Abstract

A coordinate-transform technique is proposed that enables the Fourier-transform method to analyze an interferogram that includes a closed-fringe pattern. First, the closed-fringe pattern is converted to an open-fringe pattern by transformation of the Cartesian coordinate system to a polar coordinate system. Then the phase distribution for the open-fringe interferogram is determined by the conventional Fourier-transform method. The phase distribution for the original closed-fringe pattern is obtained by inverse coordinate transformation from the polar coordinate system back to the Cartesian coordinate system. Computer simulation and experiments were performed for a closed-fringe pattern generated by interference of a spherical wave with a reference plane wave, and results are presented that demonstrate the validity of the proposed technique.

© 2001 Optical Society of America

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References

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    [CrossRef] [PubMed]
  2. See, for example, K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 94–140.
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    [CrossRef]
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    [CrossRef]
  13. T. R. Judge, C.-G. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier-transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
    [CrossRef]
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1997 (1)

1995 (1)

1992 (1)

T. R. Judge, C.-G. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier-transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
[CrossRef]

1989 (1)

Q. S. Ru, N. Ohyama, T. Honda, “Fringe scanning radial shearing interferometry with circular gratings,” Opt. Commun. 69, 189–192 (1989).
[CrossRef]

1988 (1)

1987 (2)

1986 (2)

1985 (1)

1983 (1)

1982 (1)

1974 (1)

Bachor, H.-A.

Bone, D. J.

Brangaccio, D. J.

Bruning, J. H.

Bryanston-Cross, P. J.

T. R. Judge, C.-G. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier-transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
[CrossRef]

Cheng, Y. Y.

Creath, K.

See, for example, K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 94–140.

Eiju, T.

Gallager, J. E.

Groot, P. D.

Hariharan, P.

Herriott, D. R.

Hilaire, T. P.

Honda, T.

Q. S. Ru, N. Ohyama, T. Honda, “Fringe scanning radial shearing interferometry with circular gratings,” Opt. Commun. 69, 189–192 (1989).
[CrossRef]

Ina, H.

Judge, T. R.

T. R. Judge, C.-G. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier-transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
[CrossRef]

Kinnstätter, K.

Kobayashi, S.

Kreis, T.

Lohmann, A. W.

Mutoh, K.

Nagy, G.

Ohyama, N.

Q. S. Ru, N. Ohyama, T. Honda, “Fringe scanning radial shearing interferometry with circular gratings,” Opt. Commun. 69, 189–192 (1989).
[CrossRef]

Oreb, B. F.

Pegna, J.

Quan, C.-G.

T. R. Judge, C.-G. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier-transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
[CrossRef]

Roddier, C.

Roddier, F.

Rosenfel, D. P.

Ru, Q. S.

Q. S. Ru, N. Ohyama, T. Honda, “Fringe scanning radial shearing interferometry with circular gratings,” Opt. Commun. 69, 189–192 (1989).
[CrossRef]

Sandeman, R. J.

Schwider, J.

K. Kinnstätter, A. W. Lohmann, J. Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
[CrossRef]

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1990), Vol. 28, pp. 273–359.

Streibl, N.

Takeda, M.

White, A. D.

Wyant, J. C.

Appl. Opt. (9)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

Q. S. Ru, N. Ohyama, T. Honda, “Fringe scanning radial shearing interferometry with circular gratings,” Opt. Commun. 69, 189–192 (1989).
[CrossRef]

Opt. Eng. (1)

T. R. Judge, C.-G. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier-transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
[CrossRef]

Other (2)

See, for example, K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 94–140.

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1990), Vol. 28, pp. 273–359.

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Figures (22)

Fig. 1
Fig. 1

Spherical wave front.

Fig. 2
Fig. 2

Simulated interferogram with closed fringes (moiré patterns are artifacts caused by the printer).

Fig. 3
Fig. 3

Fourier spectrum of the closed-fringe pattern (moiré patterns are artifacts caused by the printer).

Fig. 4
Fig. 4

Wave front obtained from the closed-fringe pattern by the conventional Fourier-transform method.

Fig. 5
Fig. 5

Coordinate transforms from xy space to r–θ space.

Fig. 6
Fig. 6

Transformed interferogram in r–θ space for the origin of the polar coordinate at point A (256, 256).

Fig. 7
Fig. 7

(a) Fourier spectrum of the transformed interferogram. (b) Sectional profile of Fourier spectrum shown in Fig. 7(a).

Fig. 8
Fig. 8

Spherical wave front obtained by the proposed technique.

Fig. 9
Fig. 9

Numerical errors in fringe analysis.

Fig. 10
Fig. 10

Sectional profile of numerical errors.

Fig. 11
Fig. 11

Interferogram obtained by experiment.

Fig. 12
Fig. 12

Interferogram in r–θ space.

Fig. 13
Fig. 13

Phase distribution in r–θ space obtained by the proposed method.

Fig. 14
Fig. 14

Result obtained by inverse coordinate transform.

Fig. 15
Fig. 15

Interferogram in r–θ space for the origin of the polar coordinate at point B (240, 240).

Fig. 16
Fig. 16

Phase distribution obtained by inverse coordinate transform from the interferogram shown in Fig. 15.

Fig. 17
Fig. 17

Difference between the phase distributions obtained for the different origins of the polar coordinate system.

Fig. 18
Fig. 18

Sectional profiles of the phase distributions and their difference.

Fig. 19
Fig. 19

Phase distribution in r–θ space.

Fig. 20
Fig. 20

Final result obtained by inverse coordinate transform.

Fig. 21
Fig. 21

Numerical errors in fringe analysis for a circular area that circumscribes the original square interferogram.

Fig. 22
Fig. 22

Wave front that cannot be recovered by the proposed method.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

gx, y=ax, y+bx, ycos2πfxx+2πfyy+ϕx, y,
gx, y=ax, y+cx, yexpi2πfxx+2πfyy+c*x, yexp-i2πfxx+2πfyy,
cx, y=bx, yexpiϕx, y2,
Gη, ζ=Aη, ζ+Cη-fx, ζ-fy+C*fx-η, fy-ζ,
ϕx, y=tan-1Recx, yImcx, y,
Φx, y=2πfxx+ϕx, y, Fx=12πx Φx, y=fx+12πx ϕx, y.
fx+12πx ϕx, yxD>0,
fx+12πx ϕx, yxD<0,
x=X+r cos θy=Y+r sin θ,

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