Abstract

Conic mirror reflectometers are used to measure the diffuse reflectance and total integrated scatter of surfaces. In spite of the long history of using conic mirrors for these purposes, the maximum magnification of the three primary types of conic mirror (hemisphere, hemiellipsoid, and dual paraboloid) had not been compared quantitatively. To our knowledge, an exact magnification formula has not been published for any of the three primary conic mirrors. The maximum magnification is needed for proper sizing of detectors and radiation sources used with reflectometers. Exact analytical expressions for the maximum magnification of a Coblentz hemisphere, a hemiellipsoid, and a dual-paraboloid mirror system are derived and compared.

© 1998 Optical Society of America

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Corrections

Keith A. Snail and Leonard M. Hanssen, "Magnification of conic mirror reflectometers: errata," Appl. Opt. 37, 6695-6695 (1998)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-37-28-6695

References

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    [CrossRef] [PubMed]
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    [CrossRef]
  3. R. V. Dunkle, “Spectral reflectance measurements,” in Surface Effects on Spacecraft Materials, F. J. Clauss, ed. (Wiley, New York, 1960), p. 117.
  4. S. T. Dunn, J. C. Richmond, J. A. Wiebelt, “Ellipsoidal mirror reflectometer,” J. Res. Natl. Bur. Stand. Sec. C 70, 75–88 (1966).
    [CrossRef]
  5. J. T. Neu, “Design, fabrication and performance of an ellipsoidal spectroreflectometer,” Rep. NASA-CR-73193 (National Aeronautics and Space Administration, Washington, D.C., 1968).
  6. R. P. Heinisch, F. J. Bradac, D. B. Perlick, “On the fabrication and evaluation of an integrating hemiellipsoid,” Appl. Opt. 9, 483–487 (1970).
    [CrossRef] [PubMed]
  7. B. E. Wood, J. G. Pipes, A. M. Smith, J. A. Roux, “Hemiellipsoidal mirror infrared reflectometer: development and operation,” Appl. Opt. 15, 940–950 (1976).
    [CrossRef] [PubMed]
  8. F. J. J. Clarke, “Measurement of the radiometric properties of materials for building and aerospace applications,” Proc. SPIE 234, 40–47 (1980).
    [CrossRef]
  9. W. W. Coblentz, “The diffuse reflectance power of various substances,” Bull. Bur. Stand. (V. S.) Bull. 9, 283–325 (1913).
    [CrossRef]
  10. J. M. Bennett, L. Mattson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C.1989).
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    [CrossRef]
  12. W. Richter, “Fourier transform reflectance spectrometry between 8000 cm-1 (1.25 μm) and 800 cm-1 (12.5 μm) using an integrating sphere,” Appl. Spectrosc. 37, 32–38 (1983).
    [CrossRef]
  13. U. P. Oppenheim, M. G. Turner, W. L. Wolfe, “BRDF reference standards for the infrared,” Infrared Phys. Technol. 35, 873–879 (1994).
    [CrossRef]
  14. K. A. Snail, L. M. Hanssen, “Reflectance measurements using conic mirror reflectometers,” in Applied Spectroscopy: A Compact Reference for Practitioners, J. Workman, A. Springsteen, eds. (Academic, San Diego, Calif., 1998).
  15. F. J. J. Clarke, D. J. Parry, “Helmholtz reciprocity: its validity and application to reflectometry,” Lighting Res. Technol. 17, 1–11 (1985).
    [CrossRef]
  16. W. Richter, W. Erb, “Accurate diffuse reflectance measurements in the infrared,” Appl. Opt. 26, 4620–4624 (1987).
    [CrossRef] [PubMed]
  17. W. M. Brandenberg, “Focussing properties of hemispherical and ellipsoidal mirror reflectometers,” J. Opt. Soc. Am. 54, 1235–1237 (1964).
    [CrossRef]
  18. D. K. Edwards, “Comments on reciprocal-mode 2π-sr-mirror reflectometers,” Appl. Opt. 5, 175–176 (1966).
    [CrossRef] [PubMed]
  19. P. R. Griffiths, J. A. de Haseth, Fourier Transform Infrared Spectroscopy (Wiley, New York, 1986).
  20. K. A. Snail, “Reflectometer design using nonimaging optics,” Appl. Opt. 26, 5326–5332 (1987).
    [CrossRef] [PubMed]
  21. The ray tracing was performed with the Advanced Stray Light Analysis Package (ASAP) from Breault Research Organization, Tucson, Ariz. Maximum magnification runs used rays traced in double precision at an angle of 89.9°.
  22. F. Paschen, “Ubër die Vertheilung der Energie im Spectrum des schwarzen Körpers bei nederen temperaturen,” Berl. Akad. Wiss. 22, 405–420 (1899).

1994 (1)

U. P. Oppenheim, M. G. Turner, W. L. Wolfe, “BRDF reference standards for the infrared,” Infrared Phys. Technol. 35, 873–879 (1994).
[CrossRef]

1987 (2)

1985 (1)

F. J. J. Clarke, D. J. Parry, “Helmholtz reciprocity: its validity and application to reflectometry,” Lighting Res. Technol. 17, 1–11 (1985).
[CrossRef]

1983 (1)

1980 (1)

F. J. J. Clarke, “Measurement of the radiometric properties of materials for building and aerospace applications,” Proc. SPIE 234, 40–47 (1980).
[CrossRef]

1976 (2)

1970 (1)

1966 (2)

S. T. Dunn, J. C. Richmond, J. A. Wiebelt, “Ellipsoidal mirror reflectometer,” J. Res. Natl. Bur. Stand. Sec. C 70, 75–88 (1966).
[CrossRef]

D. K. Edwards, “Comments on reciprocal-mode 2π-sr-mirror reflectometers,” Appl. Opt. 5, 175–176 (1966).
[CrossRef] [PubMed]

1964 (1)

1957 (1)

1947 (1)

1913 (1)

W. W. Coblentz, “The diffuse reflectance power of various substances,” Bull. Bur. Stand. (V. S.) Bull. 9, 283–325 (1913).
[CrossRef]

1899 (1)

F. Paschen, “Ubër die Vertheilung der Energie im Spectrum des schwarzen Körpers bei nederen temperaturen,” Berl. Akad. Wiss. 22, 405–420 (1899).

Bennett, J. M.

J. M. Bennett, L. Mattson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C.1989).

Bradac, F. J.

Brandenberg, W. M.

Clarke, F. J. J.

F. J. J. Clarke, D. J. Parry, “Helmholtz reciprocity: its validity and application to reflectometry,” Lighting Res. Technol. 17, 1–11 (1985).
[CrossRef]

F. J. J. Clarke, “Measurement of the radiometric properties of materials for building and aerospace applications,” Proc. SPIE 234, 40–47 (1980).
[CrossRef]

Coblentz, W. W.

W. W. Coblentz, “The diffuse reflectance power of various substances,” Bull. Bur. Stand. (V. S.) Bull. 9, 283–325 (1913).
[CrossRef]

de Haseth, J. A.

P. R. Griffiths, J. A. de Haseth, Fourier Transform Infrared Spectroscopy (Wiley, New York, 1986).

Derksen, W. L.

Dunkle, R. V.

R. V. Dunkle, “Spectral reflectance measurements,” in Surface Effects on Spacecraft Materials, F. J. Clauss, ed. (Wiley, New York, 1960), p. 117.

Dunn, S. T.

S. T. Dunn, J. C. Richmond, J. A. Wiebelt, “Ellipsoidal mirror reflectometer,” J. Res. Natl. Bur. Stand. Sec. C 70, 75–88 (1966).
[CrossRef]

Edwards, D. K.

Erb, W.

Griffiths, P. R.

P. R. Griffiths, J. A. de Haseth, Fourier Transform Infrared Spectroscopy (Wiley, New York, 1986).

Hanssen, L. M.

K. A. Snail, L. M. Hanssen, “Reflectance measurements using conic mirror reflectometers,” in Applied Spectroscopy: A Compact Reference for Practitioners, J. Workman, A. Springsteen, eds. (Academic, San Diego, Calif., 1998).

Heinisch, R. P.

Lawes, A. J.

Mattson, L.

J. M. Bennett, L. Mattson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C.1989).

Monahan, T. I.

Neu, J. T.

J. T. Neu, “Design, fabrication and performance of an ellipsoidal spectroreflectometer,” Rep. NASA-CR-73193 (National Aeronautics and Space Administration, Washington, D.C., 1968).

Oppenheim, U. P.

U. P. Oppenheim, M. G. Turner, W. L. Wolfe, “BRDF reference standards for the infrared,” Infrared Phys. Technol. 35, 873–879 (1994).
[CrossRef]

Parry, D. J.

F. J. J. Clarke, D. J. Parry, “Helmholtz reciprocity: its validity and application to reflectometry,” Lighting Res. Technol. 17, 1–11 (1985).
[CrossRef]

Paschen, F.

F. Paschen, “Ubër die Vertheilung der Energie im Spectrum des schwarzen Körpers bei nederen temperaturen,” Berl. Akad. Wiss. 22, 405–420 (1899).

Perlick, D. B.

Pipes, J. G.

Richmond, J. C.

S. T. Dunn, J. C. Richmond, J. A. Wiebelt, “Ellipsoidal mirror reflectometer,” J. Res. Natl. Bur. Stand. Sec. C 70, 75–88 (1966).
[CrossRef]

Richter, W.

Roux, J. A.

Sanderson, J. A.

Smith, A. M.

Snail, K. A.

K. A. Snail, “Reflectometer design using nonimaging optics,” Appl. Opt. 26, 5326–5332 (1987).
[CrossRef] [PubMed]

K. A. Snail, L. M. Hanssen, “Reflectance measurements using conic mirror reflectometers,” in Applied Spectroscopy: A Compact Reference for Practitioners, J. Workman, A. Springsteen, eds. (Academic, San Diego, Calif., 1998).

Turner, M. G.

U. P. Oppenheim, M. G. Turner, W. L. Wolfe, “BRDF reference standards for the infrared,” Infrared Phys. Technol. 35, 873–879 (1994).
[CrossRef]

Wiebelt, J. A.

S. T. Dunn, J. C. Richmond, J. A. Wiebelt, “Ellipsoidal mirror reflectometer,” J. Res. Natl. Bur. Stand. Sec. C 70, 75–88 (1966).
[CrossRef]

Willey, R. R.

Wolfe, W. L.

U. P. Oppenheim, M. G. Turner, W. L. Wolfe, “BRDF reference standards for the infrared,” Infrared Phys. Technol. 35, 873–879 (1994).
[CrossRef]

Wood, B. E.

Appl. Opt. (5)

Appl. Spectrosc. (2)

Berl. Akad. Wiss. (1)

F. Paschen, “Ubër die Vertheilung der Energie im Spectrum des schwarzen Körpers bei nederen temperaturen,” Berl. Akad. Wiss. 22, 405–420 (1899).

Bull. Bur. Stand. (V. S.) Bull. (1)

W. W. Coblentz, “The diffuse reflectance power of various substances,” Bull. Bur. Stand. (V. S.) Bull. 9, 283–325 (1913).
[CrossRef]

Infrared Phys. Technol. (1)

U. P. Oppenheim, M. G. Turner, W. L. Wolfe, “BRDF reference standards for the infrared,” Infrared Phys. Technol. 35, 873–879 (1994).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Res. Natl. Bur. Stand. Sec. C (1)

S. T. Dunn, J. C. Richmond, J. A. Wiebelt, “Ellipsoidal mirror reflectometer,” J. Res. Natl. Bur. Stand. Sec. C 70, 75–88 (1966).
[CrossRef]

Lighting Res. Technol. (1)

F. J. J. Clarke, D. J. Parry, “Helmholtz reciprocity: its validity and application to reflectometry,” Lighting Res. Technol. 17, 1–11 (1985).
[CrossRef]

Proc. SPIE (1)

F. J. J. Clarke, “Measurement of the radiometric properties of materials for building and aerospace applications,” Proc. SPIE 234, 40–47 (1980).
[CrossRef]

Other (6)

R. V. Dunkle, “Spectral reflectance measurements,” in Surface Effects on Spacecraft Materials, F. J. Clauss, ed. (Wiley, New York, 1960), p. 117.

J. T. Neu, “Design, fabrication and performance of an ellipsoidal spectroreflectometer,” Rep. NASA-CR-73193 (National Aeronautics and Space Administration, Washington, D.C., 1968).

K. A. Snail, L. M. Hanssen, “Reflectance measurements using conic mirror reflectometers,” in Applied Spectroscopy: A Compact Reference for Practitioners, J. Workman, A. Springsteen, eds. (Academic, San Diego, Calif., 1998).

J. M. Bennett, L. Mattson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C.1989).

P. R. Griffiths, J. A. de Haseth, Fourier Transform Infrared Spectroscopy (Wiley, New York, 1986).

The ray tracing was performed with the Advanced Stray Light Analysis Package (ASAP) from Breault Research Organization, Tucson, Ariz. Maximum magnification runs used rays traced in double precision at an angle of 89.9°.

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Figures (5)

Fig. 1
Fig. 1

Geometry of a typical hemispheroidal mirror reflectometer.

Fig. 2
Fig. 2

Geometry of a typical dual-paraboloidal mirror reflectometer.

Fig. 3
Fig. 3

Maximum linear magnification for hemispherical, hemiellipsoidal, and dual-paraboloidal mirrors for the geometries discussed in the text and as calculated from Eqs. (2), (4), and (5). For the hemisphere (hemiellipsoid) the ratio of the beam spot radius to the hemisphere radius (hemiellipsoid semimajor axis length) was set at 0.05.

Fig. 4
Fig. 4

Geometrical arrangement of a typical hemispherical mirror.

Fig. 5
Fig. 5

Imaging of rays in a dual-paraboloidal mirror system with paraboloids of identical focal length (f = f s = f d ) and coincident foci. Points at (x 2, y 2, z 2) and (x 3, y 3, z 3) are reflection points on the two paraboloids, whereas points (x 1, y 1) and (x 4, y 4) are the starting and end points, respectively.

Equations (39)

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M y , HS 1 1 - 2 ε 2 .
M y , HS = 2 ε 2 δ - 2 δ - 1 2 δ - 1 2 - 4 ε 2 ,
M y , HE 1 + ε 1 - ε 2 .
M y , HE = 1 - ε 2 1 - ε 2 - 2 δ 1 - ε ,
M y , DP = 1 1 - 2 ε .
s ˆ 2 = s ˆ 1 - 2 s ˆ 1 · n ˆ n ˆ ,
x 3 = x 2 + d 23 x 2 - x 1 d 12 + 2 s ˆ 1 · n ˆ x 2 R ,
y 3 = y 2 + d 23 y 2 - y 1 d 12 + 2 s ˆ 1 · n ˆ y 2 R ,
0 = z 2 + d 23 z 2 d 12 + 2 s ˆ 1 · n ˆ z 2 R ,
x 3 = x 1 R 2 2 x 2 x 1 + y 2 y 1 - R 2 ,
y 3 = y 1 R 2 2 x 2 x 1 + y 2 y 1 - R 2 .
s ˆ 1 = cos α x ˆ + cos β y ˆ + cos γ z ˆ ,
x - x 0 s x = y - y 0 s y z - z 0 s z ,
x - x 1 cos α = z cos π / 2 - α .
x 2 - x 1 cos α = z 2 sin α ,
z 2 = f - x 2 2 4 f .
s ˆ 2 = s ˆ 1 - 2 s ˆ 1 · n ˆ n ˆ ,
n ˆ = - z x   x ˆ + z ˆ z x 2 + 1 1 / 2
n ˆ = x 2 2 f   x ˆ + z ˆ x 2 2 f 2 + 1 1 / 2
s ˆ 1 · n ˆ = cos α x 2 2 f + cos π / 2 - α x 2 2 f 2 + 1 1 / 2 .
s 2 x = cos α - x 2 f cos α x 2 2 f + cos π / 2 - α x 2 2 f 2 + 1 ,
s 2 y = 0 ,
s 2 z = cos π / 2 - α - 2 cos α x 2 2 f + cos π / 2 - α x 2 2 f 2 + 1 .
x - x 2 s 2 x = z - z 2 s 2 z .
x 3 - x 2 s 2 x = z 3 - z 2 s 2 z ,
z 3 = x 3 2 4 f - f .
s ˆ 3 = s ˆ 2 - 2 s ˆ 2 · n ˆ n ˆ ,
n ˆ = x 3 2 f   x ˆ - z ˆ x 3 2 f 2 + 1 1 / 2 ,
s ˆ 2 · n ˆ = s 2 x x 3 2 f - s 2 z x 3 2 f 2 + 1 1 / 2 ,
s 3 x = s 2 x - x 3 f s 2 x x 3 2 f - s 2 z x 3 2 f 2 + 1 ,
s 3 y = 0 ,
s 3 z = s 2 z + 2 s 2 x x 3 2 f - s 2 z x 3 2 f 2 + 1 .
x 4 - x 3 s 3 x = - z 3 s 3 z .
x 4 = x 3 - z 3 s 3 x s 3 z ,
y 4 = 0 .
N x 4 = - x 1 4 α 4 + order α 5 ,
D x 4 = x 1 3 f 1 - x 1 α 4 / f + order α 5 .
M y , DP = - x 4 x 1 = - N / D x 1 = 1 x 1 x 1 4 + order α x 1 3 f - x 1 / f + order α α = 0 1 1 - x 1 / f ,
M y , DP = 1 1 - 2 ε .

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