Abstract

The purpose of this paper is to derive and summarize the focusing properties of hemispherical and ellipsoidal collector mirrors, which are often used in measuring the reflectance of imperfectly diffuse surfaces. Analytical expressions for the spherical aberration and magnification of a hemisphere and the magnification of an ellipsoid are derived. The resulting equations are applied to a rectangular and a circular source placed at one of the foci of the collector mirrors.

© 1964 Optical Society of America

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References

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  1. F. Paschen, Ber. Berlin Akad. Deut. Wiss. 27 (1899).
  2. T. Royds, Physik Z. 11, 316 (1910).
  3. W. W. Coblentz, Natl. Bur. Std. (U.S.) Bull. 9, 283 (1913).

1913 (1)

W. W. Coblentz, Natl. Bur. Std. (U.S.) Bull. 9, 283 (1913).

1910 (1)

T. Royds, Physik Z. 11, 316 (1910).

1899 (1)

F. Paschen, Ber. Berlin Akad. Deut. Wiss. 27 (1899).

Coblentz, W. W.

W. W. Coblentz, Natl. Bur. Std. (U.S.) Bull. 9, 283 (1913).

Paschen, F.

F. Paschen, Ber. Berlin Akad. Deut. Wiss. 27 (1899).

Royds, T.

T. Royds, Physik Z. 11, 316 (1910).

Ber. Berlin Akad. Deut. Wiss. (1)

F. Paschen, Ber. Berlin Akad. Deut. Wiss. 27 (1899).

Natl. Bur. Std. (U.S.) Bull. (1)

W. W. Coblentz, Natl. Bur. Std. (U.S.) Bull. 9, 283 (1913).

Physik Z. (1)

T. Royds, Physik Z. 11, 316 (1910).

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Figures (2)

F. 1
F. 1

Geometrical diagram of collector surface, source, and image.

F. 2
F. 2

Images produced by an ellipsoid (a=10.2 cm, b=10 cm) and a hemisphere (R=10cm). (A) rectangular source, (B) circular source.

Equations (19)

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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 / [ 2 ( y y 1 + x x 1 ) R 2 ]
y 2 = y 1 R 2 / [ 2 ( y y 1 + x x 1 ) R 2 ] .
m x = d x 2 / d x 1 = R 2 / ( R 2 + 2 y f )
m y = d y 2 / d y 1 = R 4 / ( R 2 + 2 y f ) 2 = m x 2 .
δ y = y 2 f = 2 y f 2 / ( R 2 + 2 y f ) .
| δ y = R | + | δ y = + R | = 4 R f 2 / ( R 2 4 f 2 ) .
[ ( X 2 + Z 2 ) / b 2 ] + ( Y 2 / a 2 ) = 1 , f 2 = a 2 b 2 ,
X = x , Y = y + , Z = z ,
Z / Y = ( Y ) / Z = ( Y / Z ) ( b 2 / a 2 ) ,
X 2 = X 1 ( a 4 f 2 Y 2 ) 2 a 2 [ Y Y 1 + ( a 2 / b 2 ) X X 1 ] ( a 4 + f 2 Y 2 ) ,
Y 2 = Y 1 ( a 4 + f 2 Y 2 ) + 2 a 2 f 2 Y [ ( X X 1 / b 2 ) 1 ] 2 a 2 [ Y Y 1 + ( a 2 / b 2 ) X X 1 ] ( a 4 + f 2 Y 2 ) .
M x = d X 2 / d X 1 ( a 2 f y ) / ( a 2 + f y ) ,
M y = d Y 2 / d Y 1 = [ ( a 2 f Y ) / ( a 2 + f Y ) ] 2 = M x 2 .