Korendyke, D. K. Prinz, D. G. Socker, R. A. Howard, J. W. Cook, A. Buffington, P. Hick, B. V. Jackson, “All-sky and high resolution coronagraphs for ‘FIRE,’” a proposal in response to NASA’s Advanced Instrument Concepts for a Near-Sun Fly-by Mission announcement of development opportunity (January1996).

Korendyke, C. M.

A. Buffington, B. V. Jackson, C. M. Korendyke, “Wide-angle stray-light reduction for a spaceborne optical hemispherical imager,” Appl. Opt. 35, 6669–6673 (1996).
[CrossRef] [PubMed]

B. V. Jackson, A. Buffington, P. P. Hick, R. Harrison, N. R. Waltham, F. M. Ipavich, A. B. Galvin, J. A. Slavin, M. H. Acuña, R. A. Howard, M. Koomin, C. M. Korendyke, D. J. Michels, “PERSPECTIVE—a deep-space mission to investigate the Sun–Earth connection,” a proposal in response to NASA’s SMEX announcement of flight opportunity (June1997).

Leinnert, C.

B. V. Jackson, C. Leinnert, “HELIOS images of solar mass ejections,” J. Geophys. Res. 90, 10,759–10,769 (1985).
[CrossRef]

Longhurst, R. S.

R. S. Longhurst, Geometrical and Physical Optics, 2nd ed. (Wiley, New York, 1967), p. 35.

Mende, S. B.

S. B. Mende, “Monochromatic imaging of the 6300 angstrom emissions from South Pole station,” Antarct. J. 19, 235–236 (1984).

S. B. Mende, R. H. Eather, E. K. Aamodt, “Instrument for the monochromatic observation of all sky auroral images,” Appl. Opt. 16, 1691–1700 (1977).
[CrossRef] [PubMed]

Michels, D. J.

B. V. Jackson, R. A. Howard, N. R. Sheeley, D. J. Michels, M. J. Koomen, R. M. E. Illing, “HELIOS spacecraft and Earth perspective observations of three looplike solar mass ejection transients,” J. Geophys. Res. 90, 5075–5081 (1985).
[CrossRef]

B. V. Jackson, A. Buffington, P. P. Hick, R. Harrison, N. R. Waltham, F. M. Ipavich, A. B. Galvin, J. A. Slavin, M. H. Acuña, R. A. Howard, M. Koomin, C. M. Korendyke, D. J. Michels, “PERSPECTIVE—a deep-space mission to investigate the Sun–Earth connection,” a proposal in response to NASA’s SMEX announcement of flight opportunity (June1997).

Minnaert, M. G. J.

M. G. J. Minnaert, The Nature of Light and Colour in the Open Air, translated by H. M. Kremer, revised by K. E. Brian Jay (Dover, New York, 1954), Section I-11.

Prinz, D. K.

C. L. Korendyke, D. K. Prinz, D. G. Socker, R. A. Howard, J. W. Cook, A. Buffington, P. Hick, B. V. Jackson, “All-sky and high resolution coronagraphs for ‘FIRE,’” a proposal in response to NASA’s Advanced Instrument Concepts for a Near-Sun Fly-by Mission announcement of development opportunity (January1996).

Sheeley, N. R.

B. V. Jackson, R. A. Howard, N. R. Sheeley, D. J. Michels, M. J. Koomen, R. M. E. Illing, “HELIOS spacecraft and Earth perspective observations of three looplike solar mass ejection transients,” J. Geophys. Res. 90, 5075–5081 (1985).
[CrossRef]

Shields, J. E.

J. E. Shields, R. W. Johnson, T. L. Koehler, “Automated whole sky imaging systems for cloud field assessment,” in Proceedings of the Fourth Symposium of Global Change Studies (American Meteorological Society, Boston, 1993), pp. 228–231.

Slavin, J. A.

B. V. Jackson, A. Buffington, P. P. Hick, R. Harrison, N. R. Waltham, F. M. Ipavich, A. B. Galvin, J. A. Slavin, M. H. Acuña, R. A. Howard, M. Koomin, C. M. Korendyke, D. J. Michels, “PERSPECTIVE—a deep-space mission to investigate the Sun–Earth connection,” a proposal in response to NASA’s SMEX announcement of flight opportunity (June1997).

Smith, W. J.

W. J. Smith, Modern Lens Design: a Resource Manual (McGraw-Hill, New York, 1992), p. 165.

W. J. Smith, Modern Optical Engineering, the Design of Optical Systems (McGraw-Hill, New York, 1966), Sect. 3.2.3, or see also the introductory discussion in Ref. 4.

Socker, D. G.

C. L. Korendyke, D. K. Prinz, D. G. Socker, R. A. Howard, J. W. Cook, A. Buffington, P. Hick, B. V. Jackson, “All-sky and high resolution coronagraphs for ‘FIRE,’” a proposal in response to NASA’s Advanced Instrument Concepts for a Near-Sun Fly-by Mission announcement of development opportunity (January1996).

Theocaris, P. S.

Waltham, N. R.

B. V. Jackson, A. Buffington, P. P. Hick, R. Harrison, N. R. Waltham, F. M. Ipavich, A. B. Galvin, J. A. Slavin, M. H. Acuña, R. A. Howard, M. Koomin, C. M. Korendyke, D. J. Michels, “PERSPECTIVE—a deep-space mission to investigate the Sun–Earth connection,” a proposal in response to NASA’s SMEX announcement of flight opportunity (June1997).

Webb, D. F.

B. V. Jackson, A. Buffington, P. L. Hick, S. W. Kahler, D. F. Webb, “A spaceborne near-Earth asteroid detection system,” Astron. Astrophys. Suppl. Ser. 108, 279–285 (1994).

D. F. Webb, B. V. Jackson, “The identification and characteristics of solar mass ejections observed in the heliosphere with the HELIOS-2 photometers,” J. Geophys. Res. 95, 20641–20661 (1990).
[CrossRef]

B. V. Jackson, A. Buffington, P. L. Hick, S. W. Kahler, R. C. Altrock, R. E. Gold, D. F. Webb, “The solar mass ejection imager,” in Solar Wind Eight, D. Winterhalter, J. T. Gosling, S. R. Habbal, W. S. Kurth, M. Neugebauer, eds., AIP Conf. Proc.382, 536–534 (1996).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th (corrected) ed. (Pergamon, New York, 1989), Sect. 5.3.

Antarct. J. (1)

S. B. Mende, “Monochromatic imaging of the 6300 angstrom emissions from South Pole station,” Antarct. J. 19, 235–236 (1984).

Appl. Opt. (3)

Astron. Astrophys. Suppl. Ser. (1)

B. V. Jackson, A. Buffington, P. L. Hick, S. W. Kahler, D. F. Webb, “A spaceborne near-Earth asteroid detection system,” Astron. Astrophys. Suppl. Ser. 108, 279–285 (1994).

J. Geophys. Res. (4)

B. V. Jackson, C. Leinnert, “HELIOS images of solar mass ejections,” J. Geophys. Res. 90, 10,759–10,769 (1985).
[CrossRef]

B. V. Jackson, R. A. Howard, N. R. Sheeley, D. J. Michels, M. J. Koomen, R. M. E. Illing, “HELIOS spacecraft and Earth perspective observations of three looplike solar mass ejection transients,” J. Geophys. Res. 90, 5075–5081 (1985).
[CrossRef]

D. F. Webb, B. V. Jackson, “The identification and characteristics of solar mass ejections observed in the heliosphere with the HELIOS-2 photometers,” J. Geophys. Res. 95, 20641–20661 (1990).
[CrossRef]

B. V. Jackson, “HELIOS spacecraft photometer observation of elongated corotating structures in the interplanetary medium,” J. Geophys. Res. 96, 11307–11318 (1991).
[CrossRef]

Phys. Ed. (1)

M. V. Berry, “Reflections on a Christmas-tree bauble,” Phys. Ed. 7, 1–6 (1972).
[CrossRef]

Publ. Astron. Soc. Pac. (2)

A. Buffington, H. S. Hudson, C. H. Booth, “A laboratory measurement of CCD photometric and dimensional stability,” Publ. Astron. Soc. Pac. 102, 688–697 (1990).
[CrossRef]

A. Buffington, C. H. Booth, H. S. Hudson, “Using image area to control CCD systematic errors in spaceborne photometric and astrometric time-series measurements,” Publ. Astron. Soc. Pac. 103, 685–693 (1991).
[CrossRef]

Sol. Phys. (1)

B. V. Jackson, “Imaging of coronal mass ejections,” Sol. Phys. 100, 563–575 (1985).
[CrossRef]

Other (15)

J. E. Shields, R. W. Johnson, T. L. Koehler, “Automated whole sky imaging systems for cloud field assessment,” in Proceedings of the Fourth Symposium of Global Change Studies (American Meteorological Society, Boston, 1993), pp. 228–231.

H. P. Brueggemann, Conic Mirrors (Focal Press, London, 1968), pp. 35–39.

B. V. Jackson, A. Buffington, P. L. Hick, S. W. Kahler, R. C. Altrock, R. E. Gold, D. F. Webb, “The solar mass ejection imager,” in Solar Wind Eight, D. Winterhalter, J. T. Gosling, S. R. Habbal, W. S. Kurth, M. Neugebauer, eds., AIP Conf. Proc.382, 536–534 (1996).

B. V. Jackson, A. Buffington, P. P. Hick, R. Harrison, N. R. Waltham, F. M. Ipavich, A. B. Galvin, J. A. Slavin, M. H. Acuña, R. A. Howard, M. Koomin, C. M. Korendyke, D. J. Michels, “PERSPECTIVE—a deep-space mission to investigate the Sun–Earth connection,” a proposal in response to NASA’s SMEX announcement of flight opportunity (June1997).

C. L. Korendyke, D. K. Prinz, D. G. Socker, R. A. Howard, J. W. Cook, A. Buffington, P. Hick, B. V. Jackson, “All-sky and high resolution coronagraphs for ‘FIRE,’” a proposal in response to NASA’s Advanced Instrument Concepts for a Near-Sun Fly-by Mission announcement of development opportunity (January1996).

W. J. Smith, Modern Optical Engineering, the Design of Optical Systems (McGraw-Hill, New York, 1966), Sect. 3.2.3, or see also the introductory discussion in Ref. 4.

M. Born, E. Wolf, Principles of Optics, 6th (corrected) ed. (Pergamon, New York, 1989), Sect. 5.3.

W. J. Smith, Modern Lens Design: a Resource Manual (McGraw-Hill, New York, 1992), p. 165.

See, for example, Ref. 24, Chap. 10.

Spectra-Tech Inc., 2 Research Drive, P.O. Box 869, Shelton, Conn., made the toroidal mirrors.

R. S. Longhurst, Geometrical and Physical Optics, 2nd ed. (Wiley, New York, 1967), p. 35.

M. G. J. Minnaert, The Nature of Light and Colour in the Open Air, translated by H. M. Kremer, revised by K. E. Brian Jay (Dover, New York, 1954), Section I-11.

See, for example, the garden globe in Mole’s garden that “reflected everything all wrong and had a very pleasing effect,” in K. Grahame, The Wind in the Willows (Scribner’s, New York1954), Chap. 5 (Dulce Domum).

See, for example, E. Hall, The Arnolfini Betrothal (U. California Press, Berkeley, Calif., 1994), describing the portrait and its wide-angle convex mirror behind, a painting from 1434 by Jan van Eyck.

J. L. Locher, ed., Escher (Thames and Hudson, London, 1982), presenting five different drawings of spherical reflectors by M. C. Escher.

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Figures (17)

Fig. 1
Fig. 1

Plot of Eq. (1), position angle Φ/Φmax versus incident angle Θ relative to the negative X axis, of light reflected in a convex hemispherical mirror M of radius R, for various distances dR between the center of curvature of M and the observer. Φmax = sin-1(R/ d) is the maximum angle subtended by M. For d/ R near unity, M covers only the rear hemisphere 90° < Θ < 180° (as a plane mirror). Coverage grows to all 0° < Θ < 180° as d/ R → ∞. The inset illustrates M, the X and Y coordinate frame in the plane Z = 0, incident light (dashed) reflecting symmetrically around the normal vector to reach the observer at point d, and angles Θ, Φ, and ζ.

Fig. 2
Fig. 2

Plot of Eq. (2), the apparent fractional position angle Φ/Φmax versus incident angle Θ of light reflected in a concave spherical mirror M of radius R for negative distances d from the center of curvature of M to the observer. Here Φmax = -tan-1(R/ d) is the half-angle subtended by the open disk of M. The effective range of Φ may be less due to self-occultation. The inset illustrates M, the XY coordinate frame, incident light (dashed) reflecting symmetrically around the normal vector and reaching the observer at d, and finally Θ and Φ redefined for Eq. (2). Angle ζ is the same as in Fig. 1. Note that M here is partially cut away to admit this particular ray of light.

Fig. 3
Fig. 3

Plot of the intersection of the tangential focal surface, [X t , Y t ]/R given by Eqs. (3), (5), (7), and (9), with the XY plane for a spherical mirror, for various d/ R. The positive domain of d/ R is plotted in the top half (Y/ R > 0) and the negative domain below. Note that the tangential focus starts at X/ R = 1/2 on the X axis for Φ = 0 and approaches M as Φ increases.

Fig. 4
Fig. 4

Plot of the intersection of the sagittal focal surface [X s , Y s ]/R with the XY plane for a spherical mirror for various d/ R. The positive domain of d/ R, Eqs. (5) and (6), is plotted in the top half (Y/ R > 0) and the negative domain, Eqs. (6) and (9), below. When Φ = 0, sagittal and tangential foci coincide on the X axis at X = R/2; as Φ increases the sagittal focus passes outside of and away from the mirror surface to the left.

Fig. 5
Fig. 5

Measured and calculated central-ray positions on a scale (left axis) placed 0.1 cm toward the camera from the surface of a garden globe and of tan Φ/tan Φmax (right axis), as a function of Θ, the incident-light angle. Here R = 12.55 and d = 56.7.

Fig. 6
Fig. 6

Locations of measured and calculated tangential and sagittal foci for the garden globe. The thick-line circle shows the reflecting part of the sphere, and the slightly larger dashed circle shows the outside of the glass.

Fig. 7
Fig. 7

Design configuration of an imager6 employing a crystal ball and toroidal mirror viewing a hemisphere and protected by a background-light-reducing corral.14 This configuration is cylindrically symmetric about the X-axis center line. This system views light from the left hemisphere. Light within ∼45° of the center line passes through the crystal ball and is viewed by the CCD/lens assembly to the right. Light from 45° to 90° enters through the circular slot between the mirror/lens assembly and the optics enclosure and is reflected from the toroidal surface of the mirror to the CCD/lens assembly.

Fig. 8
Fig. 8

Locations of measured and calculated central rays on a scale placed just in front of the glass ball for the optics illustrated in Fig. 7 as a function of Θ, the incident-light angle. Also shown is tan Φ/tan Φmax (right axis) with Φmax defined by the edge of the mirror. Measurements are shown for both positive and negative values of Θ.

Fig. 9
Fig. 9

Locations of calculated and measured tangential and sagittal foci for the optical system of Fig. 7. Here the large crosses mark the center of curvature of the crystal ball and where the toroidal mirror’s center-of-curvature circle passes through the XY plane.

Fig. 10
Fig. 10

Design configuration of an imager7 employing spherical (M1) and toroidal (M2) mirrors and a thick lens, viewing nearly the entire celestial sphere. The background-light-reducing corral14 shields the optical elements within a conical umbra that ends just outside of M2. This configuration is cylindrically symmetric about the X-axis center line. Three large crosses mark the center of curvature of M1 (middle cross) and where M2’s center-of-curvature ring intersects the XY plane (crosses above and below the X axis). For clarity the enclosure for the small lens and CCD system is not shown. The rest of the spacecraft lies beyond the corral to the right. The entire sky is viewed except within an 18° half-angle cone centered on the positive X axis (the line to the Sun). The CCD/lens assembly views sky reflected from M2 starting just outside the shadow provided by the corral and extending from 162° > Θ > 130°. The lens in the center of M2 covers an opposite portion of sky 0° < Θ < 35°, and the remainder of the sky is seen reflected in M1.

Fig. 11
Fig. 11

Locations of measured and calculated central rays on a scale placed just in front of M2, for this mirror and its encircled lens illustrated in Fig. 10, as a function of the incident-light angle Θ. Also shown is tan Φ/tan Φmax (right axis) with Φmax defined for the outside edge of M2. As in Fig. 8, sets of measurements are shown for both positive and negative values of Θ. With the full optical system of Fig. 10, M1 covers 35° < Θ < 130°, and its portion of the field of view occupies the majority of the camera’s focal-plane area.

Fig. 12
Fig. 12

Locations of calculated and measured tangential and sagittal foci for the optical system of Fig. 10. Here the measurements for the garden globe (Fig. 6) are displayed for the large spherical mirror. The sagittal focus for the lens differed insignificantly from tangential focus and is thus not displayed.

Fig. 13
Fig. 13

Plot of apparent fractional position angle Φ/Φmax versus incident angle Θ of light refracted through a homogeneous crystal ball of radius R and index of refraction n = 1.5 for various distances d from the center of the ball to the observer. Here Φmax = sin-1(R/ d) is the maximum angle subtended by the ball. The inset below the curves illustrates the coordinate frame, the observer on the X axis at +d, the path of a light ray (dashed line) incident at Θ and offset by Y 2 from the X axis, and the two normal vectors where the ray passes through the surface of the ball, as calculated by ray tracing.

Fig. 14
Fig. 14

Similar to Fig. 13, but with various indices of refraction and at fixed d/ R = 8. Note that increasing n is roughly equivalent to viewing from farther away.

Fig. 15
Fig. 15

Plot of the intersection of the tangential focal surface with the XY plane for the indicated n and values of d/ R, as calculated by ray tracing. It is easily shown16 that the curves here and in Fig. 16 must all cross the X axis at X/ R = (n/2)/(n - 1).

Fig. 16
Fig. 16

Same as Fig. 15, but for the sagittal focus using Eqs. (5) and (6).

Fig. 17
Fig. 17

Plot of the transverse offset Y 2 (see the inset in Fig. 13) versus apparent position Φ/Φmax for the indicated n and various d/ R. Unless viewed from very close, much of the front of a crystal ball can be masked off from potential stray light without interception of the incoming light over the field of view.

Equations (14)

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d   sin   Φ = R   cos Θ - Φ 2 .
- d   sin   Φ = R   sin Θ - Φ 2 .
X t R = cos   ζ + cos 2   Φ 2 sin   ζ   tan   Φ - cos   ζ = sin Θ + Φ 2 + cos 2   Φ 2 cos Θ + Φ 2 tan   Φ   - sin Θ + Φ 2 ,
ζ = 180 ° - Θ - Φ 2 ,
Y = d - X tan   Φ .
X s = d   tan   Φ tan   Φ - tan   Θ ,
X t R = cos   ζ - cos 2   Φ 2 sin   ζ   tan   Φ + cos   ζ = cos Θ + Φ 2 - cos 2   Φ 2 sin Θ + Φ 2 tan   Φ + cos Θ + Φ 2 ,
ζ = Θ + Φ 2 ,
Y = - d - X tan   Φ .
d - δ sin   Φ = d   sin   Φ - Y 0 cos   Φ = R   cos Θ - Φ 2 ,
- d + δ sin   Φ = - d   sin   Φ - Y 0 cos   Φ = R   sin Θ - Φ 2 .
d = d 2 + Y 0 2 1 / 2 ,
X s = d   tan   Φ + Y 0 tan   Θ / tan   ζ tan   Φ - tan   Θ .
f = W d ¯ 2 d ¯ tan   Φ max + W ,

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