Abstract

Any image-forming optical system is usually tested for aberrations by looking at the image plane. A point source is placed at some distance from the system (this may be a very large or small distance depending on the particular system) and its image by the optical system is studied by any of many methods available. The present paper presents the theory of a method in which a grating containing a series of concentric circles is placed at the image of the point source. Characteristic patterns are derived for the usual aberrations of optical systems and some typical photographs of these patterns are presented. Finally, this method is compared to the Ronchi test in which a series of straight lines are used for the grating.

© 1966 Optical Society of America

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References

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  1. V. Ronchi, Atti Fond. G. Ronchi 17, 93 (1962).
  2. V. Ronchi, J. Opt. Soc. Am. 3, 437 (1964).

1964

V. Ronchi, J. Opt. Soc. Am. 3, 437 (1964).

1962

V. Ronchi, Atti Fond. G. Ronchi 17, 93 (1962).

Ronchi, V.

V. Ronchi, J. Opt. Soc. Am. 3, 437 (1964).

V. Ronchi, Atti Fond. G. Ronchi 17, 93 (1962).

Atti Fond. G. Ronchi

V. Ronchi, Atti Fond. G. Ronchi 17, 93 (1962).

J. Opt. Soc. Am.

V. Ronchi, J. Opt. Soc. Am. 3, 437 (1964).

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram showing the relative positions of the exit pupil and the concentric circular grid.

Fig. 2
Fig. 2

Equally spaced concentric circles obtained with a good lens when the grid is placed outside the Gaussian focal plane.

Fig. 3
Fig. 3

Unequally spaced concentric circles obtained with primary spherical aberration.

Fig. 4
Fig. 4

Characteristic fringes obtained when the center of the grid is not on the axis of the lens having primary spherical aberration.

Fig. 5
Fig. 5

Characteristic fringes obtained with primary coma. The center of the grid is displaced in the meridional direction.

Fig. 6
Fig. 6

Characteristic fringes obtained with primary coma when the center of the grid is displaced laterally in the sagittal direction.

Fig. 7
Fig. 7

Characteristic fringes obtained with primary astigmatism. The center of the grid is at one of the foci and thus straight fringes are obtained.

Fig. 8
Fig. 8

Elliptical fringes obtained with primary astigmatism when the center of the grid is between the foci but not at the mid-point of the line joining foci.

Equations (13)

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T A x = x ¯ = R W x ;             T A y = y ¯ = R W y .
( x ¯ - ξ ) 2 + ( y ¯ - η ) 2 = n 2 ρ 1 2
( x ¯ - ξ ) 2 + ( y ¯ - η ) 2 = n ρ 1 2
( R W x - ξ ) 2 + ( R W y - η ) 2 = n 2 ρ 1 2 .
W ( x , y ) = B ( x 2 + y 2 ) 2 + z ¯ 2 R 2 ( x 2 + y 2 ) ,
R W x = 4 B R x ( x 2 + y 2 ) + z ¯ x R R W y = 4 B R y ( x 2 + y 2 ) + z ¯ y R } .
R W x = 8 B 0 F # u ( u 2 + v 2 ) + z ¯ 2 F # u R W y = 8 B 0 F # v ( u 2 + v 2 ) + z ¯ 2 F # v } ,
( 8 B 0 F # ) 2 ( u 2 + v 2 ) 3 + 8 B 0 z ¯ ( u 2 + v 2 ) 2 - 16 B 0 F # ( u 2 + v 2 ) ( ξ u + η v ) + ( z ¯ 2 F # ) 2 ( u 2 + v 2 ) - z ¯ F # ( ξ u + η v ) + ( ξ 2 + η 2 ) = n 2 ρ 1 2 } .
W ( x , y ) = F y ( x 2 + y 2 ) + z ¯ 2 R 2 ( x 2 + y 2 ) .
( 2 F 0 F # ) 2 ( u 4 + 10 u 2 v 2 + 9 v 4 ) + 6 F 0 z ¯ ( u 2 + v 2 ) v - 4 F 0 F # { 2 ξ u v + η ( u 2 + 3 v 2 ) } + ( z ¯ 2 F # ) 2 ( u 2 + v 2 ) - z ¯ F # ( ξ u + η v ) + ( ξ 2 + η 2 ) = n 2 ρ 1 2 } ,
W ( x , y ) = C ( x 2 - y 2 ) + z ¯ 2 R 2 ( x 2 + y 2 ) .
{ ( 4 C 0 F # ) 2 + ( z ¯ 2 F # ) 2 } ( u 2 + v 2 ) + 4 C 0 z ¯ ( u 2 - v 2 ) - 8 C 0 F # ( ξ u - η v ) - z ¯ F # ( ξ u + η v ) + ( ξ 2 + η 2 ) = n 2 ρ 1 2 } ,
R W x cos α + R W y sin α = n d ,

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