Abstract

A common mathematical model is established for the Ronchi and Hartmann tests and for interpretation of the Ronchigrams as level curves of the components of the transversal aberrations. With the same point of view, a Hartmanngram is regarded as two 90° crossed null Ronchi gratings. A simple and direct method is also developed for calculating Ronchigrams for the cases of centered and off-axis conic sections with the point light source at any location.

© 1992 Optical Society of America

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References

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  1. A. Cornejo-Rodriguez, “Ronchi test,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), p. 283.
  2. I. Ghozeil, “Hartmann and other screen tests,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), p. 323.
  3. A. A. Sherwood, “A quantitative analysis of the Ronchi test in terms of ray optics,” J. Br. Astron, Assoc. 68, 180–190 (1948).
  4. D. Malacara, “Geometrical Ronchi test for aspherical mirrors,” Appl. Opt. 4, 1371–1374 (1965).
    [CrossRef]
  5. A. Cordero-Davila, A. Cornejo-Rodriguez, O. Cardona-Nunez, “Null Hartmann and Ronchi–Hartmann tests,” Appl. Opt. 29, 4618–4621 (1990).
    [CrossRef] [PubMed]
  6. D. Malacara, A. Cornejo, “Null Ronchi test for aspherical surfaces,” Appl. Opt. 13, 1778–1780 (1974).
    [CrossRef] [PubMed]
  7. G. W. Hopkins, R. M. Shagam, “Null Ronchi gratings from spot diagrams,” Appl. Opt. 16, 2602–2603 (1977).
    [CrossRef] [PubMed]
  8. O. Cardona-Nunez, A. Cornejo-Rodriguez, J. R. Diaz-Uribe, A. Cordero-Davila, J. Pedraza-Contreras, “Conic that best fits an off-axis conic section,” Appl. Opt. 25, 3585–3588 (1986).
    [CrossRef] [PubMed]

1990

1986

1977

1974

1965

1948

A. A. Sherwood, “A quantitative analysis of the Ronchi test in terms of ray optics,” J. Br. Astron, Assoc. 68, 180–190 (1948).

Cardona-Nunez, O.

Cordero-Davila, A.

Cornejo, A.

Cornejo-Rodriguez, A.

Diaz-Uribe, J. R.

Ghozeil, I.

I. Ghozeil, “Hartmann and other screen tests,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), p. 323.

Hopkins, G. W.

Malacara, D.

Pedraza-Contreras, J.

Shagam, R. M.

Sherwood, A. A.

A. A. Sherwood, “A quantitative analysis of the Ronchi test in terms of ray optics,” J. Br. Astron, Assoc. 68, 180–190 (1948).

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Figures (6)

Fig. 1
Fig. 1

Reference planes with corresponding coordinates for the Ronchi and Hartmann tests.

Fig. 2
Fig. 2

Experimental setup for producing a null Ronchi ruling by means of a coarse Ronchi ruling in front of the surface being tested.

Fig. 3
Fig. 3

(a) Null Ronchi ruling obtained with the setup shown in Fig. 2. (b) Null Ronchigram obtained with the null Ronchi ruling shown in (a).

Fig. 4
Fig. 4

Local coordinates of an off-axis section centered at the point (xc′, 0, zc′) of a conic with paraxial curvature c and conic constant K.

Fig. 5
Fig. 5

Simulated Ronchigram for an off-axis parabolic mirror with a 200-cm paraxial curvature radius and 15-cm diameter. The center of the off-axis conic mirror is located 15 cm from the symmetry axis of the conic.

Fig. 6
Fig. 6

Simulated Ronchigrams for two types of mirror: the spherical type is shown on the left side with K = 0, and the parabolic type is shown on the right side with K = −1. Different light source positions with coordinates (α, β) are shown.

Equations (39)

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z = z ( x , y ) .
s ^ 1 = ( x - α , y - β , z - γ ) / [ ( x - α ) 2 + ( y - β ) 2 + ( z - γ ) 2 ] 1 / 2 ,
N ^ = ( - z x , - z y , 1 ) / ( 1 + z x 2 + z y 2 ) 1 / 2 ,
s ^ 2 = s ^ 1 - 2 ( s ^ 1 · N ^ ) N ^ .
( s ^ 2 ) x = ( x - α ) ( 1 - z x 2 + z y 2 ) - 2 z x z y [ z y ( y - β ) + ( γ - z ) ] ( 1 + z x 2 + z y 2 ) [ ( x - α ) 2 + ( y - β ) 2 + ( z - γ ) 2 ] 1 / 2 ,
( s ^ 2 ) y = ( y - β ) ( 1 + z x 2 - z y 2 ) - 2 z y z x ( x - α ) + ( γ - z ) ] ( 1 + z x 2 + z y 2 ) [ x - α ) 2 + ( y + β ) 2 + ( z - γ ) 2 ] 1 / 2 ,
( s ^ 2 ) z = ( γ - z ) ( 1 - z x 2 - z y 2 ) + 2 [ z x ( x - α ) + z y ( y - β ) ] ( 1 + z x 2 + z y 2 ) [ ( x - α ) 2 + ( y - β ) 2 + ( z - γ ) 2 ] 1 / 2
x f - x ( s ^ 2 ) x = y f - y ( s ^ 2 ) y = z f - z ( s ^ 2 ) z .
x f = x + ( z f - z ) ( s ^ 2 ) x ( s ^ 2 ) z ,
y f = y + ( z f - z ) ( s ^ 2 ) y ( s ^ 2 ) z .
x f = x + ( z f - z ) { ( x - α ) ( 1 - z x 2 + z y 2 ) - 2 z x [ z y ( y - β ) + ( γ - z ) ] ( γ - z ) ( 1 - z x 2 - z y 2 ) + 2 [ z x ( x - α ) + z y ( y - β ) ] } ,
y f = y + ( z f - z ) { ( y - β ) ( 1 + z x 2 - z y 2 ) - 2 z y [ z x ( x - α ) + ( γ - z ) ] ( γ - z ) ( 1 - z x 2 - z y 2 ) + 2 [ z x ( x - α ) + z y ( y - β ) ] } .
y f = m d ,
y f = y f ( x , y )
x f ( x , y ) = μ i ,
y f ( x , y ) = γ i .
sin θ = c x c ( 1 - k c 2 x c 2 ) 1 / 2 ,
z = D B + ( B 2 - A D ) 1 / 2 ,
A = c ( 1 + k cos 2 θ ) ,
B = 1 ( 1 + k sin z θ ) 1 / 2 - c k sin θ cos θ x ,
D = c ( 1 + k sin 2 θ ) x 2 + c y 2 .
Z x = [ B ( B 2 - A D ) 1 / 2 + B 2 - A D / 2 ] D x - [ D ( B 2 - A D ) 1 / 2 + B D ] B x [ B + ( B 2 - A D ) 1 / 2 ] 2 ( B 2 - A D ) 1 / 2 ,
Z y = [ B ( B 2 - A D ) 1 / 2 + B 2 - A D / 2 ] D y [ B + ( B 2 - A D ) 1 / 2 ] 2 ( B 2 - A D ) 1 / 2 ,
B x = - c k sin θ cos θ ,
D x = 2 c ( 1 + k sin 2 θ ) x ,
D y = 2 c y .
Δ Z T = 2 β 2 r ,
z = z ( S ) ,
S = ( x 2 + y 2 ) 1 / 2 .
x f = [ { ( γ + z f - 2 z ) ( 1 - z c 2 ) + 2 z c [ S - ( γ - z ) ( z f - z ) S ] } x - 2 z c S [ 1 - ( z f - z ) z c S ] ( α x 2 + β x y ) - α ( 1 + z c 2 ) ( z f - z ) ] / [ ( γ - z ) ( 1 - z c 2 ) + 2 s z c - 2 z c S ( α x + β y ) ] ,
y f = ( { ( γ + z f - 2 z ) ( 1 - z c 2 ) + 2 z c [ S - ( γ - z ) ( z f - z ) S ] } y - 2 z c S [ 1 - ( z f - z ) z c S ] ( α x y + β y 2 ) - β ( 1 + z c z ) ( z f - z ) ) / [ ( γ - z ) ( 1 - z c z ) + 2 s z c - 2 z c S ( α x + β y ) ] ,
z c = d z d S .
y = α A x 2 + ( B + α C ) x + ( D + α C ) - β ( A x + C ) ,
A = 2 z c S [ 1 - ( z f - z ) S z c ] ,
B = - ( γ + z f - 2 z ) ( 1 - z c 2 ) - 2 z c [ S - ( z f - z ) ( γ - z ) S ] ,
C = 2 x f z c S ,
D = x f [ ( γ - z ) ( 1 - z c 2 ) + 2 s z c ] ,
E = ( 1 + z c 2 ) ( z f - z ) .
( α 2 + β 2 ) A 2 x 4 + 2 A [ α B + ( α 2 + β 2 ) C ] x 3 + [ ( B + α C ) 2 + 2 α A ( D + α E ) + β 2 ( C 2 - A 2 S 2 ) ] x 2 + 2 [ ( B + α C ) ( D + α E ) - β 2 A C S 2 ] x + [ ( D + α C ) 2 - β 2 C 2 S 2 ] = 0.

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