Abstract

The characteristics of an all-sky camera with a concave mirror are analyzed. A differential equation for a concave aspheric mirror with constant angular magnification is derived for the general dependence of the camera image height on the camera field angle. This equation is solved in parametric form for the case of a concave mirror with a constant angular magnification. The explicit equations for the shape of the aspheric mirror are given for some particular values of the angular magnification. Parametric equations of the surface shape for sevenfold angular magnification are developed into a power series that is used to analyze the imaging performance of such a mirror. The performance of the concave aspheric mirror is compared with that of a spherical mirror. The minimal camera-to-mirror distance is determined as a function of the blur allowed and the camera lens aperture. Some characteristics of convex mirrors are also presented for comparison.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Ž Andreić, “Simple 180° field of view F-theta all-sky camera,” in Innovative Optics and Phase Conjugate Optics, R. Ahlers, T. T. Tschudi, eds. Proc. Soc. Photo-Opt. Instrum. Eng.1500, 293–304 (1991).
  2. N. Radić, Ž Andreić, “Aspheric mirror with constant angular magnification,” Appl. Opt. 31, 5915–5917 (1992).
    [CrossRef]
  3. Ž Andreić, N. Radić, “Aspheric mirror with constant angular magnification II,” Appl. Opt. 33, 4179–4183 (1994).
    [CrossRef]
  4. C. Schur, “Experiments with all-sky photography,” Sky Telesc. 63, 621–624 (1982).

1994 (1)

1992 (1)

1982 (1)

C. Schur, “Experiments with all-sky photography,” Sky Telesc. 63, 621–624 (1982).

Appl. Opt. (2)

Sky Telesc. (1)

C. Schur, “Experiments with all-sky photography,” Sky Telesc. 63, 621–624 (1982).

Other (1)

Ž Andreić, “Simple 180° field of view F-theta all-sky camera,” in Innovative Optics and Phase Conjugate Optics, R. Ahlers, T. T. Tschudi, eds. Proc. Soc. Photo-Opt. Instrum. Eng.1500, 293–304 (1991).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Geometry of an all-sky camera with a concave mirror. Symbols are explained in the text.

Fig. 2
Fig. 2

Distortion of a concave spherical mirror with a paraxial magnification of m = 7 compared with the distortion in a convex spherical mirror with the same m.

Fig. 3
Fig. 3

Shape of the surface containing the circles of least confusion determined from spot diagrams. It can be seen that the required depth of field of the camera is larger for aspheric mirrors. The camera-to-mirror distance was 1000 mm in this case and the angular magnification M = 7.

Fig. 4
Fig. 4

Minimal camera-to-mirror distance in an all-sky camera with a concave aspheric mirror (M = 7) necessary to produce the blur of the star images smaller than the tolerance over the whole 180° FOV. The lens aperture is a curve parameter.

Fig. 5
Fig. 5

Minimal camera-to-mirror distance in an all-sky camera with a concave spherical mirror (M = 7) necessary to produce the blur of the star images smaller than the tolerance over the whole 180° FOV. The lens aperture is a curve parameter.

Fig. 6
Fig. 6

Minimal camera-to-mirror distance in an all-sky camera with a convex aspheric mirror (M = 7) required to produce the blur of the star images smaller than the tolerance over the whole 180° FOV. The lens aperture is a curve parameter.

Fig. 7
Fig. 7

Minimal camera-to-mirror distance in an all-sky camera with a convex spherical mirror (M = 7) necessary to produce the blur of the star images smaller than the tolerance over the whole 180° FOV. The lens aperture is a curve parameter.

Tables (1)

Tables Icon

Table 1 Spot Sizes Observed in a Model of the All-Sky Camera with a Convex Aspheric Mirror a

Equations (31)

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

h = 1 M β ,
h = g ( α ) ,
g ( α ) = 1 M β .
g ( α ) = F tan ( α ) ,
g ( α ) F α
β = 2 δ + α
δ = φ α .
tan ( σ ) = 1 d f ( x ) d x
σ = 90 ° + φ ,
d f ( x ) d x = tan ( δ + α ) .
d f ( x ) d x = tan ( β + α 2 )
d y d x = tan [ M g ( α ) + α 2 ] .
α = arctan ( x d y ) .
d y d x = tan [ M 1 2 arctan ( x d y ) ] .
X = x d ,
Y = y d ,
Z = 1 Y ,
U = X Z ,
N = M 1 2 .
d Z Z = tan [ ( N + 1 ) arctan ( U ) ] 1 + U tan [ ( N + 1 ) arctan ( U ) ] d U
x ( α ) = d sin ( α ) [ cos ( M 1 2 α ) ] 2 / ( M 1 ) ,
y ( α ) = d d cos ( α ) [ cos ( M 1 2 α ) ] 2 / ( M 1 ) ,
T N { ( 1 Y ) [ X 2 + ( 1 Y ) 2 ] 1 / 2 } = [ X 2 + ( 1 Y ) 2 ] N / 2 ,
M = 3 , Y = 1 1 + ( 1 4 X 2 ) 1 / 2 2 ,
M = 5 , Y = 1 [ 1 2 X 2 + ( 1 8 X 2 ) 1 / 2 2 ] 1 / 2 .
M = 7 , Y = 2 X 2 + 7 X 4 + 140 3 X 6 + 4187 15 X 8 + O ( X 10 ) ,
R = 2 d m + 1 ,
R = 2 d m 1 ,
β = 2 arcsin ( m 1 2 sin α ) + α ,
β = 2 arcsin ( m + 1 2 sin α ) α .
DIST = β m α m α .

Metrics