Abstract

We describe a generalized phase-shifting interferometry for which the reference phases are directly evaluated at each time that the interference fringe data are read. The reference phases are obtained from the additional straight fringes on the interfering plane by the fast-Fourier-transform method. According to error estimation, the repeatabilities in the measurements of optical surfaces are λ/500 rms, when the generalized algorithm with eight data acquisitions is used.

© 1991 Optical Society of America

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References

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  1. J. H. Bruning, D. R. Herriott, 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, 2693–2703 (1974).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
    [CrossRef]
  4. Y. Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
    [CrossRef] [PubMed]
  5. N. Ohyama, S. Kinoshita, A. Cornejo-Rodriguez, T. Honda, J. Tsujiuchi, “Accuracy of phase determination with unequal reference shift,” J. Opt. Soc. Am. A 5, 2019–2025 (1988).
    [CrossRef]
  6. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error compensation phase calculation,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  7. K. Kinnstaetter, A. W. Lohmann, J. Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
    [CrossRef] [PubMed]
  8. G. Lai, M. Kobayashi, T. Yatagai, “Fizeau and Young’s fringe analysis using FFT method and its applications,” submitted to Appl. Opt.
  9. J. Chen, Y. Ishii, K. Murata, “Heterodyne interferometry with a frequency-modulated laser diode,” Appl. Opt. 27, 124–128 (1988).
    [CrossRef] [PubMed]
  10. Y. Suematsu, “Advances in semiconductor lasers,” Phys. Today 38(5), 32–34 (1985).
    [CrossRef]
  11. P. E. Ciddor, R. M. Duffy, “Two-mode frequency-stabilized He–Ne (633 nm) lasers: studies of short- and long-term stability,”J. Phys. E 16, 1223–1227 (1983).
    [CrossRef]

1988

1987

1985

1984

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
[CrossRef]

1983

P. E. Ciddor, R. M. Duffy, “Two-mode frequency-stabilized He–Ne (633 nm) lasers: studies of short- and long-term stability,”J. Phys. E 16, 1223–1227 (1983).
[CrossRef]

1982

1974

Brangaccio, D. J.

Bruning, J. H.

Chen, J.

Cheng, Y. Y.

Ciddor, P. E.

P. E. Ciddor, R. M. Duffy, “Two-mode frequency-stabilized He–Ne (633 nm) lasers: studies of short- and long-term stability,”J. Phys. E 16, 1223–1227 (1983).
[CrossRef]

Cornejo-Rodriguez, A.

Duffy, R. M.

P. E. Ciddor, R. M. Duffy, “Two-mode frequency-stabilized He–Ne (633 nm) lasers: studies of short- and long-term stability,”J. Phys. E 16, 1223–1227 (1983).
[CrossRef]

Eiju, T.

Gallagher, J. E.

Greivenkamp, J. E.

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
[CrossRef]

Hariharan, P.

Herriott, D. R.

Honda, T.

Ishii, Y.

Kinnstaetter, K.

Kinoshita, S.

Kobayashi, M.

G. Lai, M. Kobayashi, T. Yatagai, “Fizeau and Young’s fringe analysis using FFT method and its applications,” submitted to Appl. Opt.

Lai, G.

G. Lai, M. Kobayashi, T. Yatagai, “Fizeau and Young’s fringe analysis using FFT method and its applications,” submitted to Appl. Opt.

Lohmann, A. W.

Morgan, C. J.

Murata, K.

Ohyama, N.

Oreb, B. F.

Rosenfeld, D. P.

Schwider, J.

Streibl, N.

Suematsu, Y.

Y. Suematsu, “Advances in semiconductor lasers,” Phys. Today 38(5), 32–34 (1985).
[CrossRef]

Tsujiuchi, J.

White, A. D.

Wyant, J. C.

Yatagai, T.

G. Lai, M. Kobayashi, T. Yatagai, “Fizeau and Young’s fringe analysis using FFT method and its applications,” submitted to Appl. Opt.

Appl. Opt.

J. Opt. Soc. Am. A

J. Phys. E

P. E. Ciddor, R. M. Duffy, “Two-mode frequency-stabilized He–Ne (633 nm) lasers: studies of short- and long-term stability,”J. Phys. E 16, 1223–1227 (1983).
[CrossRef]

Opt. Eng.

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
[CrossRef]

Opt. Lett.

Phys. Today

Y. Suematsu, “Advances in semiconductor lasers,” Phys. Today 38(5), 32–34 (1985).
[CrossRef]

Other

G. Lai, M. Kobayashi, T. Yatagai, “Fizeau and Young’s fringe analysis using FFT method and its applications,” submitted to Appl. Opt.

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Figures (10)

Fig. 1
Fig. 1

Optical setup used to realize the generalized phase-shifting interferometry with direct reference phase evaluation from straight fringes.

Fig. 2
Fig. 2

Interferogram on the viewing plane. The upper part shows interfering fringes used in the measurement of the optical wave front, and the lower part shows the straight fringes used to evaluate the shifted reference phase.

Fig. 3
Fig. 3

Intensity distribution of straight fringes obtained from an output of one video line.

Fig. 4
Fig. 4

2-D phase distribution calculated by the generalized algorithm, using the calibration data of a PZT. The reference phase is shifted by eight steps.

Fig. 5
Fig. 5

1-D phase distribution taken for the calculation of the measurement repeatability.

Fig. 6
Fig. 6

Matrix A in Eq. (5) with random phase in 5 steps (top), 10 steps (center), and 20 steps (bottom).

Fig. 7
Fig. 7

2-D phase distribution calculated by the generalized phase-shifting algorithm with 10 steps of random reference phases.

Fig. 8
Fig. 8

Phase calculation error in rms versus the original phase from the stepwise algorithm: (a) 4 steps, (b) 8 steps, (c) 16 steps.

Fig. 9
Fig. 9

Phase calculation error in rms versus the original phase from randomly shifted reference phases.

Fig. 10
Fig. 10

Effect of errors in digitizing the intensity with limited gray levels: (a) error versus fringe contrast, keeping the levels at 128, (b) error versus gray levels with the fringe contrast of 0.8.

Equations (20)

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I k ( x , y ) = I 0 ( x , y ) { 1 + α ( x , y ) cos [ ϕ ( x , y ) + ψ k ] } ,
I k = I 0 + I 0 α cos ϕ cos ψ k - I 0 α sin ϕ sin ψ k = a 0 + a 1 cos ψ k + a 2 sin ψ k ,
E = k = 0 N - 1 ( I k - I ^ k ) 2 = k = 0 N - 1 ( a 0 + a 1 cos ψ k + a 2 sin ψ k - I ^ k ) 2 .
A ( ψ k ) a = b ( ψ k ) ,
A ( ψ k ) = [ N k = 0 N - 1 cos ψ k k = 0 N - 1 sin ψ k k = 0 N - 1 cos ψ k k = 0 N - 1 cos 2 ψ k k = 0 N - 1 cos ψ k sin ψ k k = 0 N - 1 sin ψ k k = 0 N - 1 cos ψ k sin ψ k k = 0 N - 1 sin 2 ψ k ] ,
a = ( a 0 a 1 a 2 ) ,
b ( ψ k ) = [ k = 0 N - 1 I ^ k k = 0 N - 1 I ^ k cos ψ k k = 0 N - 1 I ^ k sin ψ k ] .
a = A - 1 ( ψ k ) b ( ψ k ) .
ϕ = - tan - 1 a 2 a 1 .
I ( x ) = a ( x ) [ 1 + γ ( x ) cos ( 2 π f x + ψ ) ] ,
I ^ ( ν ) = - I ( x ) exp ( - 2 π i ν x ) d x = a ^ ( ν ) + c ^ ( ν - f ) exp ( i ψ ) + c ^ ( ν + f ) exp ( - i ψ ) ,
a ^ ( ν ) = - a ( x ) exp ( - 2 π i ν x ) d x ,
c ^ ( ν ) = 1 2 - a ( x ) γ ( x ) exp ( - 2 π i ν x ) d x
I ^ ( ν ) = c ^ ( ν - f ) exp ( i ψ ) .
log [ I ^ ( f ) ] = log [ c ^ ( 0 ) ] + i ψ .
ψ = Im { log [ I ^ ( f ) ] } - constant .
δ ψ = 2 π L δ λ λ 0 2 .
I ( ψ k ) = a [ 1 + γ cos ( ϕ + ψ k + e k ) ] ,
I ( ψ k ) = M / ( 1 + γ ) [ 1 + γ cos ( ϕ + ψ k ) ] + Δ k ,
P ( Δ ) = { 1 - 0.5 < Δ < 0.5 0 otherwise .

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