Abstract

It is well known that the Ronchi test has two equivalent interpretations, Physical, as an interferometer, or geometrical, as if the fringes were just shadows from the fringes on the ruling. The second interpretation is nearly always used in practice because it is simpler. However, the disadvantage is that the irradiance profile of the fringes cannot be calculated with this theory. Here, the interferometric interpretation of the test will be used to obtain the irradiance profile and the sharpness of the fringes.

© 1990 Optical Society of America

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References

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  1. D. Malacara, A. Cornejo, “The Talbot Effect in the Ronchi Test,” Bol. Inst. Tonantzintla 1, 193–195 (1974).
  2. A. Cornejo, “Ronchi Test,” in Optical Shop Testing, D. Malacara Ed. (Wiley, New York, 1978), pp. 283.

1974

D. Malacara, A. Cornejo, “The Talbot Effect in the Ronchi Test,” Bol. Inst. Tonantzintla 1, 193–195 (1974).

Cornejo, A.

D. Malacara, A. Cornejo, “The Talbot Effect in the Ronchi Test,” Bol. Inst. Tonantzintla 1, 193–195 (1974).

A. Cornejo, “Ronchi Test,” in Optical Shop Testing, D. Malacara Ed. (Wiley, New York, 1978), pp. 283.

Malacara, D.

D. Malacara, A. Cornejo, “The Talbot Effect in the Ronchi Test,” Bol. Inst. Tonantzintla 1, 193–195 (1974).

Bol. Inst. Tonantzintla

D. Malacara, A. Cornejo, “The Talbot Effect in the Ronchi Test,” Bol. Inst. Tonantzintla 1, 193–195 (1974).

Other

A. Cornejo, “Ronchi Test,” in Optical Shop Testing, D. Malacara Ed. (Wiley, New York, 1978), pp. 283.

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Figures (8)

Fig. 1
Fig. 1

Transmittance in a Ronchi ruling. The parameter K defines the width of the clear slits with respect to the dark zones; the most frequent value is K = 0.5.

Fig. 2
Fig. 2

Diagram showing one of the first-order diffracted beams in a Ronchi ruling. The wavefront is projected back to the position of the incident wavefront, with its correct relative phase.

Fig. 3
Fig. 3

Interferometric addition of the diffracted wavefronts.

Fig. 4
Fig. 4

Fringe irradiance profiles for the Ronchi fringes in a parabolic mirror. The optical axis is at the left edge of each plot: (A) outside of focus at the Rayleigh distance (l = −218.66 mm); (B) outside of focus at three-fourths of the Rayleigh distance (l = −166.69 mm); (C) outside of focus at half of the Rayleigh distance (l = −113.04 mm); (D) outside of focus at l = −100.0 mm; (E) outside of focus at one-fourth of the Rayleigh distance (l = −57.54 mm); and (F) outside of focus at l = −50.0 mm.

Fig. 5
Fig. 5

Fringe irradiance profiles for the Ronchi fringes in a parabolic mirror. The optical axis is at the left edge of each plot: (A) outside of focus at l = −45.0 mm; (B) outside of focus at l = −40.0 mm; (C) outside of focus at l = −35.0 mm; (D) outside of focus at l = −30.0 mm; (E) outside of focus at l = −25.0 mm; and (F) outside of focus at l = −20.0 mm.

Fig. 6
Fig. 6

Fringe irradiance profiles for the Ronchi fringes in a parabolic mirror. The optical axis is at the left edge of each plot: (A) outside of focus at l = −15.0 mm; (B) outside of focus at l = −10.0 mm; (C) outside of focus at l = −5.0 mm; (D) inside of focus at l = 5.0 mm; (E) inside of focus at l = 10.0 mm; and (F) inside of focus at l = 15.0 mm.

Fig. 7
Fig. 7

Fringe irradiance profiles for the Ronchi fringes in a parabolic mirror. The optical axis is at the left edge of each plot: (A) inside of focus at l = 20.0 mm; (B) inside of focus at l = 25.0 mm; (C) inside of focus at l = 30.0 mm; (D) inside of focus at l = 35.0 mm; (E) inside of focus at l = 40.0 mm; and (F) inside of focus at l = 45.0 mm.

Fig. 8
Fig. 8

Fringe irradiance profiles for the Ronchi fringes in a parabolic mirror. The optical axis is at the left edge of each plot: (A) inside of focus at l = 50.0 mm; (B) inside of focus at one-fourth of the Rayleigh distance (l = 59.84 mm); (C) inside of focus at l = 100.0 mm; (D) inside of focus at half of the Rayleigh distance (l = 122.28 mm); (E) inside of focus at three-fourths of the Rayleigh distance (l = 187.70 mm); and (F) inside of focus at the Rayleigh distance (l = 256.54).

Equations (30)

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B n = ( - 1 ) n n π sin ( n π K ) .
f ( x ) = n = - B n exp [ i ( 2 π n x d ) ] .
G ( x , y ) = n = - B n exp [ i ϕ ( x + n S , y ) ] ,
ϕ ( x + n S , y ) = 2 π λ W ( x + n S , y ) ,
S = λ ( r - 1 ) d ,
ϕ ( x + n s , y ) = ϕ 0 + ϕ odd + ϕ even .
G ( x , y ) = exp ( i ϕ 0 ) n = - B n i ( ϕ even + ϕ odd ) ,
= exp ( i ϕ 0 ) [ B 0 + 2 n = 1 B n ( cos ϕ odd ) exp ( - i ϕ even ) ] .
I ( x , y ) = G ( x , y ) G * ( x , y ) .
α = λ l n r d ,
OPD = λ 2 l ( r - l ) n 2 2 r d 2 .
ϕ even = 2 π λ OPD = π λ l ( r - l ) n 2 r d 2 ,
ϕ odd = 2 π λ α x = 2 π l n x r d .
W ( x + n S , y ) = i = 0 k j = 0 i A i j [ x + n S ] j y i - j ,
W ( x + n S , y ) = i = 0 k J = 0 i r = 0 J A i j [ j r ] x J - r y i - j ( n S ) r ,
[ j r ] = j ! r ! ( j - r ) ! .
W ( x + n S , y ) = D [ x + n S ) 2 + y 2 ] + E [ ( x + n S ) 2 + y 2 ] 2 .
W ( x + n S , y ) = D n 2 S 2 + D y 2 + S y 4 + 2 E y 2 n 2 S 2 + E n 4 S 4 + [ 2 D n S + 4 E y 2 n s + 4 E n 3 S 3 ] x + [ D + 2 E y 2 + 6 E n 2 S 2 ] x 2 + 4 E n S x 3 + E x 4 .
W ( x + n S , y ) = W 0 + W even + W odd ,
W 0 = D y 2 + S y 4 + [ D + 2 E y 2 ] x 2 + E x 4 , W even = D n 2 S 2 + 2 E y 2 n 2 S 2 + E n 4 S 4 + 6 E n 2 S 2 x 2 , W odd = [ 2 D n S + 4 E y 2 n s + 4 E n 3 S 3 ] x + 4 E n S x 3 .
W even = D n 2 S 2 + E n 4 S 4 + 6 E n 2 S 2 x 2 , W odd = [ 2 D n S + 4 E n 3 S 3 ] x .
ϕ even = 2 π n 2 S 2 [ D + E n 2 S 2 ] / λ , ϕ odd = 4 π x n S [ D + 2 E n 2 S 2 ] / λ .
D = l 2 r ( r - l ) ,
E = 1 4 r 3 .
ϕ even = π λ l ( r - l ) n 2 r d 2 [ 1 + n 2 λ 2 ( r - l ) 3 2 r 2 d 2 l ] ,
ϕ odd = 2 π l n x r d [ 1 + n 2 λ 2 ( r - l ) 3 r 2 d 2 l ] .
ϕ even = π λ l ( r - l ) n 2 r d 2 [ 1 + n 2 λ 2 r 2 d 2 l ] ,
ϕ odd = 2 π l n x r d [ 1 + n 2 λ 2 r d 2 l ] .
λ l ( r - l ) π r d 2 = 2 M π .
2 λ l ( r - l ) r d 2 = ± 1.

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