Abstract

Use of synchronous phase detection in the conventional Ronchi test is discussed to measure a large amount of wavefront aberration or an aspheric wavefront. This method provides a simple but powerful tool for aspheric surface evaluation. By moving the Ronchi grating sideways, a periodic phase shift in the Ronchigram is introduced for synchronous phase detection. A theoretical interpretation of the Ronchigram and the procedure for an analysis are presented, and experimental results are shown.

© 1988 Optical Society of America

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References

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  1. A. J. MacGovern, J. C. Wyant, “Computer Generated Holograms for Testing Optical Elements,” Appl. Opt. 10, 619 (1971).
    [CrossRef] [PubMed]
  2. J. C. Wyant, V. P. Bennett, “Using Computer Generated Holograms to Test Aspheric Wavefronts,” Appl. Opt. 11, 2833 (1972).
    [CrossRef] [PubMed]
  3. T. Yatagai, H. Saito, “Interferometric Testing with Computer-Generated Holograms: Aberration Balancing Method and Error Analysis,” Appl. Opt. 17, 558 (1978).
    [CrossRef] [PubMed]
  4. J. C. Wyant, “Double Frequency Grating Lateral Shearing Interferometer,” Appl. Opt. 12, 2057 (1973).
    [CrossRef] [PubMed]
  5. D. Nyyssonen, J. M. Jerke, “Lens Testing with a Simple Wavefront Shearing Interferometer,” Appl. Opt. 12, 2061 (1973).
    [CrossRef] [PubMed]
  6. M. V. K. Murty, in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 81–148.
  7. T. Yatagai, T. Kanou, “Aspherical Surface Testing with Shearing Interferometer Using Fringe Scanning Detection Method,” Opt. Eng. 23, 357 (1984).
    [CrossRef]
  8. D. Malacara, A. Cornejo, “Null Ronchi Test for Aspherical Surfaces,” Appl. Opt. 13, 1778 (1974).
    [CrossRef] [PubMed]
  9. J. H. Bruning, D. R. Heriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital Wavefront Measuring Interferometer for Testing Optical Surfaces and Lenses,” Appl. Opt. 13, 2693 (1974).
    [CrossRef] [PubMed]
  10. T. Yatagai, “Fringe Scanning Ronchi Test for Aspherical Surfaces,” Appl. Opt. 23, 3676 (1984).
    [CrossRef] [PubMed]
  11. A. Corenjo, in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 283–436.
  12. C. L. Koliopoulos, “Radial Grating Lateral Shear Heterodyne Interferometer,” Appl. Opt. 19, 1523 (1980).
    [CrossRef] [PubMed]
  13. M. P. Rimmer, “Evaluating Lateral Shearing Interferograms,” Appl. Opt. 13, 623 (1974).
    [CrossRef] [PubMed]
  14. J. C. Wyant, “Use of an ac Heterodyne Lateral Shear Inteferometer with Real-Time Wavefront Correction Systems,” Appl. Opt. 14, 2622 (1975).
    [CrossRef] [PubMed]
  15. T. Yatagai, to be published.

1984

T. Yatagai, T. Kanou, “Aspherical Surface Testing with Shearing Interferometer Using Fringe Scanning Detection Method,” Opt. Eng. 23, 357 (1984).
[CrossRef]

T. Yatagai, “Fringe Scanning Ronchi Test for Aspherical Surfaces,” Appl. Opt. 23, 3676 (1984).
[CrossRef] [PubMed]

1980

1978

1975

1974

1973

1972

1971

Bennett, V. P.

Brangaccio, D. J.

Bruning, J. H.

Corenjo, A.

A. Corenjo, in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 283–436.

Cornejo, A.

Gallagher, J. E.

Heriott, D. R.

Jerke, J. M.

Kanou, T.

T. Yatagai, T. Kanou, “Aspherical Surface Testing with Shearing Interferometer Using Fringe Scanning Detection Method,” Opt. Eng. 23, 357 (1984).
[CrossRef]

Koliopoulos, C. L.

MacGovern, A. J.

Malacara, D.

Murty, M. V. K.

M. V. K. Murty, in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 81–148.

Nyyssonen, D.

Rimmer, M. P.

Rosenfeld, D. P.

Saito, H.

White, A. D.

Wyant, J. C.

Yatagai, T.

Appl. Opt.

A. J. MacGovern, J. C. Wyant, “Computer Generated Holograms for Testing Optical Elements,” Appl. Opt. 10, 619 (1971).
[CrossRef] [PubMed]

J. C. Wyant, V. P. Bennett, “Using Computer Generated Holograms to Test Aspheric Wavefronts,” Appl. Opt. 11, 2833 (1972).
[CrossRef] [PubMed]

T. Yatagai, H. Saito, “Interferometric Testing with Computer-Generated Holograms: Aberration Balancing Method and Error Analysis,” Appl. Opt. 17, 558 (1978).
[CrossRef] [PubMed]

J. C. Wyant, “Double Frequency Grating Lateral Shearing Interferometer,” Appl. Opt. 12, 2057 (1973).
[CrossRef] [PubMed]

D. Nyyssonen, J. M. Jerke, “Lens Testing with a Simple Wavefront Shearing Interferometer,” Appl. Opt. 12, 2061 (1973).
[CrossRef] [PubMed]

D. Malacara, A. Cornejo, “Null Ronchi Test for Aspherical Surfaces,” Appl. Opt. 13, 1778 (1974).
[CrossRef] [PubMed]

J. H. Bruning, D. R. Heriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital Wavefront Measuring Interferometer for Testing Optical Surfaces and Lenses,” Appl. Opt. 13, 2693 (1974).
[CrossRef] [PubMed]

T. Yatagai, “Fringe Scanning Ronchi Test for Aspherical Surfaces,” Appl. Opt. 23, 3676 (1984).
[CrossRef] [PubMed]

C. L. Koliopoulos, “Radial Grating Lateral Shear Heterodyne Interferometer,” Appl. Opt. 19, 1523 (1980).
[CrossRef] [PubMed]

M. P. Rimmer, “Evaluating Lateral Shearing Interferograms,” Appl. Opt. 13, 623 (1974).
[CrossRef] [PubMed]

J. C. Wyant, “Use of an ac Heterodyne Lateral Shear Inteferometer with Real-Time Wavefront Correction Systems,” Appl. Opt. 14, 2622 (1975).
[CrossRef] [PubMed]

Opt. Eng.

T. Yatagai, T. Kanou, “Aspherical Surface Testing with Shearing Interferometer Using Fringe Scanning Detection Method,” Opt. Eng. 23, 357 (1984).
[CrossRef]

Other

T. Yatagai, to be published.

A. Corenjo, in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 283–436.

M. V. K. Murty, in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 81–148.

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Figures (10)

Fig. 1
Fig. 1

Typical geometry of the Ronchi test.

Fig. 2
Fig. 2

Schematic of the phase measuring Ronchi test.

Fig. 3
Fig. 3

Flow diagram of a software system.

Fig. 4
Fig. 4

(a) Ronchigram and (b) its intensity profile along a central cross section.

Fig. 5
Fig. 5

(a) Principal value of the phases and (b) its profile along a central cross section.

Fig. 6
Fig. 6

(a) Unwrapped phase and (b) its profile along a central cross section.

Fig. 7
Fig. 7

(a) Sectional wavefront aberration for one direction obtained by numerical integration of the phase and (b) its profile along a central cross section.

Fig. 8
Fig. 8

Sectional wavefront aberration for (a) the x direction and (b) the y direction obtained by numerical integration of the phase.

Fig. 9
Fig. 9

Two-dimensional profile of a wavefront aberration reconstructed from the profile shown in Fig. 8.

Fig. 10
Fig. 10

Detected phase error estimation: (a) the 1-D phase profile of Fig. 6(a) along a central cross section and (b) its polynomial interpolated phase profile.

Equations (28)

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F 0 ( x 0 , y 0 ) = exp [ i 2 π w ( x , y ) ] ,
F ˜ 0 ( ν x , ν y ) = F [ F 0 ( x 0 , y 0 ) ] ,
ν x = x r / λ f , ν y = y r / λ f . }
M ( ν x , ν y ) = n = B n exp ( i 2 π d f n ν x ) ,
B n = 1 2 sinc ( n π 2 ) .
G ( x , y ) = F [ F ˜ 0 ( ν x , ν y ) · M ( ν x , ν y ) ] .
G ( x , y ) = F 0 ( x , y ) * F [ M ( ν x , ν y ) ] ,
M ( ν x , ν y ) = n = B n exp ( i n Δ ) exp ( i 2 π d λ f n ν x ) ,
Δ = 2 π d Δ x r .
G ( x , y , Δ ) = n = B n exp ( i n Δ ) exp [ i 2 π λ w ( x + λ f d n , y ) ] .
I ( x , y , Δ ) = | G ( x , y , Δ ) | 2 = n = | B n | 2 + n = ( n m ) m = B n B * m cos { 2 π λ [ w ( x + n s , y ) w ( x + m s , y ) ] ( n m ) Δ } ,
s = λ f d .
Δ x r j = d N j ( j = 0 , 1 , 2 , N 1 ) ,
Δ j = 2 π N j .
C ( x , y ) = j = 0 N 1 I ( x , y , Δ j ) cos 2 π N j = j = 0 N 1 n = ( n m ) m = B n B * m { cos 2 π λ [ w ( x + n s , y ) w ( x + m s , y ) ] ( n m ) Δ } × cos 2 π N j ,
S ( x , y ) = j = 0 N 1 I ( x , y , Δ j ) sin 2 π N j = j = 0 N 1 n = ( n m ) m = B n B * m { cos 2 π λ [ w ( x + n s , y ) w ( x + m s , y ) ] ( n m ) Δ } × sin 2 π N j .
C ( x , y ) = 2 π cos π λ [ 2 w ( x , y ) w ( x + s , y ) w ( x s , y ) ] × cos π λ [ w ( x + s , y ) w ( x s , y ) ] ,
S ( x , y ) = 2 π cos π λ [ 2 w ( x , y ) w ( x + s , y ) w ( x s , y ) ] × sin π λ [ w ( x + s , y ) w ( x s , y ) ] .
ϕ = tan 1 S ( x , y ) C ( x , y ) ,
γ = C 2 ( x , y ) + S 2 ( x , y ) .
tan 1 S C = 2 π λ [ w ( x + s , y ) w ( x s , y ) ] ,
C 2 + S 2 = 2 π λ [ 2 w ( x , y ) w ( x + s , y ) w ( x s , y ) ] .
tan 1 S C = 2 π λ ( s w x + s 3 3 ! 3 w x 3 + ) ,
C 2 + S 2 = 2 π λ ( s 2 2 ! 2 w x 2 + s 4 4 ! 4 w x 4 + ) .
w x = λ 2 π s tan 1 S C ,
2 w x 2 = λ π s 2 C 2 + S 2 .
W x ( x , y ) = w n d x ,
W y ( x , y ) = w y d y .

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