Abstract

A method for accurate phase determination in holographic interferometry using a one- or two-dimensional Fourier transform is described. The method calculates the interference phase pointwise, even between fringe extrema, and thus has advantages over conventional fringe-finding and -tracking methods. Only one interference pattern may be used, although the use of two patterns reconstructed with a mutual phase shift permits an easier phase unwrapping and determination of nonmonotonic fringe-order variations. Additionally, the method offers a means for filtering out disturbances such as speckle noise and background variations.

© 1986 Optical Society of America

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  1. R. J. Pryputniewicz, K. A. Stetson, “Holographic strain analysis: extension of fringe vector method to include perspective,” Appl. Opt. 15, 725–728 (1976).
    [CrossRef] [PubMed]
  2. R. J. Pryputniewicz, “Holographic strain analysis: an experimental implementation of the fringe vector theory,” Appl. Opt. 17, 3613–3618 (1978).
    [CrossRef] [PubMed]
  3. H. Kreitlow, Th. Kreis, “Entwicklung eines Gerätesystems zur automatisierten statischen und dynamischen Auswertung holografischer Interferenzmuster,” in Proceedings of the Laser ’79 Optoelectronics Conference (IPC Science and Technology, Guildford, UK, 1979), pp. 426–436.
  4. W. Jüptner, “Automatisierte Auswertung holografischer Interferogramme mit dem Zeilen-Scan-Verfahren,” presented at the DPG/DGaO-Frühjahrsschule, Holografische Interferometrie in Technik und Medizin, Hannover, April 4–7, 1978).
  5. W. Jüptner, Th. Kreis, H. Kreitlow, “Automatic evaluation of holographic interferograms by reference beam phase shifting,” Proc. Soc. Photo-Opt. Instrum. Eng. 398, 22–29 (1983).
  6. Th. Kreis, B. Fischer, W. Jüptner, G. Sepold, “Automatisierte Auswertung holografischer Interferenzmuster bei der Untersuchung von Zugproben,” in Proceedings of the Laser ’81 Optoelectronics Conference (Springer-Verlag, Berlin, 1981), pp. 105–110.
  7. P. Hariharan, B. F. Oreb, N. Brown, “Real-time holographic interferometry: a microcomputer system for the measurement of vector displacement,” Appl. Opt. 22, 876–880 (1983).
    [CrossRef] [PubMed]
  8. R. Dändliker, R. Thalmann, J.-F. Willemin, “Fringe interpolation by two-reference-beam holographic interferometry: reducing sensitivity to hologram misalignment,” Opt. Commun. 42, 301–306 (1982).
    [CrossRef]
  9. J. B. Schemm, C. M. Vest, “Fringe pattern recognition and interpolation using nonlinear regression analysis,” Appl. Opt. 22, 2850–2853 (1983).
    [CrossRef] [PubMed]
  10. Th. Kreis, H. Kreitlow, “Quantitative evaluation of holographic interference patterns under image processing aspects,” Proc. Soc. Photo-Opt. Instrum. Eng. 210, 196–202 (1979).
  11. Th. Kreis, H. Kreitlow, “Digital processing of holographic interference patterns,” in Digest of Topical Meeting on Holographic Interferometry and Speckle Metrology (Optical Society of America, Washington, D.C., 1980).
  12. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,”J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  13. W. W. Macy, “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22, 3898–3901 (1983).
    [CrossRef] [PubMed]
  14. K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
    [CrossRef]
  15. W. Jüptner, K. Ringer, H. Welling, “Auswertung von Interferenzstreifensystemen bei holografischer Translations- und Dehnungsmessung,” Optik 38, 437–448 (1973).

1984 (1)

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

1983 (4)

1982 (2)

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,”J. Opt. Soc. Am. 72, 156–160 (1982).
[CrossRef]

R. Dändliker, R. Thalmann, J.-F. Willemin, “Fringe interpolation by two-reference-beam holographic interferometry: reducing sensitivity to hologram misalignment,” Opt. Commun. 42, 301–306 (1982).
[CrossRef]

1979 (1)

Th. Kreis, H. Kreitlow, “Quantitative evaluation of holographic interference patterns under image processing aspects,” Proc. Soc. Photo-Opt. Instrum. Eng. 210, 196–202 (1979).

1978 (1)

1976 (1)

1973 (1)

W. Jüptner, K. Ringer, H. Welling, “Auswertung von Interferenzstreifensystemen bei holografischer Translations- und Dehnungsmessung,” Optik 38, 437–448 (1973).

Brown, N.

Dändliker, R.

R. Dändliker, R. Thalmann, J.-F. Willemin, “Fringe interpolation by two-reference-beam holographic interferometry: reducing sensitivity to hologram misalignment,” Opt. Commun. 42, 301–306 (1982).
[CrossRef]

Fischer, B.

Th. Kreis, B. Fischer, W. Jüptner, G. Sepold, “Automatisierte Auswertung holografischer Interferenzmuster bei der Untersuchung von Zugproben,” in Proceedings of the Laser ’81 Optoelectronics Conference (Springer-Verlag, Berlin, 1981), pp. 105–110.

Hariharan, P.

Ina, H.

Jüptner, W.

W. Jüptner, Th. Kreis, H. Kreitlow, “Automatic evaluation of holographic interferograms by reference beam phase shifting,” Proc. Soc. Photo-Opt. Instrum. Eng. 398, 22–29 (1983).

W. Jüptner, K. Ringer, H. Welling, “Auswertung von Interferenzstreifensystemen bei holografischer Translations- und Dehnungsmessung,” Optik 38, 437–448 (1973).

Th. Kreis, B. Fischer, W. Jüptner, G. Sepold, “Automatisierte Auswertung holografischer Interferenzmuster bei der Untersuchung von Zugproben,” in Proceedings of the Laser ’81 Optoelectronics Conference (Springer-Verlag, Berlin, 1981), pp. 105–110.

W. Jüptner, “Automatisierte Auswertung holografischer Interferogramme mit dem Zeilen-Scan-Verfahren,” presented at the DPG/DGaO-Frühjahrsschule, Holografische Interferometrie in Technik und Medizin, Hannover, April 4–7, 1978).

Kobayashi, S.

Kreis, Th.

W. Jüptner, Th. Kreis, H. Kreitlow, “Automatic evaluation of holographic interferograms by reference beam phase shifting,” Proc. Soc. Photo-Opt. Instrum. Eng. 398, 22–29 (1983).

Th. Kreis, H. Kreitlow, “Quantitative evaluation of holographic interference patterns under image processing aspects,” Proc. Soc. Photo-Opt. Instrum. Eng. 210, 196–202 (1979).

H. Kreitlow, Th. Kreis, “Entwicklung eines Gerätesystems zur automatisierten statischen und dynamischen Auswertung holografischer Interferenzmuster,” in Proceedings of the Laser ’79 Optoelectronics Conference (IPC Science and Technology, Guildford, UK, 1979), pp. 426–436.

Th. Kreis, H. Kreitlow, “Digital processing of holographic interference patterns,” in Digest of Topical Meeting on Holographic Interferometry and Speckle Metrology (Optical Society of America, Washington, D.C., 1980).

Th. Kreis, B. Fischer, W. Jüptner, G. Sepold, “Automatisierte Auswertung holografischer Interferenzmuster bei der Untersuchung von Zugproben,” in Proceedings of the Laser ’81 Optoelectronics Conference (Springer-Verlag, Berlin, 1981), pp. 105–110.

Kreitlow, H.

W. Jüptner, Th. Kreis, H. Kreitlow, “Automatic evaluation of holographic interferograms by reference beam phase shifting,” Proc. Soc. Photo-Opt. Instrum. Eng. 398, 22–29 (1983).

Th. Kreis, H. Kreitlow, “Quantitative evaluation of holographic interference patterns under image processing aspects,” Proc. Soc. Photo-Opt. Instrum. Eng. 210, 196–202 (1979).

Th. Kreis, H. Kreitlow, “Digital processing of holographic interference patterns,” in Digest of Topical Meeting on Holographic Interferometry and Speckle Metrology (Optical Society of America, Washington, D.C., 1980).

H. Kreitlow, Th. Kreis, “Entwicklung eines Gerätesystems zur automatisierten statischen und dynamischen Auswertung holografischer Interferenzmuster,” in Proceedings of the Laser ’79 Optoelectronics Conference (IPC Science and Technology, Guildford, UK, 1979), pp. 426–436.

Macy, W. W.

Oreb, B. F.

Pryputniewicz, R. J.

Ringer, K.

W. Jüptner, K. Ringer, H. Welling, “Auswertung von Interferenzstreifensystemen bei holografischer Translations- und Dehnungsmessung,” Optik 38, 437–448 (1973).

Schemm, J. B.

Sepold, G.

Th. Kreis, B. Fischer, W. Jüptner, G. Sepold, “Automatisierte Auswertung holografischer Interferenzmuster bei der Untersuchung von Zugproben,” in Proceedings of the Laser ’81 Optoelectronics Conference (Springer-Verlag, Berlin, 1981), pp. 105–110.

Stetson, K. A.

Takeda, M.

Thalmann, R.

R. Dändliker, R. Thalmann, J.-F. Willemin, “Fringe interpolation by two-reference-beam holographic interferometry: reducing sensitivity to hologram misalignment,” Opt. Commun. 42, 301–306 (1982).
[CrossRef]

Vest, C. M.

Welling, H.

W. Jüptner, K. Ringer, H. Welling, “Auswertung von Interferenzstreifensystemen bei holografischer Translations- und Dehnungsmessung,” Optik 38, 437–448 (1973).

Willemin, J.-F.

R. Dändliker, R. Thalmann, J.-F. Willemin, “Fringe interpolation by two-reference-beam holographic interferometry: reducing sensitivity to hologram misalignment,” Opt. Commun. 42, 301–306 (1982).
[CrossRef]

Womack, K. H.

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

Appl. Opt. (5)

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

R. Dändliker, R. Thalmann, J.-F. Willemin, “Fringe interpolation by two-reference-beam holographic interferometry: reducing sensitivity to hologram misalignment,” Opt. Commun. 42, 301–306 (1982).
[CrossRef]

Opt. Eng. (1)

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

Optik (1)

W. Jüptner, K. Ringer, H. Welling, “Auswertung von Interferenzstreifensystemen bei holografischer Translations- und Dehnungsmessung,” Optik 38, 437–448 (1973).

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

Th. Kreis, H. Kreitlow, “Quantitative evaluation of holographic interference patterns under image processing aspects,” Proc. Soc. Photo-Opt. Instrum. Eng. 210, 196–202 (1979).

W. Jüptner, Th. Kreis, H. Kreitlow, “Automatic evaluation of holographic interferograms by reference beam phase shifting,” Proc. Soc. Photo-Opt. Instrum. Eng. 398, 22–29 (1983).

Other (4)

Th. Kreis, B. Fischer, W. Jüptner, G. Sepold, “Automatisierte Auswertung holografischer Interferenzmuster bei der Untersuchung von Zugproben,” in Proceedings of the Laser ’81 Optoelectronics Conference (Springer-Verlag, Berlin, 1981), pp. 105–110.

H. Kreitlow, Th. Kreis, “Entwicklung eines Gerätesystems zur automatisierten statischen und dynamischen Auswertung holografischer Interferenzmuster,” in Proceedings of the Laser ’79 Optoelectronics Conference (IPC Science and Technology, Guildford, UK, 1979), pp. 426–436.

W. Jüptner, “Automatisierte Auswertung holografischer Interferogramme mit dem Zeilen-Scan-Verfahren,” presented at the DPG/DGaO-Frühjahrsschule, Holografische Interferometrie in Technik und Medizin, Hannover, April 4–7, 1978).

Th. Kreis, H. Kreitlow, “Digital processing of holographic interference patterns,” in Digest of Topical Meeting on Holographic Interferometry and Speckle Metrology (Optical Society of America, Washington, D.C., 1980).

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Figures (10)

Fig. 1
Fig. 1

Set of phase distributions stemming from the same intensity distribution in holographic interferometry.

Fig. 2
Fig. 2

Interference-phase determination: a, intensity distribution; b, amplitude spectrum; c, amplitude spectrum after bandpass filtering; d, interference phase modulo 2π.

Fig. 3
Fig. 3

Image enhancement by filtering: a, intensity distribution; b, amplitude spectrum; c, filtered amplitude spectrum; d, interference phase modulo 2π; e, intensity distribution according to phase of d.

Fig. 4
Fig. 4

Interference-phase determination: a, nonmonotonic phase distribution; b, intensity distribution; c, amplitude spectrum; d, filtered amplitude spectrum; e, interference phase modulo 2π; f, continuous interference phase.

Fig. 5
Fig. 5

Interference-phase determination: a, intensity distribution; b, interference phase modulo 2π; c, continuous interference phase; d, phase shift calculated from Figs. 4b and 5a; e, sign-corrected phase modulo 2π; f, sign-corrected continuous interference phase.

Fig. 6
Fig. 6

Simulated test interferogram.

Fig. 7
Fig. 7

Two-dimensional amplitude spectrum.

Fig. 8
Fig. 8

Bandpass filters with passband in a, +u half-plane; b, +v half-plane.

Fig. 9
Fig. 9

Interference-phase distributions modulo 2π determined with bandpass filters of a, Fig. 8a and b, Fig. 8b.

Fig. 10
Fig. 10

Plot of continuous interference-phase distribution.

Equations (14)

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i ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) ] ,
cos ϕ = cos ( s ϕ + 2 π n ) ,             n Z ,             s { - 1 , + 1 } .
i ( x , y ) = a ( x , y ) + c ( x , y ) + c * ( x , y ) ,
c ( x , y ) = ½ b ( x , y ) exp [ j ϕ ( x , y ) ] ,
i ( u , y ) = A ( u , y ) + C ( u , y ) + C * ( u , y ) ;
I ( x , v ) = A ( x , v ) + C ( x , v ) + C * ( x , v ) ;
I ( u , v ) = A ( u , v ) + C ( u , v ) + C * ( u , v ) .
ϕ ( x , y ) = arctan Im [ c ( x , y ) ] Re [ c ( x , y ) ] ,
n ( x 1 ) = 0 , n ( x i ) = { n ( x i - 1 ) if ϕ ( x i ) - ϕ ( x i - 1 ) < π n ( x i - 1 ) + 1 if ϕ ( x i ) - ϕ ( x i - 1 ) π ,             i = 2 , 3 , , ϕ contin ( x i ) = ϕ ( x i ) + 2 π n ( x i ) ,             i = 1 , 2 , .
i 1 ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) ] , i 2 ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) + α ] .
c 1 ( x , y ) = ½ b ( x , y ) exp [ j ϕ ( x , y ) ] , c 2 ( x , y ) = ½ b ( x , y ) exp [ j ϕ ( x , y ) + j α ( x , y ) ] .
α ( x , y ) = arctan Re c 1 Im c 2 - Im c 1 Re c 2 Re c 1 Re c 2 + Im c 1 Im c 2 .
n ( x 1 ) = 0 , n ( x i ) = { n ( x i - 1 ) if ϕ ( x i ) - ϕ ( x i - 1 ) < π n ( x i - 1 ) + 1 if ϕ ( x i ) - ϕ ( x i - 1 ) - π n ( x i - 1 ) - 1 if ϕ ( x i ) - ϕ ( x i - 1 ) π ,             i = 2 , 3 , , ϕ contin ( x i ) = ϕ ( x i ) + 2 π n ( x i ) ,             i = 1 , 2 , .
n ( x 1 , y 1 ) = 0 , n ( x 1 , y i ) = { n ( x i , y i - 1 ) if ϕ ( x 1 , y i ) - ϕ ( x 1 , y i - 1 ) π n ( x 1 , y i - 1 ) + 1 if ϕ ( x 1 , y i ) - ϕ ( x 1 , y i - 1 ) - π n ( x 1 , y i - 1 ) - 1 if ϕ ( x 1 , y 1 ) - ϕ ( x 1 , y i - 1 ) π ,             i = 2 , 3 , , n ( x j , y i ) = { n ( x j - 1 , y i ) if ϕ ( x j , y i ) - ϕ ( x j - 1 , y i ) < π n ( x j - 1 , y i ) + 1 if ϕ ( x j , y i ) - ϕ ( x j - 1 , y i ) - π n ( x j - 1 , y i ) - 1 if ϕ ( x j , y i ) - ϕ ( x j - 1 , y i ) π             j = 2 , 3 , , ϕ contin ( x j , y j ) = ϕ ( x j , y i ) + 2 π n ( x j , y i ) ,             i , j = 1 , 2 , .

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