Abstract

Differential equations for a constant-angular-magnification aspheric-mirror surface shape are derived for a general dependence of the camera image height on the camera field angle. The explicit equations of the constant-angular-magnification mirror surface are given for some particular values of the angular magnification. For a series of odd integer values of the angular magnification, 10th-order polynomial approximations of the mirror surface are presented. The imaging performance of such a mirror with an angular magnification of 7 is analyzed and compared with a spherical mirror. The main cause of image blur in all-sky cameras at the edge of the field of view was found to be a strong image curvature. We show that increasing the camera-to-mirror distance and/or stopping down the camera lens reduces the image blur.

© 1994 Optical Society of America

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References

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  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. A. Dyer, “Seeking the best 35 mm camera,” Astronomy 21, 74–79 (1993).

1993 (1)

A. Dyer, “Seeking the best 35 mm camera,” Astronomy 21, 74–79 (1993).

1992 (1)

Andreic, Ž.

N. Radić, Ž. Andreić, “Aspheric mirror with constant angular magnification,” Appl. Opt. 31, 5915–5917 (1992).
[CrossRef]

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).

Dyer, A.

A. Dyer, “Seeking the best 35 mm camera,” Astronomy 21, 74–79 (1993).

Radic, N.

Appl. Opt. (1)

Astronomy (1)

A. Dyer, “Seeking the best 35 mm camera,” Astronomy 21, 74–79 (1993).

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).

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Figures (7)

Fig. 1
Fig. 1

All-sky camera imaging the complete celestial hemisphere above the horizon. During the imaging process the observer’s spherical coordinate system is transformed into a plane polar coordinate system. In the imaging process the azimuth AZ is left intact and is equal to the polar angle ϕ, but the zenith distance ZD is transformed into radius vector ρ. The idea behind this research is to produce a camera that would satisfy ZD = constant × ρ, which requires use of an aspheric mirror whose properties are discussed below.

Fig. 2
Fig. 2

Geometry of an all-sky camera. An ordinary photographic camera looks down onto a convex mirror and photographs the virtual image of the sky produced by the mirror. All signs are positive as shown.

Fig. 3
Fig. 3

Spot diagrams of a spherical mirror with paraxial region M = 7. The y-axis parameter is an object angle (in degrees), and the x-axis parameter is defocused (in millimeters) from the spherical image surface (R = 330.3 mm).

Fig. 4
Fig. 4

Spot diagrams of a constant-angular-magnification aspheric mirror with M = 7. The y-axis parameter is an object angle (in degrees), and the x-axis parameter is defocused (in millimeters) from the spherical image surface (R = 330.3 mm).

Fig. 5
Fig. 5

Spot diagrams of a parabolic mirror with paraxial region M = 7. The y-axis parameter is an object angle (in degrees), and the x-axis parameter is defocused (in millimeters) from the spherical image surface (R = 330.3 mm).

Fig. 6
Fig. 6

Shape of the surface of 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 was M = 7. S, spherical mirror; P, parabolic mirror; A, aspheric mirror.

Fig. 7
Fig. 7

Depth of field of a 50-mm focal-length camera lens as a function of the camera-to-mirror distance and the lens aperture (solid curves). A blur circle of 50 μm was used in the calculations. The depth of field required for an all-sky camera is indicated by the dashed curves for a spherical (S), parabolic (P), and aspheric (A) mirror (M = 7).

Equations (16)

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h = g ( α ) ,
h = F tan ( α ) ,
h = 1 M β
d y d x = - tan [ M g ( α ) - α 2 ]
α = - arctan ( x y - d ) .
d y d x = - tan [ M - 1 2 arctan ( x d - y ) ]
d y d x = - tan [ M x 2 ( d - y ) - 1 2 arctan ( x d - y ) ] .
x ( α ) = d sin ( α ) { cos [ ( N + 1 ) α ] } 1 / ( N + 1 ) , y ( α ) = d - d cos ( α ) { cos [ ( N + 1 ) α ] } 1 / ( N + 1 ) ,
T N + 1 { 1 - Y [ X 2 + ( 1 - Y ) 2 ] 1 / 2 } = 1 [ X 2 + ( 1 - Y ) 2 ] ( N + 1 ) / 2 .
M = 3 ,             Y = 1 - ( 1 + X 2 ) 1 / 2 ,
M = 7 ,             Y = 1 - [ ( 1 + 8 X 4 ) 1 / 2 + 3 X 2 ] 1 / 2 .
M = 3 ,             Y = - X 2 ( 1 2 - X 2 8 + X 4 16 - 5 X 6 128 + 35 X 8 1286 ) + O ( X 12 ) ,
M = 5 ,             Y = - X 2 ( 1 - X 4 3 + X 6 3 ) + O ( X 12 ) ,
M = 7 ,             Y = - X 2 ( 3 2 + 7 X 2 8 - 19 X 4 16 - 203 X 6 128 + 3381 X 8 256 ) + O ( X 12 ) ,
M = 9 ,             Y = - X 2 ( 2 + 3 X 2 - 21 X 6 - 212 X 8 5 ) + O ( X 12 ) ,
M = 11 ,             Y = - X 2 ( 5 2 + 55 X 2 8 + 583 X 4 48 + 158977 X 6 384 ) + O ( X 10 ) ,

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