Abstract

In the fringe-scanning methods such as the three- and four-bucket methods, the phase of a reference wave is generally changed at equal intervals from 0 to 2π. In practice, however, it is difficult or almost impossible for some interferometers, such as the dynamic zone plate interferometer, to realize an equal phase shift and to make a full scanning of 2π, which are necessary for discrete Fourier transform analysis. We show that even in such a case, phases in an interferogram can be obtained by using a Fourier analysis in a continuous space, and we analyze the accuracy of the phase determination in case of additive noise.

© 1988 Optical Society of America

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References

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  1. J. H. Bruning, D. H. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  2. J. H. Bruning, “Fringe scanning interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978).
  3. Y.-Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
    [CrossRef] [PubMed]
  4. M. Chang, C. P. Hu, P. Lam, J. C. Wyant, “High-precision deformation measurement by digital phase shifting holographic interferometry,” Appl. Opt. 24, 3780–3783 (1985).
    [CrossRef]
  5. N. Ohyama, T. Shimano, J. Tsujiuchi, T. Honda, “An analysis of systematic phase errors due to nonlinearity in fringe scanning system,” Opt. Commun. 58, 223–225 (1986).
    [CrossRef]
  6. P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
    [CrossRef]
  7. N. Ohyama, I. Ichimura, I. Yamaguchi, T. Honda, J. Tsujiuchi, “The dynamic zone plate interferometer for measuring aspherical surfaces,” Opt. Commun. 56, 369–373 (1986).
    [CrossRef]
  8. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
    [CrossRef]

1986 (2)

N. Ohyama, T. Shimano, J. Tsujiuchi, T. Honda, “An analysis of systematic phase errors due to nonlinearity in fringe scanning system,” Opt. Commun. 58, 223–225 (1986).
[CrossRef]

N. Ohyama, I. Ichimura, I. Yamaguchi, T. Honda, J. Tsujiuchi, “The dynamic zone plate interferometer for measuring aspherical surfaces,” Opt. Commun. 56, 369–373 (1986).
[CrossRef]

1985 (2)

1984 (1)

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

1974 (1)

1966 (1)

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Brangccio, D. J.

Bruning, J. H.

Carre, P.

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Chang, M.

Cheng, Y.-Y.

Gallagher, J. E.

Greivenkamp, J. E.

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

Herriott, D. H.

Honda, T.

N. Ohyama, T. Shimano, J. Tsujiuchi, T. Honda, “An analysis of systematic phase errors due to nonlinearity in fringe scanning system,” Opt. Commun. 58, 223–225 (1986).
[CrossRef]

N. Ohyama, I. Ichimura, I. Yamaguchi, T. Honda, J. Tsujiuchi, “The dynamic zone plate interferometer for measuring aspherical surfaces,” Opt. Commun. 56, 369–373 (1986).
[CrossRef]

Hu, C. P.

Ichimura, I.

N. Ohyama, I. Ichimura, I. Yamaguchi, T. Honda, J. Tsujiuchi, “The dynamic zone plate interferometer for measuring aspherical surfaces,” Opt. Commun. 56, 369–373 (1986).
[CrossRef]

Lam, P.

Ohyama, N.

N. Ohyama, T. Shimano, J. Tsujiuchi, T. Honda, “An analysis of systematic phase errors due to nonlinearity in fringe scanning system,” Opt. Commun. 58, 223–225 (1986).
[CrossRef]

N. Ohyama, I. Ichimura, I. Yamaguchi, T. Honda, J. Tsujiuchi, “The dynamic zone plate interferometer for measuring aspherical surfaces,” Opt. Commun. 56, 369–373 (1986).
[CrossRef]

Rosenfeld, D. P.

Shimano, T.

N. Ohyama, T. Shimano, J. Tsujiuchi, T. Honda, “An analysis of systematic phase errors due to nonlinearity in fringe scanning system,” Opt. Commun. 58, 223–225 (1986).
[CrossRef]

Tsujiuchi, J.

N. Ohyama, T. Shimano, J. Tsujiuchi, T. Honda, “An analysis of systematic phase errors due to nonlinearity in fringe scanning system,” Opt. Commun. 58, 223–225 (1986).
[CrossRef]

N. Ohyama, I. Ichimura, I. Yamaguchi, T. Honda, J. Tsujiuchi, “The dynamic zone plate interferometer for measuring aspherical surfaces,” Opt. Commun. 56, 369–373 (1986).
[CrossRef]

White, A. D.

Wyant, J. C.

Yamaguchi, I.

N. Ohyama, I. Ichimura, I. Yamaguchi, T. Honda, J. Tsujiuchi, “The dynamic zone plate interferometer for measuring aspherical surfaces,” Opt. Commun. 56, 369–373 (1986).
[CrossRef]

Appl. Opt. (3)

Metrologia (1)

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Opt. Commun. (2)

N. Ohyama, I. Ichimura, I. Yamaguchi, T. Honda, J. Tsujiuchi, “The dynamic zone plate interferometer for measuring aspherical surfaces,” Opt. Commun. 56, 369–373 (1986).
[CrossRef]

N. Ohyama, T. Shimano, J. Tsujiuchi, T. Honda, “An analysis of systematic phase errors due to nonlinearity in fringe scanning system,” Opt. Commun. 58, 223–225 (1986).
[CrossRef]

Opt. Eng. (1)

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

Other (1)

J. H. Bruning, “Fringe scanning interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978).

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Figures (2)

Fig. 1
Fig. 1

Condition number versus the scanning area. The sampling positions are assumed to be equally spaced within the scanning area. The numbers of sampled data m are 5 and 50.

Fig. 2
Fig. 2

Graph of the expected phase error anticipated by the theory. M represents the number of samples that are equally spaced within the scan period. For example, the value 1.0 represents the full scanning of one period, and a value of 0.6 indicates that all M sampled data are taken in 60% of the area of one period and the remaining 40% are missing. The division of the ordinate by the factor S converts its units into degrees of arc. For instance, when the contrast ratio γ is 0.5 and σ is 0.05, a value of 50 on the ordinate scale corresponds to 5 deg. Note that the expected phase error varies according to the phase to be derived; thus two lines, the maximum and the minimum, are shown for each case.

Equations (63)

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i ( r , x k ) = a ( r ) + b ( r ) cos [ ϕ ( r ) + x k ] ;
i s ( x ) = k = 1 m f k δ ( x x k ) ,
f k = a + b cos ( ϕ + x k ) ,
f k = a + ( b / 2 ) exp [ j ( ϕ + x k ) ] + ( b / 2 ) exp [ j ( ϕ + x k ) ] .
I s ( u ) = 0 2 π i s ( x ) exp ( j u x ) d x = k = 1 m f k exp ( j u x k ) ,
I s ( u ) = a k = 1 m exp ( j u x k ) + ( b / 2 ) e j ϕ k = 1 m exp [ j ( u + 1 ) x k ] + ( b / 2 ) e j ϕ k = 1 m exp [ j ( u 1 ) x k ] .
k = 1 m f k exp ( j u x k ) = a k = 1 m exp ( j u x k ) + B k = 1 m exp [ j ( u + 1 ) x k ] + C k = 1 m exp [ j ( u 1 ) x k ] .
ϕ = arctan [ Im ( B ) / Re ( B ) ] ,
ϕ = arctan [ Im ( C ) / Re ( C ) ]
ϕ = 1 2 arctan ( C / B ) .
H [ a B C ] = [ ( 1 / m ) k = 1 m f k exp ( j u 1 x k ) ( 1 / m ) k = 1 m f k exp ( j u 2 x k ) ( 1 / m ) k = 1 m f k exp ( j u 3 x k ) ] ,
H = [ ( 1 / m ) k exp ( j u 1 x k ) ( 1 / m ) k exp [ j ( u 1 + 1 ) x k ] ( 1 / m ) k exp [ j ( u 1 1 ) x k ] ( 1 / m ) k exp ( j u 2 x k ) ( 1 / m ) k exp [ j ( u 2 + 1 ) x k ] ( 1 / m ) k exp [ j ( u 2 1 ) x k ] ( 1 / m ) k exp ( j u 3 x k ) ( 1 / m ) k exp [ j ( u 3 + 1 ) x k ] ( 1 / m ) k exp [ j ( u 3 1 ) x k ] ] .
[ a B C ] = H 1 [ ( 1 / m ) k = 1 m f k exp ( j u 1 x k ) ( 1 / m ) k = 1 m f k exp ( j u 2 x k ) ( 1 / m ) k = 1 m f k exp ( j u 3 x k ) ] .
H = [ 1 ( 1 / m ) k exp ( j x k ) ( 1 / m ) k exp ( j x k ) ( 1 / m ) k exp ( j x k ) 1 ( 1 / m ) k exp ( 2 j x k ) ( 1 / m ) k exp ( j x k ) ( 1 / m ) k exp ( 2 j x k ) 1 ] .
H e i = λ i e i ( i = 1 , 3 ) .
H v 0 = f 0 .
H v = f .
v = v 0 + Δ v , f = f 0 + Δ f .
H ( v 0 + Δ v ) = f 0 + Δ f ,
H Δ v = Δ f .
Δ f = 1 / m [ i i exp ( j u 1 x i ) i i exp ( j u 2 x i ) i i exp ( j u 3 x i ) ] t ,
Δ v v 0 λ max λ min Δ f f 0 ,
v 0 = i α i e i ,
Δ v = i Δ α i e i .
H Δ v = H i Δ α i e i = i Δ α i H e i .
H Δ v = i Δ α i λ i e i = Δ f .
Δ α i = Δ f · e i * / λ i ,
Δ v = i fe i * λ i e i .
Δ B = Δ v 2 = i f · e i * λ i e i 2 .
Δ B = i Δ fe i * λ i e i 2 = i e i 2 λ i Δ fe i * ,
Δ B = i e i 2 λ i ( Δ f 1 e i 1 * + Δ f 2 e i 2 * + Δ f 3 e i 3 * ) = i e i 2 λ i n Δ f n e i n * .
Δ f n e i n * = k k exp ( j u n x k ) e i n * = k e i n * exp ( j u n x k ) k = 0.
Δ B = 0.
Δ B = ( 1 / 2 ) exp ( j ϕ ) Δ b j ( b / 2 ) exp ( j ϕ ) Δ ϕ ,
Δ b = 0 , Δ ϕ = 0.
Δ B 2 = i Δ fe i * λ i e i 2 n Δ fe n * λ n e n 2 = i n e i 2 e n 2 λ i λ n ( Δ fe i * ) ( Δ fe n * ) .
( Δ fe i * ) ( Δ fe n * ) = s Δ f s e i s * t Δ f t e n t * = s t e i s * e n t * Δ f s Δ f t .
Δ f s Δ f t = ( 1 / m 2 ) k k exp ( j u s x k ) k k exp ( j u t x k ) = ( 1 / m 2 ) k k exp [ j ( u s x k + u t x k ) ] k k .
Δ f s Δ f t = ( 1 / m 2 ) k exp [ j ( u s + u t ) x k ] k 2 .
Δ B 2 = ( 1 / m 2 ) i n s t k = 1 m e i 2 e n 2 λ i λ n × e i s * e n t * exp [ j ( u s + u t ) x k ] σ 2 .
| Δ B | 2 = ( 1 / m 2 ) i n s t k = 1 m e i 2 e n 2 * λ i λ n × e i s * e n t exp [ j ( u s u t ) x k ] σ 2 .
Δ B 2 = ( 1 / 4 ) exp ( 2 j ϕ ) Δ b 2 ( b 2 / 4 ) exp ( 2 j ϕ ) Δ ϕ 2 ( j b / 2 ) exp ( 2 j ϕ ) Δ b Δ ϕ
| Δ B | 2 = ( 1 / 4 ) Δ b 2 + ( b 2 / 4 ) Δ ϕ 2 .
Δ ϕ 2 = ( 2 / b 2 ) { | Δ B | 2 Re [ exp ( 2 j ϕ ) Δ B 2 ] } .
x k = 2 π k / m
H = [ 1 0 1 0 1 0 0 0 1 ] .
Δ B 2 = α σ 2 / m 2
| Δ B | 2 = β σ 2 / m 2 ,
E = ( Δ ϕ 2 Δ ϕ 2 ) 1 / 2 = 2 1 / 2 σ b m { β Re [ α exp ( 2 j ϕ ) ] } 1 / 2 ,
E = 2 1 / 2 σ γ m m 1 / 2 = 2 1 / 2 σ m 1 / 2 γ .
E n = k = 1 m | f k [ a + B exp ( j x k ) + C exp ( j x k ) ] | 2 exp ( j u n x k )
E n / a = 2 { k = 1 m f k exp ( j u n x k ) a k = 1 m exp ( j u n x k ) B k = 1 m exp [ j ( u n + 1 ) x k ] C k = 1 m exp [ j ( u n 1 ) x k ] } .
a k = 1 m exp ( j u n x k ) + B k = 1 m exp [ j ( u n + 1 ) x k ] + C k = 1 m exp [ j ( u n 1 ) x k ] = k = 1 m f k exp ( j u n x k ) ( n = 1 , 3 ) .
E = k = 1 m | f k [ a + B exp ( j x k ) + C exp ( j x k ) ] | 2 .
E / a = 2 k = 1 m [ f k a B exp ( j x k ) C exp ( j x k ) ] ,
E / B = 2 k = 1 m exp ( j x k ) × [ f k a B exp ( j x k ) C exp ( j x k ) ] ,
E / C = 2 k = 1 m exp ( j x k ) × [ f k a B exp ( j x k ) C exp ( j x k ) ] .
a + 1 / m k exp ( j x k ) B + ( 1 / m ) k exp ( j x k ) C = ( 1 / m ) k f k ,
( 1 / m ) k exp ( j x k ) a + B + ( 1 / m ) k exp ( 2 j x k ) C = ( 1 / m ) k f k exp ( j x k ) ,
( 1 / m ) k exp ( j x k ) a + ( 1 / m ) k exp ( 2 j x k ) B + C = ( 1 / m ) k f k exp ( j x k ) .
Δ v = H 1 Δ f .
Δ B 2 = [ ( 1 / m ) i = 1 3 e i k = 1 m k exp ( j u i x k ) ] 2 = i k i k 1 / m 2 e i e i exp [ j ( u i x k + u i x k ) ] k k = i i k ( 1 / m 2 ) e i e i exp [ j ( u i + u i ) x k ] σ 2 ,
| Δ B | 2 = i i k ( 1 / m 2 ) e i e i * exp [ j ( u i u i ) ] x k σ 2 .

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