Abstract

A modified holographic interferometer is described which enables automatic fringe pattern analysis to be achieved when recording transient events. The introduction of a phase grating enables the method of phase-stepping fringe analysis to be applied to a wide range of measurement problems. The theory is developed, and an analysis of the sources of error is presented with the aid of computer simulations. Initial experimental results confirm the utility of the method and its potential application in pulsed laser holography. The method can in principle be applied to electronic speckle pattern interferometry.

© 1988 Optical Society of America

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  1. J. H. Bruning, J. G. Gallagher, D. P. Rosenfeld, D. A. White, D. J. Brangaccio, D. R. Herriott, “Digital Wavefront Measuring Interferometer for Testing Optical Surfaces and Lenses,” Appl. Opt. 13, 2693 (1974).
    [CrossRef] [PubMed]
  2. P. Hariharan, B. F. Oreb, N. Brown, “A Digital Phase-Measurement System for Real-Time Holographic Interferometry,” Opt. Commun. 14, 393 (1982).
    [CrossRef]
  3. R. Dandliker, R. Thalmann, J. F. Willemin, “Fringe Interpolation by Two-Reference-Beam Holographic Interferometry: Reducing Sensitivity to Hologram Misalignment,” Opt. Commun. 42, 301 (1982).
    [CrossRef]
  4. P. Hariharan, B. F. Oreb, N. Brown, “Real-Time Holography Interferometry: a Microcomputer System for the Measurement of Vector Displacements,” Appl. Opt. 22, 876 (1983).
    [CrossRef] [PubMed]
  5. M. Chang, C.-P. Hu, J. C. Wyant, “Phase Shifting Holographic Interferometry (PSHI),” Proc. Soc. Photo-Opt. Instrum. Eng. 599, 149 (1985).
  6. R. Smythe, R. Moore, “Instantaneous Phase Measuring Interferometry,” Opt. Eng. 23, 361 (1984).
    [CrossRef]
  7. O. Y. Kwon, D. M. Shough, “Multichannel Grating Phase-Shift Interferometers,” Proc. Soc. Photo-Opt. Instrum. Eng. 599, 273 (1985).
  8. R. Dandliker, R. Thalmann, “Heterodyne and Quasi-heterodyne Holographic Interferometry,” Opt. Eng. 24, 824 (1985).
  9. M. C. Hutley, Diffraction Gratings (Academic, New York, 1982).
  10. D. W. Robinson, “Automatic Fringe Analysis with a Computer Image-Processing System,” Appl. Opt. 22, 2169 (1983).
    [CrossRef] [PubMed]
  11. D. W. Robinson, D. C. Williams, “Digital Phase Stepping Speckle Interferometry,” Opt. Commun. 57, 26 (1986).
    [CrossRef]
  12. J. Schwider, R. Burrow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital Wave-Front Measuring Interferometry: Some Systematic Error Sources,” Appl. Opt. 22, 3421 (1983).
    [CrossRef] [PubMed]
  13. Y. Cheng, J. C. Wyant, “Phase Shifter Calibration in Phase-Shifting Interferometry,” Appl. Opt. 24, 3079 (1985).
    [CrossRef]

1986 (1)

D. W. Robinson, D. C. Williams, “Digital Phase Stepping Speckle Interferometry,” Opt. Commun. 57, 26 (1986).
[CrossRef]

1985 (4)

M. Chang, C.-P. Hu, J. C. Wyant, “Phase Shifting Holographic Interferometry (PSHI),” Proc. Soc. Photo-Opt. Instrum. Eng. 599, 149 (1985).

O. Y. Kwon, D. M. Shough, “Multichannel Grating Phase-Shift Interferometers,” Proc. Soc. Photo-Opt. Instrum. Eng. 599, 273 (1985).

R. Dandliker, R. Thalmann, “Heterodyne and Quasi-heterodyne Holographic Interferometry,” Opt. Eng. 24, 824 (1985).

Y. Cheng, J. C. Wyant, “Phase Shifter Calibration in Phase-Shifting Interferometry,” Appl. Opt. 24, 3079 (1985).
[CrossRef]

1984 (1)

R. Smythe, R. Moore, “Instantaneous Phase Measuring Interferometry,” Opt. Eng. 23, 361 (1984).
[CrossRef]

1983 (3)

1982 (2)

P. Hariharan, B. F. Oreb, N. Brown, “A Digital Phase-Measurement System for Real-Time Holographic Interferometry,” Opt. Commun. 14, 393 (1982).
[CrossRef]

R. Dandliker, R. Thalmann, J. F. Willemin, “Fringe Interpolation by Two-Reference-Beam Holographic Interferometry: Reducing Sensitivity to Hologram Misalignment,” Opt. Commun. 42, 301 (1982).
[CrossRef]

1974 (1)

Brangaccio, D. J.

Brown, N.

P. Hariharan, B. F. Oreb, N. Brown, “Real-Time Holography Interferometry: a Microcomputer System for the Measurement of Vector Displacements,” Appl. Opt. 22, 876 (1983).
[CrossRef] [PubMed]

P. Hariharan, B. F. Oreb, N. Brown, “A Digital Phase-Measurement System for Real-Time Holographic Interferometry,” Opt. Commun. 14, 393 (1982).
[CrossRef]

Bruning, J. H.

Burrow, R.

Chang, M.

M. Chang, C.-P. Hu, J. C. Wyant, “Phase Shifting Holographic Interferometry (PSHI),” Proc. Soc. Photo-Opt. Instrum. Eng. 599, 149 (1985).

Cheng, Y.

Dandliker, R.

R. Dandliker, R. Thalmann, “Heterodyne and Quasi-heterodyne Holographic Interferometry,” Opt. Eng. 24, 824 (1985).

R. Dandliker, R. Thalmann, J. F. Willemin, “Fringe Interpolation by Two-Reference-Beam Holographic Interferometry: Reducing Sensitivity to Hologram Misalignment,” Opt. Commun. 42, 301 (1982).
[CrossRef]

Elssner, K.-E.

Gallagher, J. G.

Grzanna, J.

Hariharan, P.

P. Hariharan, B. F. Oreb, N. Brown, “Real-Time Holography Interferometry: a Microcomputer System for the Measurement of Vector Displacements,” Appl. Opt. 22, 876 (1983).
[CrossRef] [PubMed]

P. Hariharan, B. F. Oreb, N. Brown, “A Digital Phase-Measurement System for Real-Time Holographic Interferometry,” Opt. Commun. 14, 393 (1982).
[CrossRef]

Herriott, D. R.

Hu, C.-P.

M. Chang, C.-P. Hu, J. C. Wyant, “Phase Shifting Holographic Interferometry (PSHI),” Proc. Soc. Photo-Opt. Instrum. Eng. 599, 149 (1985).

Hutley, M. C.

M. C. Hutley, Diffraction Gratings (Academic, New York, 1982).

Kwon, O. Y.

O. Y. Kwon, D. M. Shough, “Multichannel Grating Phase-Shift Interferometers,” Proc. Soc. Photo-Opt. Instrum. Eng. 599, 273 (1985).

Merkel, K.

Moore, R.

R. Smythe, R. Moore, “Instantaneous Phase Measuring Interferometry,” Opt. Eng. 23, 361 (1984).
[CrossRef]

Oreb, B. F.

P. Hariharan, B. F. Oreb, N. Brown, “Real-Time Holography Interferometry: a Microcomputer System for the Measurement of Vector Displacements,” Appl. Opt. 22, 876 (1983).
[CrossRef] [PubMed]

P. Hariharan, B. F. Oreb, N. Brown, “A Digital Phase-Measurement System for Real-Time Holographic Interferometry,” Opt. Commun. 14, 393 (1982).
[CrossRef]

Robinson, D. W.

D. W. Robinson, D. C. Williams, “Digital Phase Stepping Speckle Interferometry,” Opt. Commun. 57, 26 (1986).
[CrossRef]

D. W. Robinson, “Automatic Fringe Analysis with a Computer Image-Processing System,” Appl. Opt. 22, 2169 (1983).
[CrossRef] [PubMed]

Rosenfeld, D. P.

Schwider, J.

Shough, D. M.

O. Y. Kwon, D. M. Shough, “Multichannel Grating Phase-Shift Interferometers,” Proc. Soc. Photo-Opt. Instrum. Eng. 599, 273 (1985).

Smythe, R.

R. Smythe, R. Moore, “Instantaneous Phase Measuring Interferometry,” Opt. Eng. 23, 361 (1984).
[CrossRef]

Spolaczyk, R.

Thalmann, R.

R. Dandliker, R. Thalmann, “Heterodyne and Quasi-heterodyne Holographic Interferometry,” Opt. Eng. 24, 824 (1985).

R. Dandliker, R. Thalmann, J. F. Willemin, “Fringe Interpolation by Two-Reference-Beam Holographic Interferometry: Reducing Sensitivity to Hologram Misalignment,” Opt. Commun. 42, 301 (1982).
[CrossRef]

White, D. A.

Willemin, J. F.

R. Dandliker, R. Thalmann, J. F. Willemin, “Fringe Interpolation by Two-Reference-Beam Holographic Interferometry: Reducing Sensitivity to Hologram Misalignment,” Opt. Commun. 42, 301 (1982).
[CrossRef]

Williams, D. C.

D. W. Robinson, D. C. Williams, “Digital Phase Stepping Speckle Interferometry,” Opt. Commun. 57, 26 (1986).
[CrossRef]

Wyant, J. C.

M. Chang, C.-P. Hu, J. C. Wyant, “Phase Shifting Holographic Interferometry (PSHI),” Proc. Soc. Photo-Opt. Instrum. Eng. 599, 149 (1985).

Y. Cheng, J. C. Wyant, “Phase Shifter Calibration in Phase-Shifting Interferometry,” Appl. Opt. 24, 3079 (1985).
[CrossRef]

Appl. Opt. (5)

Opt. Commun. (3)

D. W. Robinson, D. C. Williams, “Digital Phase Stepping Speckle Interferometry,” Opt. Commun. 57, 26 (1986).
[CrossRef]

P. Hariharan, B. F. Oreb, N. Brown, “A Digital Phase-Measurement System for Real-Time Holographic Interferometry,” Opt. Commun. 14, 393 (1982).
[CrossRef]

R. Dandliker, R. Thalmann, J. F. Willemin, “Fringe Interpolation by Two-Reference-Beam Holographic Interferometry: Reducing Sensitivity to Hologram Misalignment,” Opt. Commun. 42, 301 (1982).
[CrossRef]

Opt. Eng. (2)

R. Smythe, R. Moore, “Instantaneous Phase Measuring Interferometry,” Opt. Eng. 23, 361 (1984).
[CrossRef]

R. Dandliker, R. Thalmann, “Heterodyne and Quasi-heterodyne Holographic Interferometry,” Opt. Eng. 24, 824 (1985).

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

O. Y. Kwon, D. M. Shough, “Multichannel Grating Phase-Shift Interferometers,” Proc. Soc. Photo-Opt. Instrum. Eng. 599, 273 (1985).

M. Chang, C.-P. Hu, J. C. Wyant, “Phase Shifting Holographic Interferometry (PSHI),” Proc. Soc. Photo-Opt. Instrum. Eng. 599, 149 (1985).

Other (1)

M. C. Hutley, Diffraction Gratings (Academic, New York, 1982).

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Figures (14)

Fig. 1
Fig. 1

Holographic setup producing three instantaneous interferograms: ΣO, ΣR, object and reference beams, respectively; M, mirrors; BS, beam splitter.

Fig. 2
Fig. 2

Schematic representation of diffraction of the object beam: Σ0 caused by the diffraction grating without and with the lateral shift x0; G, the linear diffraction grating; Σ0, Σ±1 and 0, Σ ± 1′, the object beams behind the grating (The prime designates the beams propagating behind the grating translated by x0.)

Fig. 3
Fig. 3

Real-time holographic arrangement with a hologram H illuminated by three object beams, +1,0,−1, propagating behind the grating G. θG, θH are the angles of the first diffraction orders of the hologram and grating, respectively. R O * 1 , R O * 0 , R O * 1 the primary self-reconstructions; (k,l), the object beams obtained behind the hologram due to its multibeam illumination; k,l, the diffraction orders from the grating and hologram, respectively.

Fig. 4
Fig. 4

Geometrical relationships of the object beams in the multichannel holographic interferometer: π,π′, the object and detector planes (conjugate); CL, the camera lens; G, the diffraction grating; ΔΔ′, the separation of the 0 and ±1th diffraction order images in the object and detector plane, respectively.

Fig. 5
Fig. 5

Analysis of computer generated interferograms for the phase function ϕ(x,y) with a coefficient value of F = 5.0 and no error introduced: (a)–(c) interferograms with the phase shift, 120, 0, +120°, respectively; (d) unwrapped and (f) wrapped phase; (e) look-up table.

Fig. 6
Fig. 6

Map of phase differences between the theoretical phase function and the phase function produced by analysis of the computer simulated interferograms [Figs. 5(a)(c)] (S = 200).

Fig. 7
Fig. 7

Map of phase errors obtained when the phase shift error between interferograms is −5° (S = 200).

Fig. 8
Fig. 8

Map of errors obtained for two-pixel mismatch in the x direction introduced in the ±1 diffraction-order interferograms (S = 200).

Fig. 9
Fig. 9

Unwrapped phase map calculated for phase functions with the coefficient: (a) F = 15.0, (b) F = 25.0, and 6-pixel mismatch in the x direction.

Fig. 10
Fig. 10

Phase error map obtained for the interferogram with a distortion function Ψ(x,y) (A = 0.0002) introduced (S = 200).

Fig. 11
Fig. 11

Unwrapped phase map calculated for phase functions with the coefficients (a) F = 5.0, (b) F = 15.0, and the distortion function (A = 0.0006) introduced.

Fig. 12
Fig. 12

Wrapped phase map (a), (c) and the phase error map (b), (d) obtained for 50% (S = 50) and 10% (S = 200) mean intensity differences between ±1 and 0 diffraction-order interferograms, respectively.

Fig. 13
Fig. 13

Interferograms obtained simultaneously in a realtime holographic interferometer (a)–(c), wrapped, and unwrapped phase maps (d), (f) and look-up table (e).

Fig. 14
Fig. 14

Interferograms obtained in a real-time holographic interferometer for larger distortion of the object (a)–(c), wrapped and unwrapped phase maps (d), (f), and look-up table (e).

Equations (21)

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I ( x , y ) = A ( x , y ) { 1 + m ( x , y ) cos [ Δ Φ ( x , y ) ] } ,
I ( x , y ) = A ( x , y ) { 1 + m ( x , y ) cos [ Δ Φ ( x , y ) + k δ ) } ,
U ( x , y ) = O ( x , y ) T ( x , y ) = O 1 ( x , y ) exp [ i Φ ( x , y ) ] [ C 0 + C 1 exp ( i 2 π ω x ) + C 1 exp ( i 2 π ω x ) ] ,
δ = 2 π x 0 d .
U ( x , y ) = O ( x , y ) T ( x , y ) = O 1 ( x , y ) exp [ ( i Φ ( x , y ) ] { C 0 + C 1 exp [ i ( 2 π ω x δ ) ] + C 1 exp [ i ( 2 π ω x + δ ) ] } .
I k ( x , y ) [ U k ( x , y ) + U k ( x , y ) ] · [ U k ( x , y ) + U k ( x , y ) ] * ,
I 1 ( x , y ) = A 1 ( x , y ) { 1 + m 1 ( x , y ) cos [ Δ Φ ( x , y ) + δ ] } ,
I 0 ( x , y ) = A 0 ( x , y ) { 1 + m 0 ( x , y ) cos [ Δ Φ ( x , y ) ] } ,
I 1 ( x , y ) = A 1 ( x , y ) { 1 + m 1 ( x , y ) cos [ Δ Φ ( x , y ) δ ] } ,
Δ Φ = tan 1 3 ( I 1 I 1 ) / ( 2 I 0 I 1 I 1 ) ,
I ( x , y ) = ( R + O 1 + O 0 + O 1 ) ( R + O 1 + O 0 + O 1 ) * = K + R O * 1 + R O * 0 + R O * 1 + R * O 1 + R * O 0 + R * O 1 + O 1 O * 0 + O 1 O * 1 + O 0 O * 1 + O 0 O * 1 + O 1 O * 1 + O 1 O * 0 ,
θ H > 2 θ G + α 0 ,
Δ = s z g s f f λ d ,
( δ Φ ) 2 = ( 2 / 3 m 2 ) ( Δ I / I 2 ) + ( 1 / 2 ) ( Δ Φ k ) 2 ,
I k ( x , y ) = a k ( x , y ) { 1 + m k ( x , y ) × cos [ Φ ( x , y ) + Φ k + Ψ k ( x , y ) ] } ,
Φ ( x , y ) = F [ x + ( x 2 + y 2 ) ] ,
tan Φ ( x , y ) = ( I 3 I 2 ) cos Φ 1 + ( I 1 I 3 ) + ( I 2 I 1 ) cos Φ + 1 ( I 3 I 1 ) sin Φ 1 + ( I 1 I 3 ) + ( I 2 I 1 ) sin Φ + 1 .
tan Φ ( x , y ) = ( I 3 + I 2 ) sin ( 120 ) + ( I 1 I 3 ) + ( I 2 I 1 ) sin ( 120 ) ( I 3 + I 1 ) cos ( 120 ) + ( I 1 I 3 ) + ( I 2 I 1 ) cos ( 120 ) .
Δ p < 1 4 p min .
Ψ ± 1 ( x , y ) = A [ ( x x 0 ) 2 + ( y y 0 ) 2 ] ,
( δ ϕ ) 2 = ( 2 / 3 m 2 ) ( Δ I / I ) 2 .

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