Abstract

The requirements are given for absolute calibration of a spectroradiometer by means of an extended standard source. In some cases the extended source may be used without auxiliary optics, but in many cases a concave mirror is necessary or desirable. For such a mirror, a relation between incident and reflected steradiancy is derived. Brandenberg’s focusing relations for a concave spherical mirror are extended and applied to four practical cases, and it is shown that no loss in accuracy of calibration will result from the use of such mirrors provided that the source and the mirror are large enough. Experimental verification is given.

© 1965 Optical Society of America

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References

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  1. R. W. Patch, J. Quant. Spectry. Radiative Transfer (to be published).
  2. R. E. Danielson, J. E. Gaustad, M. Schwarzschild, H. F. Weaver, and N. J. Woolf, Astron. J. 69, 344 (1964).
    [Crossref]
  3. M. G. Dreyfus and D. T. Hilleary, Aerospace Engr. 21, 28 (1962).
  4. R. Stair, R. G. Johnston, and E. W. Halbach, J. Res. Natl. Bur. Std. (U.S.) 64A, 291 (1960).
    [Crossref]
  5. R. W. Patch, Ph.D. thesis, California Institute of Technology, Pasadena (1964), pp. 173–175.
  6. W. M. Brandenberg, J. Opt. Soc. Am. 54, 1235 (1964).
    [Crossref]
  7. S. Chandrasekhar, Radiative Transfer (Dover Publications, New York, 1960), p. 9 (Chandrasekhar uses Iν to represent Bν).
  8. H. E. Bennett and W. F. Koehler, J. Opt. Soc. Am. 50, 1 (1960).
    [Crossref]

1964 (2)

R. E. Danielson, J. E. Gaustad, M. Schwarzschild, H. F. Weaver, and N. J. Woolf, Astron. J. 69, 344 (1964).
[Crossref]

W. M. Brandenberg, J. Opt. Soc. Am. 54, 1235 (1964).
[Crossref]

1962 (1)

M. G. Dreyfus and D. T. Hilleary, Aerospace Engr. 21, 28 (1962).

1960 (2)

R. Stair, R. G. Johnston, and E. W. Halbach, J. Res. Natl. Bur. Std. (U.S.) 64A, 291 (1960).
[Crossref]

H. E. Bennett and W. F. Koehler, J. Opt. Soc. Am. 50, 1 (1960).
[Crossref]

Bennett, H. E.

Brandenberg, W. M.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover Publications, New York, 1960), p. 9 (Chandrasekhar uses Iν to represent Bν).

Danielson, R. E.

R. E. Danielson, J. E. Gaustad, M. Schwarzschild, H. F. Weaver, and N. J. Woolf, Astron. J. 69, 344 (1964).
[Crossref]

Dreyfus, M. G.

M. G. Dreyfus and D. T. Hilleary, Aerospace Engr. 21, 28 (1962).

Gaustad, J. E.

R. E. Danielson, J. E. Gaustad, M. Schwarzschild, H. F. Weaver, and N. J. Woolf, Astron. J. 69, 344 (1964).
[Crossref]

Halbach, E. W.

R. Stair, R. G. Johnston, and E. W. Halbach, J. Res. Natl. Bur. Std. (U.S.) 64A, 291 (1960).
[Crossref]

Hilleary, D. T.

M. G. Dreyfus and D. T. Hilleary, Aerospace Engr. 21, 28 (1962).

Johnston, R. G.

R. Stair, R. G. Johnston, and E. W. Halbach, J. Res. Natl. Bur. Std. (U.S.) 64A, 291 (1960).
[Crossref]

Koehler, W. F.

Patch, R. W.

R. W. Patch, Ph.D. thesis, California Institute of Technology, Pasadena (1964), pp. 173–175.

R. W. Patch, J. Quant. Spectry. Radiative Transfer (to be published).

Schwarzschild, M.

R. E. Danielson, J. E. Gaustad, M. Schwarzschild, H. F. Weaver, and N. J. Woolf, Astron. J. 69, 344 (1964).
[Crossref]

Stair, R.

R. Stair, R. G. Johnston, and E. W. Halbach, J. Res. Natl. Bur. Std. (U.S.) 64A, 291 (1960).
[Crossref]

Weaver, H. F.

R. E. Danielson, J. E. Gaustad, M. Schwarzschild, H. F. Weaver, and N. J. Woolf, Astron. J. 69, 344 (1964).
[Crossref]

Woolf, N. J.

R. E. Danielson, J. E. Gaustad, M. Schwarzschild, H. F. Weaver, and N. J. Woolf, Astron. J. 69, 344 (1964).
[Crossref]

Aerospace Engr. (1)

M. G. Dreyfus and D. T. Hilleary, Aerospace Engr. 21, 28 (1962).

Astron. J. (1)

R. E. Danielson, J. E. Gaustad, M. Schwarzschild, H. F. Weaver, and N. J. Woolf, Astron. J. 69, 344 (1964).
[Crossref]

J. Opt. Soc. Am. (2)

J. Res. Natl. Bur. Std. (U.S.) (1)

R. Stair, R. G. Johnston, and E. W. Halbach, J. Res. Natl. Bur. Std. (U.S.) 64A, 291 (1960).
[Crossref]

Other (3)

R. W. Patch, Ph.D. thesis, California Institute of Technology, Pasadena (1964), pp. 173–175.

R. W. Patch, J. Quant. Spectry. Radiative Transfer (to be published).

S. Chandrasekhar, Radiative Transfer (Dover Publications, New York, 1960), p. 9 (Chandrasekhar uses Iν to represent Bν).

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Figures (4)

Fig. 1
Fig. 1

Various optical arrangements for absolute calibrations of spectroradiometers. Case A corresponds to an entrance slit just inside the spectroradiometer. In case B the spectroradiometer is focused at infinity. The instrument in case C is focused on a plane outside the instrument case; if, for some reason, the standard source cannot be located at this plane, then the arrangement in case D may be necessary.

Fig. 2
Fig. 2

Pencils of light rays incident upon and reflected from a specular reflector of arbitrary curvature. This diagram is used in deriving a relation between the incident spectral steradiancy Bν and the reflected spectral steradiancy Bν′.

Fig. 3
Fig. 3

Diagrams of the hemispherical reflector and coordinates for three different optical arrangements. In case I the source and the image are both in the xy plane. In case II the image is below the xy plane, so that the source is above it. In case IV we consider only rays from the source that produce collimated light with rays parallel to the yz plane and at an angle φ to the z axis.

Fig. 4
Fig. 4

Minimum required sources for four arrangements useful in calibrating spectroradiometers. In cases I, II, and III a desired image was selected, and the required source was calculated. In case IV a desired aperture was selected, and the required source was calculated.

Tables (1)

Tables Icon

Table I Over-all dimensions of minimum required sources.

Equations (24)

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B ν cos i d A d ω d ν .
B ν cos i d A d ω d ν .
B ν cos i d A d ω d ν = r ν B ν cos i d A d ω d ν .
d ω = d ω .
B ν = r ν B ν .
d 1 2 = ( x - x 1 ) 2 + ( y - y 1 ) 2 + z 2 ,
d 2 2 = ( x - x 2 ) 2 + ( y - y 2 ) 2 + z 2 .
r 1 2 = R 2 + d 1 2 - 2 d 1 R cos α ,
r 2 2 = R 2 + d 2 2 - 2 d 2 R cos α .
x 1 / r 1 = - x 2 / r 2 ,
y 1 / r 1 = - y 2 / r 2 ,
R 2 = x 2 + y 2 + z 2 .
x 2 = - x 1 R 2 / [ R 2 - 2 ( x x 1 + y y 1 ) ] ,
y 2 = - y 1 R 2 / [ R 2 - 2 ( x x 1 + y y 1 ) ] ,
( x 3 - x 1 ) / ( x - x 1 ) = z 3 / z ,
( y 3 - y 1 ) / ( y - y 1 ) = z 3 / z ,
( x 4 - x 2 ) / ( x - x 2 ) = z 4 / z ,
( y 4 - y 2 ) / ( y - y 2 ) = z 4 / z .
x 1 = ( x 3 - x z 3 / z ) / ( 1 - z 3 / z ) ,
y 1 = ( y 3 - y z 3 / z ) / ( 1 - z 3 / z ) .
x 4 = x z 4 / z - x 1 R 2 ( 1 - z 4 / z ) / [ R 2 - 2 ( x x 1 + y y 1 ) ] .
y 4 = y z 4 / z - y 1 R 2 ( 1 - z 4 / z ) / [ R 2 - 2 ( x x 1 + y y 1 ) ] .
sin φ = ( y - y 1 ) / [ ( y - y 1 ) 2 + z 2 ] 1 2 .
y = y 1 cos 2 φ + sin φ ( R 2 - x 1 2 - y 1 2 cos 2 φ ) 1 2 .