Abstract

Sinusoidal gratings made by either the spatial filtering method or holography are used in a Ronchi setup to test lenses and mirrors. The interference between the strong zero-order and two weak first-order wavefronts gives a three-beam Ronchigram in which elliptically shaped moire fringes appear. The ellipses may be used to determine quantitatively the primary aberration, such as the coefficient of spherical aberration, coma, or astigmatism.

© 1990 Optical Society of America

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References

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  1. V. Ronchi, “Forty Years of History of a Grating Interferometer,” Appl. Opt. 3, 437–451 (1964).
    [Crossref]
  2. A. Cornejo-Rodriguez, “Ronchi Test,” in Optical Shop Testing, D. Malacara, Ed (Wiley, New York, 1978).
  3. K. Patorski, A. Cornejo-Rodriguez, “Ronchi Test with Daylight Illumination,” Appl. Opt. 25, 2031–2032 (1986).
    [Crossref] [PubMed]
  4. J. D. Briers, “Ronchi Test Formulae. 1. Theory,” Opt. Laser Technol. 11, 189–196 (1979).
    [Crossref]
  5. J. D. Briers, “Ronchi Test Formulae. 2. Practical Formulae and Experimental Verification,” Opt. Laser Technol. 11, 245–257 (1979).
    [Crossref]
  6. T. Yatagai, “Fringe Scanning Ronchi Test for Aspherical Surfaces,” Appl. Opt. 23, 3676–3679 (1984).
    [Crossref] [PubMed]
  7. V. A. Komissaruk, “Investigation of Wave Front Aberration of Optical Systems Using Three-Beam Interference,” Opt. Spectrosc. 16, 571–574 (1964).
  8. J. A. Lin, J. M. Cowley, “Calibration of Operating Parameters for the HB5 STEM Instrument,” Ultramicroscopy 19, 31–42 (1986).
    [Crossref]
  9. J.-A. Lin, J. M. Cowley, “Aberration Analysis by Three-Beam Interferograms,” Appl. Opt. 25, 2245–2246 (1986).
    [Crossref] [PubMed]
  10. R. Kingslake, “The Interferometer Patterns Due to the Primary Aberrations,” Trans. Opt. Soc. 27, 94–105 (1925–1926).
  11. J.-A. Lin, W. T. Yeh, S. W. Hsu, “Testing Zone Plate with Grating Interferometer,” submitted to Appl. Opt.
  12. S. Yokozeki, T. Suzuki, “Shearing Interferometer Using the Grating as the Beam Splitter,” Appl. Opt. 10, 1575–1580 (1971).
    [Crossref] [PubMed]
  13. J. E. Harvey, R. V. Shack, “Aberrations of Diffracted Wave Fields,” Appl. Opt. 17, 3003–3009 (1978).
    [Crossref] [PubMed]
  14. Y. Cohen-Sabban, D. Joyeux, “Aberration-Free Nonparaxial Self-Imaging,” J. Opt. Soc. Am. 73, 707–719 (1983).
    [Crossref]

1986 (3)

1984 (1)

1983 (1)

1979 (2)

J. D. Briers, “Ronchi Test Formulae. 1. Theory,” Opt. Laser Technol. 11, 189–196 (1979).
[Crossref]

J. D. Briers, “Ronchi Test Formulae. 2. Practical Formulae and Experimental Verification,” Opt. Laser Technol. 11, 245–257 (1979).
[Crossref]

1978 (1)

1971 (1)

1964 (2)

V. A. Komissaruk, “Investigation of Wave Front Aberration of Optical Systems Using Three-Beam Interference,” Opt. Spectrosc. 16, 571–574 (1964).

V. Ronchi, “Forty Years of History of a Grating Interferometer,” Appl. Opt. 3, 437–451 (1964).
[Crossref]

Briers, J. D.

J. D. Briers, “Ronchi Test Formulae. 1. Theory,” Opt. Laser Technol. 11, 189–196 (1979).
[Crossref]

J. D. Briers, “Ronchi Test Formulae. 2. Practical Formulae and Experimental Verification,” Opt. Laser Technol. 11, 245–257 (1979).
[Crossref]

Cohen-Sabban, Y.

Cornejo-Rodriguez, A.

K. Patorski, A. Cornejo-Rodriguez, “Ronchi Test with Daylight Illumination,” Appl. Opt. 25, 2031–2032 (1986).
[Crossref] [PubMed]

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

Cowley, J. M.

J.-A. Lin, J. M. Cowley, “Aberration Analysis by Three-Beam Interferograms,” Appl. Opt. 25, 2245–2246 (1986).
[Crossref] [PubMed]

J. A. Lin, J. M. Cowley, “Calibration of Operating Parameters for the HB5 STEM Instrument,” Ultramicroscopy 19, 31–42 (1986).
[Crossref]

Harvey, J. E.

Hsu, S. W.

J.-A. Lin, W. T. Yeh, S. W. Hsu, “Testing Zone Plate with Grating Interferometer,” submitted to Appl. Opt.

Joyeux, D.

Kingslake, R.

R. Kingslake, “The Interferometer Patterns Due to the Primary Aberrations,” Trans. Opt. Soc. 27, 94–105 (1925–1926).

Komissaruk, V. A.

V. A. Komissaruk, “Investigation of Wave Front Aberration of Optical Systems Using Three-Beam Interference,” Opt. Spectrosc. 16, 571–574 (1964).

Lin, J. A.

J. A. Lin, J. M. Cowley, “Calibration of Operating Parameters for the HB5 STEM Instrument,” Ultramicroscopy 19, 31–42 (1986).
[Crossref]

Lin, J.-A.

J.-A. Lin, J. M. Cowley, “Aberration Analysis by Three-Beam Interferograms,” Appl. Opt. 25, 2245–2246 (1986).
[Crossref] [PubMed]

J.-A. Lin, W. T. Yeh, S. W. Hsu, “Testing Zone Plate with Grating Interferometer,” submitted to Appl. Opt.

Patorski, K.

Ronchi, V.

Shack, R. V.

Suzuki, T.

Yatagai, T.

Yeh, W. T.

J.-A. Lin, W. T. Yeh, S. W. Hsu, “Testing Zone Plate with Grating Interferometer,” submitted to Appl. Opt.

Yokozeki, S.

Appl. Opt. (6)

J. Opt. Soc. Am. (1)

Opt. Laser Technol. (2)

J. D. Briers, “Ronchi Test Formulae. 1. Theory,” Opt. Laser Technol. 11, 189–196 (1979).
[Crossref]

J. D. Briers, “Ronchi Test Formulae. 2. Practical Formulae and Experimental Verification,” Opt. Laser Technol. 11, 245–257 (1979).
[Crossref]

Opt. Spectrosc. (1)

V. A. Komissaruk, “Investigation of Wave Front Aberration of Optical Systems Using Three-Beam Interference,” Opt. Spectrosc. 16, 571–574 (1964).

Trans. Opt. Soc. (1)

R. Kingslake, “The Interferometer Patterns Due to the Primary Aberrations,” Trans. Opt. Soc. 27, 94–105 (1925–1926).

Ultramicroscopy (1)

J. A. Lin, J. M. Cowley, “Calibration of Operating Parameters for the HB5 STEM Instrument,” Ultramicroscopy 19, 31–42 (1986).
[Crossref]

Other (2)

J.-A. Lin, W. T. Yeh, S. W. Hsu, “Testing Zone Plate with Grating Interferometer,” submitted to Appl. Opt.

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

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Figures (11)

Fig. 1
Fig. 1

Grating lines, parallel to the y-axis, make an angle ϕ relative to the y0-axis.

Fig. 2
Fig. 2

Ray diagram illustrating the Ronchi test.

Fig. 3
Fig. 3

Spatial filtering setup for making the sinusoidal grating: G is the Ronchi ruling and f could be the focal length or any other suitable distance.

Fig. 4
Fig. 4

Holographic setup for making the sinusoidal grating: BS, M, L, SP, and G represent beam splitter, mirror, microscope objective, spatial filter, and grating, respectively.

Fig. 5
Fig. 5

Diffraction patterns of (a) sinusoidal grating (32.3 lines/mm) made by the spatial filtering method; (b) sinusoidal grating (39.1 lines/mm) made by holography; (c) a Ronchi ruling of 19.69 lines/mm served as a standard to calibrate the frequency of the sinusoidal grating.

Fig. 6
Fig. 6

Ronchigrams produced from (a) a sine grating of 32.3 lines/mm and (b) a Ronchi ruling of 19.69 lines/mm.

Fig. 7
Fig. 7

Ronchigrams produced by a biconvex lens (f = 17.6 cm, D = 5 cm) by placing the 39.1-lines/mm sinusoidal grating at (a) 11.0-mm underfocus, (c) 0.3-mm underfocus, and (e) 10.5-mm overfocus, respectively, from the point source image. Diffraction patterns (b), (d), and (f) are of the same grating at the positions of (a), (c), and (e) for shear calibration.

Fig. 8
Fig. 8

Ray diagram illustrating the relationship between the semiangle of illumination ϕ, the diffraction angle β, and the normalized shear s for the grating at exact focus.

Fig. 9
Fig. 9

Ronchigrams produced by a biconvex lens (f = 17.6 cm, D = 5 cm) (a) with or (b) without rotating the sine grating of 39.1 lines/mm by 90°. (c) Diffraction patterns of the same grating at the positions of (a) and (b). (d) Diffraction pattern at exact focus.

Fig. 10
Fig. 10

Ray diagram illustrating the testing of a mirror.

Fig. 11
Fig. 11

Ronchigrams and diffraction spots from the mirror testing: (a) Δ = 2.32 mm; (b) Δ = −2.58 mm; (c) Δ = 2.32 mm; (d) Δ = −2.58 mm; (e) Δ = −0.08 mm, uniform focus.

Tables (1)

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Table I Results of the Mirror Testing

Equations (33)

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W ( x 0 , y 0 ) = A ( x 0 2 + y 0 2 ) 2 + B y 0 ( x 0 2 + y 0 2 ) + C ( x 0 2 + 3 y 0 2 ) + D ( x 0 2 + y 0 2 ) + E y 0 + F x 0 ,
x 0 = x cos ϕ + y sin ϕ ,
y 0 = - x sin ϕ + y cos ϕ .
W ( x , y ) = A ( x 2 + y 2 ) 2 + ( B 1 x + B 2 y ) ( x 2 + y 2 ) + C 1 x 2 + C 2 y 2 + J 12 x y + J 1 x + J 2 y ,
B 1 = - B sin ϕ , B 2 = B cos ϕ ; C 1 = C + D + 2 C sin 2 ϕ C 2 = C + D + 2 C cos 2 ϕ ; J 1 = - E sin ϕ + F cos ϕ , J 2 = E cos ϕ + F sin ϕ ; J 12 = - 2 C sin 2 ϕ .
t ( x , y ) = 1 + 2 σ cos ( 2 π g s x + θ ) ,
I ( x , y ; s ) = A ( x , y ) exp [ i 2 π W ( x , y ) / λ ] + σ exp ( i θ ) A ( x + s , y ) × exp [ i 2 π W ( x + s , y ) / λ ] + σ exp ( - i θ ) A ( x - s , y ) × exp [ i 2 π W ( x - s , y ) / λ ] 2 = A 2 ( x , y ) + σ 2 [ A 2 ( x + s , y ) + A 2 ( x - s , y ) ] + 2 σ 2 A ( x + s , y ) A ( x - s , y ) cos [ 4 π O ( x , y ; s ) / λ + 2 θ ] + 2 σ A ( x , y ) A ( x - s , y ) × cos { 2 π [ E ( x , y ; s ) - O ( x , y ; s ) ] / λ - θ } + 2 σ A ( x , y ) A ( x + s , y ) × cos { 2 π [ E ( x , y ; s ) ] + O ( x , y ; s ) ] / λ + θ } ,
E ( x , y ; s ) = [ W ( x + s , y ) - W ( x , y ) ] even in s = A s 4 + 6 A s 2 x 2 + 2 A s 2 y 2 + 3 B 1 x s 2 + B 2 y s 2 + C 1 s 2 ,
O ( x , y ; s ) = [ W ( x + s , y ) - W ( x , y ) ] odd in s = 4 A s x ( x 2 + y 2 ) + ( 4 A s 3 + 2 C 1 s ) x + 3 B 1 s x 2 + B 1 s y 2 + 2 B 2 s x y + J 12 s y + J 1 s + B 1 s 3 ,
E ( x , y ; s ) = A s 4 + 6 A s 2 x 2 + 2 A s 2 y 2 + D s 2 , and
O ( x , y ; s ) = 2 s x [ ( 2 A s 2 + D ) + 2 A ( x 2 + y 2 ) ] .
Δ = 2 D q 2 / R 2 .
I ( x , y ; s ) 1 + 4 σ cos [ 2 π E ( x , y ; s ) / λ ] cos [ 2 π O ( x , y ; s ) / λ + θ ] .
cos [ 2 π E ( x , y ; s ) / λ ] = 0 ,
E ( x , y ; s ) = ( 2 n + 1 ) λ / 4 ,
( x - h ) 2 a n 2 + ( y - k ) 2 b n 2 = 1 ,
b n 2 = 3 a n 2 = ( 2 n + 1 ) λ / 4 + 3 B 1 2 s 2 / 8 A + B 2 2 s 2 / 8 A - C 1 s 2 - A s 4 2 A s 2 .
A = λ 4 s 2 ( b n + 1 2 - b n 2 ) ,
B 1 = - 4 A h ,
B 2 = - 4 A k .
C 1 - C 2 = ( B 1 2 - B 2 2 ) 4 A + 2 A ( b n 2 - b n 2 ) ,
s = 0.246 ,
b n + 1 2 - b n 2 = ( b n + 5 2 - b n 2 ) / 5 = 0.142.
A = λ 4 s 2 ( b n + 1 2 - b n 2 ) = 29.1 λ .
s = 0.196 ,             b n + 1 2 - b n 2 = 0.214 , A = 30.2 λ .
g 0 = sin ϕ / λ = sin [ tan - 1 ( A O ¯ / z ) ] / λ = sin [ tan - 1 ( tan β / s ) ] / λ = sin ( tan - 1 { tan [ sin - 1 ( λ g s ) ] / s } ) / λ = 175.1 lines / mm ,
Δ u = 2 D q 2 / R 2 = 2 ( - 2 A s 2 ) / tan 2 ϕ = - 0.30 mm .
( h , k ) = ( 0.190 , 0 ) , B 1 = - 4 A h = - 17.9 λ , B 2 = - 4 A k = 0.
b n = 0.633 ,             b n = 0.724 , C 1 - C 2 = ( B 1 2 - B 2 2 ) 4 A + 2 A ( b n 2 - b n 2 ) = 9.2 λ .
B = 17.9 λ and C = 4.6 λ for ϕ = π 2 .
W ( x 0 , y 0 ) = 24.0 λ ( x 0 2 + y 0 2 ) 2 + 17.9 λ y 0 ( x 0 2 + y 0 2 ) + 2.31 λ ( x 0 2 + 3 y 0 2 ) + .
A = λ + 2 ( W 20 + W 31 s ) ( b n + 1 2 - b n 2 ) 4 s 2 ( b n + 1 2 - b n 2 ) ,
W 20 = Δ tan 2 ϕ ( λ g s ) 2 / 4 , and W 31 = - Δ tan 3 ϕ ( λ g s ) / 2 ,

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