Abstract

The optical test for the flatness of a plane mirror in conjunction with a concave spherical mirror has been analyzed in terms of the hyperboloid to which it can be thought to be referenced. This geometrical approach renders obvious the requirement to make two tests, the second after rotating the plane mirror 90° about its normal. The expression for the separation of the astigmatic foci for a long radius of curvature, nearly flat, mirror is in agreement with previous derivations.

© 1970 Optical Society of America

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References

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  1. A. A. Common, Roy. Astron. Soc. Monthly Not. 48, 105 (1888).
  2. G. W. Ritchey, Smithsonian Contributions to Knowledge No. 1459 (1904).
  3. J. M. Pierce, in Amateur Telescope Making, A. G. Ingalls, Ed. (Scientific American, Inc., New York, 1962), Book 1, Part 10, Chap. 3.
  4. A. Danjon, A. Couder, Lunette et Télescopes (Editions de la revue d’optique theorique et instrumentale, Paris, 1935), pp. 500–501.
  5. D. D. Maksutow, Technologie der Astronomischen Optik (Veb Verlag Technike, Berlin, 1954), pp. 197–199.
  6. J. H. Hindle, Ref. 3, pp. 225–228.

1888 (1)

A. A. Common, Roy. Astron. Soc. Monthly Not. 48, 105 (1888).

Common, A. A.

A. A. Common, Roy. Astron. Soc. Monthly Not. 48, 105 (1888).

Couder, A.

A. Danjon, A. Couder, Lunette et Télescopes (Editions de la revue d’optique theorique et instrumentale, Paris, 1935), pp. 500–501.

Danjon, A.

A. Danjon, A. Couder, Lunette et Télescopes (Editions de la revue d’optique theorique et instrumentale, Paris, 1935), pp. 500–501.

Hindle, J. H.

J. H. Hindle, Ref. 3, pp. 225–228.

Maksutow, D. D.

D. D. Maksutow, Technologie der Astronomischen Optik (Veb Verlag Technike, Berlin, 1954), pp. 197–199.

Pierce, J. M.

J. M. Pierce, in Amateur Telescope Making, A. G. Ingalls, Ed. (Scientific American, Inc., New York, 1962), Book 1, Part 10, Chap. 3.

Ritchey, G. W.

G. W. Ritchey, Smithsonian Contributions to Knowledge No. 1459 (1904).

Roy. Astron. Soc. Monthly Not. (1)

A. A. Common, Roy. Astron. Soc. Monthly Not. 48, 105 (1888).

Other (5)

G. W. Ritchey, Smithsonian Contributions to Knowledge No. 1459 (1904).

J. M. Pierce, in Amateur Telescope Making, A. G. Ingalls, Ed. (Scientific American, Inc., New York, 1962), Book 1, Part 10, Chap. 3.

A. Danjon, A. Couder, Lunette et Télescopes (Editions de la revue d’optique theorique et instrumentale, Paris, 1935), pp. 500–501.

D. D. Maksutow, Technologie der Astronomischen Optik (Veb Verlag Technike, Berlin, 1954), pp. 197–199.

J. H. Hindle, Ref. 3, pp. 225–228.

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Figures (7)

Fig. 1
Fig. 1

Geometry of Common test. M—mirror tested; S—spherical mirror, center at F; F—autostigmatic test point; α—angle of incidence at center of M.

Fig. 2
Fig. 2

(a) Test geometry for M concave or convex. (b) Hindle test, α = 0.

Fig. 3
Fig. 3

Analytical geometry axes and coordinates.

Fig. 4
Fig. 4

Parameters of tested mirror.

Fig. 5
Fig. 5

Hyperboloid parameters a, e derived from computed RT and measured FA, α.

Fig. 6
Fig. 6

Contours α = constant on tested mirror.

Fig. 7
Fig. 7

Geometry for obtaining separation, ΔP, of astigmatic images in terms of coordinate transformation x, y to x, y.

Equations (22)

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( x 2 / a 2 ) [ ( y 2 + z 2 ) / a 2 ( e 2 1 ) ] = 1 ,
R T 2 = e 2 y 2 + R 0 2 , R M = ( 1 + y ) 3 2 / y = R T 3 / R 0 2 , R 0 = a ( e 2 1 ) ,
R M R T = ( R T R 0 ) 2 = 1 + e 2 ( e 2 1 ) 2 y 2 a 2 .
a 0 , e ,
R M / R T = ˙ 1 + ( y 2 / e 2 a 2 ) = ˙ sec 2 α
C M = C T cos 2 α .
δ 1 = ( r 1 2 / 2 ) ( C 1 C M 1 ) = S 1 S M 1 for the meridional plane , δ 2 = ( r 2 2 / 2 ) ( C 2 C T 1 ) = S 2 S T 1 for the transverse plane ,
S M 1 = ( r 1 2 / 2 ) C M 1 = [ ( r 1 cos α ) 2 / 2 ] C T 1 = ( r 1 cos α / r 2 ) 2 S T 1 ,
S 1 ( r 1 cos α / r 2 ) 2 S 2 = [ δ 1 δ 2 ( r 1 cos α / r 2 ) 2 ] .
δ 3 = ( r 2 2 / 2 ) ( C 2 C M 2 ) = S 2 S M 2 for the meridional plane , δ 4 = ( r 1 2 / 2 ) ( C 1 C T 2 ) = S 1 S T 2 for the transverse plane .
S M 2 = ( r 2 cos α / r 1 ) 2 S T 2 ,
( r 2 cos α / r 1 ) 2 S 1 S 2 = [ δ 3 + ( r 2 cos α / r 1 ) 2 δ 4 ] .
Δ W = 4 δ cos α = ( Δ m ) λ,
δ = ( λ / 4 cos α ) Δ m .
K 12 = ( λ / 4 cos α ) [ Δ m 1 ( r 1 cos α / r 2 ) 2 Δ m 2 ] , K 34 = ( λ / 4 cos α ) [ Δ m 3 + ( r 2 cos α / r 1 ) 2 Δ m 4 ] ,
( 1 cos 4 α ) S 1 = K 12 ( r 1 cos α r 2 ) 2 K 34 = λ 4 cos α × { Δ m 1 ( r 1 cos α r 2 ) 2 ( Δ m 2 Δ m 3 ) ( cos 4 α ) Δ m 4 } , ( 1 cos 4 α ) S 2 = ( r 2 cos α r 1 ) 2 K 12 K 34 = λ 4 cos α { Δ m 3 ( r 2 cos α r 1 ) 2 ( Δ m 4 Δ m 1 ) ( cos 4 α ) Δ m 2 } .
Δ P = 2 F 2 C 0 sin 2 α / cos 3 α ( see Appendix ) ,
C T = C 0 sec 2 α or Δ C T = C 0 tan 2 α .
C T = 1 / e F sec α = x / F 2 sec 2 α ,
Δ C T / C T = Δ x / x 2 ( Δ F / F ) .
Δ x = ( Δ P / 2 ) sec α , Δ F = ( Δ P / 2 ) cos α ,
Δ C T C T = Δ P 2 ( sec α x + 2 cos α F ) = ˙ Δ P 2 sec α x , F x , Δ P = 2 Δ C T ( x / C T sec α ) = 2 F 2 C 0 tan 2 α sec α .

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