Abstract

Phase-stepping techniques for the measurement of the difference and sum of phases in holographic-moiré-based interferometers are described. The phase-recovering algorithms for the extraction of this phase information is relatively insensitive to the inaccuracies of the phase-shifting device. This development promises not only to extend the domain of measurements but also to improve significantly the accuracy that can be achieved in deformation analysis and nondestructive testing of diffuse objects.

© 1993 Optical Society of America

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References

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  1. P. Hariharan, B. F. Oreb, N. Brown, “A digital phase-measurement system for real-time holographic interferometry,” Opt. Commun. 41, 393–396 (1982).
    [CrossRef]
  2. P. Hariharan, “Quasi-heterodyne holographic interferometry,” Opt. Eng. 24, 632–638 (1985).
  3. P. Hariharan, B. F. Oreb, “Stroboscopic holographic interferometry: application of digital techniques,” Opt. Commun. 59, 83–86 (1986).
    [CrossRef]
  4. P. Hariharan, B. F. Oreb, “Two index holographic contouring: application of digital techniques,” Opt. Commun. 51, 142–144 (1984).
    [CrossRef]
  5. K. Creath, “Phase-measurement interferometry,” in Progress in Optics XXVI, E. Wolf, ed. (North-Holland, Amsterdam, 1987), pp. 349–393.
  6. K. Creath, “Holographic contour and deformation measurement using a 1.4 million element detector array,” Appl. Opt. 28, 2170–2175 (1989).
    [CrossRef] [PubMed]
  7. P. K. Rastogi, L. Pflug, “A holographic technique featuring broad range sensitivity to contour diffuse objects,” J. Mod. Opt. 38, 1673–1683 (1991).
    [CrossRef]
  8. P. K. Rastogi, E. Denarié, “Visualization of in-plane displacement fields using phase shifting holographic moiré: application to crack detection and propagation,” Appl. Opt. 31, 2402–2404 (1992).
    [CrossRef] [PubMed]
  9. P. K. Rastogi, M. Barillot, G. Kaufmann, “Comparative phase shifting holographic interferometry,” Appl. Opt. 30, 722–728 (1991).
    [CrossRef] [PubMed]
  10. P. K. Rastogi, “Visualization and measurement of slope and curvature fields using holographic interferometry: an application to flaw detection,” J. Mod. Opt. 38, 1251–1263 (1991).
    [CrossRef]
  11. P. K. Rastogi, “Phase shifting four wave holographic interferometry,” J. Mod. Opt. 39, 677–680 (1992).
    [CrossRef]
  12. P. K. Rastogi, “Phase shifting applied to four-wave holographic interferometers,” Appl. Opt. 31, 1680–1681 (1992).
    [CrossRef] [PubMed]
  13. C. Joenathan, B. Pfister, H. J. Tiziani, “Contouring by electronic speckle pattern interferometry employing dual beam illumination,” Appl. Opt. 29, 1905–1911 (1990).
    [CrossRef] [PubMed]
  14. K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24, 3053–3058 (1985).
    [CrossRef] [PubMed]
  15. A. A. M. Maas, H. A. Vrooman, “In-plane strain measurement by digital phase shifting speckle interferometry,” in Laser Interferometry: Quantitative Analysis of Interferograms: Third in a Series, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1162, 248–256 (1989).
  16. F. M. Santoyo, M. C. Shellabear, J. R. Tyrer, “Whole field in-plane vibration analysis using pulsed phase-stepped ESPI,” Appl. Opt. 30, 717–721 (1991).
    [CrossRef] [PubMed]
  17. C. Joenathan, “Vibration fringes by phase stepping on an electronic speckle pattern interferomoter: an analysis,” Appl. Opt. 30, 4658–4665 (1991).
    [CrossRef] [PubMed]
  18. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  19. P. K. Rastogi, M. Spajer, J. Monneret, “In-plane deformation measurement using holographic moiré,” Opt. Lasers Eng. 2, 79–103 (1981).
    [CrossRef]
  20. C. A. Sciammarella, P. K. Rastogi, P. Jacquot, R. Narayanan, “Holographic moiré in real time,” Exp. Mech. 22, 52–63 (1982).
    [CrossRef]
  21. K. Kinnstaetter, A. W. Lohmann, J. Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
    [CrossRef] [PubMed]

1992 (3)

1991 (5)

1990 (1)

1989 (1)

1988 (1)

1987 (1)

1986 (1)

P. Hariharan, B. F. Oreb, “Stroboscopic holographic interferometry: application of digital techniques,” Opt. Commun. 59, 83–86 (1986).
[CrossRef]

1985 (2)

P. Hariharan, “Quasi-heterodyne holographic interferometry,” Opt. Eng. 24, 632–638 (1985).

K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24, 3053–3058 (1985).
[CrossRef] [PubMed]

1984 (1)

P. Hariharan, B. F. Oreb, “Two index holographic contouring: application of digital techniques,” Opt. Commun. 51, 142–144 (1984).
[CrossRef]

1982 (2)

P. Hariharan, B. F. Oreb, N. Brown, “A digital phase-measurement system for real-time holographic interferometry,” Opt. Commun. 41, 393–396 (1982).
[CrossRef]

C. A. Sciammarella, P. K. Rastogi, P. Jacquot, R. Narayanan, “Holographic moiré in real time,” Exp. Mech. 22, 52–63 (1982).
[CrossRef]

1981 (1)

P. K. Rastogi, M. Spajer, J. Monneret, “In-plane deformation measurement using holographic moiré,” Opt. Lasers Eng. 2, 79–103 (1981).
[CrossRef]

Barillot, M.

Brown, N.

P. Hariharan, B. F. Oreb, N. Brown, “A digital phase-measurement system for real-time holographic interferometry,” Opt. Commun. 41, 393–396 (1982).
[CrossRef]

Creath, K.

Denarié, E.

Eiju, T.

Hariharan, P.

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
[CrossRef] [PubMed]

P. Hariharan, B. F. Oreb, “Stroboscopic holographic interferometry: application of digital techniques,” Opt. Commun. 59, 83–86 (1986).
[CrossRef]

P. Hariharan, “Quasi-heterodyne holographic interferometry,” Opt. Eng. 24, 632–638 (1985).

P. Hariharan, B. F. Oreb, “Two index holographic contouring: application of digital techniques,” Opt. Commun. 51, 142–144 (1984).
[CrossRef]

P. Hariharan, B. F. Oreb, N. Brown, “A digital phase-measurement system for real-time holographic interferometry,” Opt. Commun. 41, 393–396 (1982).
[CrossRef]

Jacquot, P.

C. A. Sciammarella, P. K. Rastogi, P. Jacquot, R. Narayanan, “Holographic moiré in real time,” Exp. Mech. 22, 52–63 (1982).
[CrossRef]

Joenathan, C.

Kaufmann, G.

Kinnstaetter, K.

Lohmann, A. W.

Maas, A. A. M.

A. A. M. Maas, H. A. Vrooman, “In-plane strain measurement by digital phase shifting speckle interferometry,” in Laser Interferometry: Quantitative Analysis of Interferograms: Third in a Series, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1162, 248–256 (1989).

Monneret, J.

P. K. Rastogi, M. Spajer, J. Monneret, “In-plane deformation measurement using holographic moiré,” Opt. Lasers Eng. 2, 79–103 (1981).
[CrossRef]

Narayanan, R.

C. A. Sciammarella, P. K. Rastogi, P. Jacquot, R. Narayanan, “Holographic moiré in real time,” Exp. Mech. 22, 52–63 (1982).
[CrossRef]

Oreb, B. F.

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
[CrossRef] [PubMed]

P. Hariharan, B. F. Oreb, “Stroboscopic holographic interferometry: application of digital techniques,” Opt. Commun. 59, 83–86 (1986).
[CrossRef]

P. Hariharan, B. F. Oreb, “Two index holographic contouring: application of digital techniques,” Opt. Commun. 51, 142–144 (1984).
[CrossRef]

P. Hariharan, B. F. Oreb, N. Brown, “A digital phase-measurement system for real-time holographic interferometry,” Opt. Commun. 41, 393–396 (1982).
[CrossRef]

Pfister, B.

Pflug, L.

P. K. Rastogi, L. Pflug, “A holographic technique featuring broad range sensitivity to contour diffuse objects,” J. Mod. Opt. 38, 1673–1683 (1991).
[CrossRef]

Rastogi, P. K.

P. K. Rastogi, E. Denarié, “Visualization of in-plane displacement fields using phase shifting holographic moiré: application to crack detection and propagation,” Appl. Opt. 31, 2402–2404 (1992).
[CrossRef] [PubMed]

P. K. Rastogi, “Phase shifting four wave holographic interferometry,” J. Mod. Opt. 39, 677–680 (1992).
[CrossRef]

P. K. Rastogi, “Phase shifting applied to four-wave holographic interferometers,” Appl. Opt. 31, 1680–1681 (1992).
[CrossRef] [PubMed]

P. K. Rastogi, L. Pflug, “A holographic technique featuring broad range sensitivity to contour diffuse objects,” J. Mod. Opt. 38, 1673–1683 (1991).
[CrossRef]

P. K. Rastogi, “Visualization and measurement of slope and curvature fields using holographic interferometry: an application to flaw detection,” J. Mod. Opt. 38, 1251–1263 (1991).
[CrossRef]

P. K. Rastogi, M. Barillot, G. Kaufmann, “Comparative phase shifting holographic interferometry,” Appl. Opt. 30, 722–728 (1991).
[CrossRef] [PubMed]

C. A. Sciammarella, P. K. Rastogi, P. Jacquot, R. Narayanan, “Holographic moiré in real time,” Exp. Mech. 22, 52–63 (1982).
[CrossRef]

P. K. Rastogi, M. Spajer, J. Monneret, “In-plane deformation measurement using holographic moiré,” Opt. Lasers Eng. 2, 79–103 (1981).
[CrossRef]

Santoyo, F. M.

Schwider, J.

Sciammarella, C. A.

C. A. Sciammarella, P. K. Rastogi, P. Jacquot, R. Narayanan, “Holographic moiré in real time,” Exp. Mech. 22, 52–63 (1982).
[CrossRef]

Shellabear, M. C.

Spajer, M.

P. K. Rastogi, M. Spajer, J. Monneret, “In-plane deformation measurement using holographic moiré,” Opt. Lasers Eng. 2, 79–103 (1981).
[CrossRef]

Streibl, N.

Tiziani, H. J.

Tyrer, J. R.

Vrooman, H. A.

A. A. M. Maas, H. A. Vrooman, “In-plane strain measurement by digital phase shifting speckle interferometry,” in Laser Interferometry: Quantitative Analysis of Interferograms: Third in a Series, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1162, 248–256 (1989).

Appl. Opt. (10)

P. K. Rastogi, E. Denarié, “Visualization of in-plane displacement fields using phase shifting holographic moiré: application to crack detection and propagation,” Appl. Opt. 31, 2402–2404 (1992).
[CrossRef] [PubMed]

P. K. Rastogi, M. Barillot, G. Kaufmann, “Comparative phase shifting holographic interferometry,” Appl. Opt. 30, 722–728 (1991).
[CrossRef] [PubMed]

P. K. Rastogi, “Phase shifting applied to four-wave holographic interferometers,” Appl. Opt. 31, 1680–1681 (1992).
[CrossRef] [PubMed]

C. Joenathan, B. Pfister, H. J. Tiziani, “Contouring by electronic speckle pattern interferometry employing dual beam illumination,” Appl. Opt. 29, 1905–1911 (1990).
[CrossRef] [PubMed]

K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24, 3053–3058 (1985).
[CrossRef] [PubMed]

F. M. Santoyo, M. C. Shellabear, J. R. Tyrer, “Whole field in-plane vibration analysis using pulsed phase-stepped ESPI,” Appl. Opt. 30, 717–721 (1991).
[CrossRef] [PubMed]

C. Joenathan, “Vibration fringes by phase stepping on an electronic speckle pattern interferomoter: an analysis,” Appl. Opt. 30, 4658–4665 (1991).
[CrossRef] [PubMed]

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
[CrossRef] [PubMed]

K. Creath, “Holographic contour and deformation measurement using a 1.4 million element detector array,” Appl. Opt. 28, 2170–2175 (1989).
[CrossRef] [PubMed]

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

Exp. Mech. (1)

C. A. Sciammarella, P. K. Rastogi, P. Jacquot, R. Narayanan, “Holographic moiré in real time,” Exp. Mech. 22, 52–63 (1982).
[CrossRef]

J. Mod. Opt. (3)

P. K. Rastogi, L. Pflug, “A holographic technique featuring broad range sensitivity to contour diffuse objects,” J. Mod. Opt. 38, 1673–1683 (1991).
[CrossRef]

P. K. Rastogi, “Visualization and measurement of slope and curvature fields using holographic interferometry: an application to flaw detection,” J. Mod. Opt. 38, 1251–1263 (1991).
[CrossRef]

P. K. Rastogi, “Phase shifting four wave holographic interferometry,” J. Mod. Opt. 39, 677–680 (1992).
[CrossRef]

Opt. Commun. (3)

P. Hariharan, B. F. Oreb, N. Brown, “A digital phase-measurement system for real-time holographic interferometry,” Opt. Commun. 41, 393–396 (1982).
[CrossRef]

P. Hariharan, B. F. Oreb, “Stroboscopic holographic interferometry: application of digital techniques,” Opt. Commun. 59, 83–86 (1986).
[CrossRef]

P. Hariharan, B. F. Oreb, “Two index holographic contouring: application of digital techniques,” Opt. Commun. 51, 142–144 (1984).
[CrossRef]

Opt. Eng. (1)

P. Hariharan, “Quasi-heterodyne holographic interferometry,” Opt. Eng. 24, 632–638 (1985).

Opt. Lasers Eng. (1)

P. K. Rastogi, M. Spajer, J. Monneret, “In-plane deformation measurement using holographic moiré,” Opt. Lasers Eng. 2, 79–103 (1981).
[CrossRef]

Other (2)

A. A. M. Maas, H. A. Vrooman, “In-plane strain measurement by digital phase shifting speckle interferometry,” in Laser Interferometry: Quantitative Analysis of Interferograms: Third in a Series, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1162, 248–256 (1989).

K. Creath, “Phase-measurement interferometry,” in Progress in Optics XXVI, E. Wolf, ed. (North-Holland, Amsterdam, 1987), pp. 349–393.

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Figures (5)

Fig. 1
Fig. 1

Schematic of the phase-shifting holographic moiré arrangement for the measurement of out-of-plane and in-plane displacements.

Fig. 2
Fig. 2

Assumed form of phase functions (a) φ1 and (b) φ2.

Fig. 3
Fig. 3

a, Plot of the phase function ψ (or φ1 + φ2). b, Modulo 2π phase ψ obtained by phase-shifting holographic moiré.

Fig. 4
Fig. 4

a, Plot of the phase function φ (or, φ1 − φ2). b, Modulo 2π phase φ obtained by phase-shifting holographic moiré.

Fig. 5
Fig. 5

Plots of phase errors versus ψ for the inaccuracies of 2° and 5° introduced in the phase step values assumed to be initially adjusted at 90° and 128.173°, respectively. The dashed and solid curves correspond to the algorithms shown in Eqs. (11) and (13), respectively. The maximum phase errors introduced when using algorithm (13) are ~3 times higher than those obtained when using algorithm (11). Analogous phase error plots are obtained for the measurement of φ.

Equations (52)

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A 1 = A 1 exp ( i φ 1 ) A 2 = A 2 exp ( i φ 2 ) ,
I ( P ) = ( A + A ) ( A * + A * ) .
A 1 A 2 * = A 1 A 2 * = A 1 A 2 * = A 1 A 2 * = A 2 A 1 * = A 2 A 1 * = A 2 A 1 * = A 2 A 1 * = 0 | A 1 | 2 = | A 1 | 2 = I 1 ; | A 2 | 2 = | A 2 | 2 = I 2 .
I ( P ) = 2 ( I 1 + I 2 + I 1 V 1 cos φ 1 + I 2 V 2 cos φ 2 ) .
I ( P ) = 4 I 1 { 1 + ( V / 2 ) [ ( cos φ 1 ) + ( cos φ 2 ) ] } .
I ( P ) = I 0 [ 1 + V 2 cos ( φ 1 φ 2 2 ) cos ( φ 1 + φ 2 2 ) ] .
φ ( P ) = φ 1 ( P ) φ 2 ( P ) 2 = s · K 2 K 1 2 ,
ψ ( P ) = φ 1 ( P ) + φ 2 ( P ) 2 = s · ( K 0 K 1 + K 2 2 ) .
φ ( P ) = 2 π s x ( P ) sin θ λ
ψ ( P ) = 2 π λ s z ( P ) ( 1 + cos θ ) ,
I 1 = I 0 ( P ) ( 1 + ( V / 2 ) { cos [ φ 1 ( P ) 2 α ] + cos [ φ 2 ( P ) 2 α ] } ) ,
I 2 = I 0 ( P ) ( 1 + ( V / 2 ) { cos [ φ 1 ( P ) α ] + cos [ φ 2 ( P ) α ] } ) ,
I 3 = I 0 ( P ) ( 1 + ( V / 2 ) { cos [ φ 1 ( P ) ] + cos [ φ 2 ( P ) ] } ) ,
I 4 = I 0 ( P ) ( 1 + ( V / 2 ) { cos [ φ 1 ( P ) + α ] + cos [ φ 2 ( P ) + α ] } ) ,
I 5 = I 0 ( P ) ( 1 + ( V / 2 ) { cos [ φ 1 ( P ) + 2 α ] + cos [ φ 2 ( P ) + 2 α ] } ) ,
tan ( ψ 2 ) sin α 1 cos ( 2 α ) = I 2 I 4 2 I 3 I 1 I 5
tan ( ψ 2 ) sin ( 2 α ) ( 1 cos α ) = I 1 I 5 2 I 3 I 2 I 4 .
ψ = 2 tan 1 [ 2 ( I 2 I 4 ) 2 I 3 I 1 I 5 ] .
( cos 3 α ) 2 ( cos 2 α ) + 1 = 0 ,
cos α = 1 , ( 1 ± 5 ) / 2 .
cos α = ( 1 5 ) / 2 ,
ψ = 2 tan 1 [ 1 . 665 ( I 1 I 5 ) 2 I 3 I 2 I 4 ] .
tan ( ψ 2 ) cot ( α 2 ) = I 2 I 4 2 I 3 I 2 I 4 ,
tan ( ψ 2 ) ( cot α ) = I 1 I 5 2 I 3 I 1 I 5 .
ψ ( P ) = 2 tan 1 [ 3 ( I 2 I 4 ) 2 I 3 I 2 I 4 ] = 2 tan 1 [ 3 ( I 5 I 1 ) 2 I 3 I 1 I 5 ] .
I 1 = I 0 ( P ) ( 1 + ( V / 2 ) { cos [ φ 1 ( P ) 2 α ] + cos [ φ 2 ( P ) + 2 α ] } ) ,
I 2 = I 0 ( P ) ( 1 + ( V / 2 ) { cos [ φ 1 ( P ) α ] + cos [ φ 2 ( P ) + α ] } ) ,
I 3 = I 0 ( P ) ( 1 + ( V / 2 ) { cos [ φ 1 ( P ) ] + cos [ φ 2 ( P ) ] } ) ,
I 4 = I 0 ( P ) ( 1 + ( V / 2 ) { cos [ φ 1 ( P ) + α ] + cos [ φ 2 ( P ) α ] } ) ,
I 5 = I 0 ( P ) ( 1 + ( V / 2 ) { cos [ φ 1 ( P ) + 2 α ] + cos [ φ 2 ( P ) 2 α ] } ) ,
tan ( φ 2 ) sin α 1 cos ( 2 α ) = I 2 I 4 2 I 3 I 1 I 5 ,
tan ( φ 2 ) sin 2 α 1 cos α = I 1 I 5 2 I 3 I 2 I 4 .
φ ( P ) = 2 tan 1 [ 2 ( I 2 I 4 ) 2 I 3 I 1 I 5 ] ,
φ ( P ) = 2 tan 1 [ 1 . 665 ( I 1 I 5 ) 2 I 3 I 2 I 4 ] .
I 1 = I 0 ( P ) ( 1 + ( V / 2 ) { cos [ φ 1 ( P ) 2 α ] + cos [ φ 2 ( P ) 2 α ] } ) ,
I 2 = I 0 ( P ) ( 1 + ( V / 2 ) { cos [ φ 1 ( P ) α ] + cos [ φ 2 ( P ) + α ] } ) ,
I 3 = I 0 ( P ) ( 1 + ( V / 2 ) { cos [ φ 1 ( P ) α ] + cos [ φ 2 ( P ) α ] } ) ,
I 4 = I 0 ( P ) ( 1 + ( V / 2 ) { cos [ φ 1 ( P ) ] + cos φ 2 ( P ) } ) ,
I 5 = I 0 ( P ) ( 1 + ( V / 2 ) { cos [ φ 1 ( P ) + α ] + cos [ φ 2 ( P ) + α ] } ) ,
I 6 = I 0 ( P ) ( 1 + ( V / 2 ) { cos [ φ 1 ( P ) + α ] + cos [ φ 2 ( P ) α ] } ) ,
I 7 = I 0 ( P ) ( 1 + ( V / 2 ) { cos [ φ 1 ( P ) + 2 α ] + cos [ φ 2 ( P ) + 2 α ] } ) .
tan ( φ 2 ) sin α 1 cos ( 2 α ) = I 2 I 6 2 I 4 I 1 I 7 ,
tan ( ψ 2 ) sin α 1 cos ( 2 α ) = I 3 I 5 2 I 4 I 1 I 7 .
φ ( P ) = 2 tan 1 [ 2 ( I 2 I 6 ) 2 I 4 I 1 I 7 ] ,
ψ ( P ) = 2 tan 1 [ 2 ( I 3 I 5 ) 2 I 4 I 1 I 7 ] .
α = ( π / 2 ) + ,
tan ( φ 2 ) = ( 1 2 2 ) tan ( φ 2 ) .
φ φ = ( 2 / 2 ) sin φ .
ψ ψ = ( 2 / 2 ) sin ψ .
tan ( φ 2 ) = ( 1 + 1 . 57 2 ) tan ( φ 2 ) .
φ φ = 1 . 57 2 sin φ,
ψ ψ = 1 . 57 2 sin ψ .

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