Abstract

An algorithm is presented for calculating the phase difference that is obtained with rapid-switching, double-pulsed holographic interferometry. Imperfect switching between reference beams in between two laser pulses causes a distortion in the phase-stepped interferograms that results in erroneous phase calculation when conventional phase-calculating algorithms are used. Instead of estimating the error that is involved, an extra step is added to a more conventional algorithm such as Carré’s algorithm, so that this leakage effect is compensated for. Thus the desired phase difference is found. Also, a measure is found for excluding unreliable calculations, based on the light intensities and phase steps for each pixel and the leakage ratios.

© 1998 Optical Society of America

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  3. 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]
  4. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  12. B. H. Timmerman, D. W. Watt, “Tomographic high-speed digital holographic interferometry,” Meas. Sci. Technol. 6, 1270–1277 (1995).
    [CrossRef]
  13. T. A. W. M. Lanen, “Digital holographic interferometry in compressible flow research,” Ph.D. dissertation (Delft University of Technology, Delft, The Netherlands, 1992).
  14. T. A. W. M. Lanen, C. Nebbeling, J. L. van Ingen, “Phase-stepping holographic interferometry in studying transparent density fields around 2-D objects of arbitrary shape,” Opt. Commun. 76, 268–276 (1990).
    [CrossRef]
  15. P. Hariharan, Optical Interferometry (Academic, Sydney, Australia, 1985).
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    [CrossRef] [PubMed]

1995 (1)

B. H. Timmerman, D. W. Watt, “Tomographic high-speed digital holographic interferometry,” Meas. Sci. Technol. 6, 1270–1277 (1995).
[CrossRef]

1994 (1)

1993 (1)

H.-H. Bartels-Lehnhoff, P. H. Baumann, B. Bretthauer, G. E. A. Meier, “Computer aided evaluation of interferograms,” Exp. Fluids 16, 46–53 (1993).
[CrossRef]

1992 (1)

1990 (1)

T. A. W. M. Lanen, C. Nebbeling, J. L. van Ingen, “Phase-stepping holographic interferometry in studying transparent density fields around 2-D objects of arbitrary shape,” Opt. Commun. 76, 268–276 (1990).
[CrossRef]

1988 (1)

1987 (2)

1985 (1)

1983 (1)

1974 (1)

1966 (1)

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

Bartels-Lehnhoff, H.-H.

H.-H. Bartels-Lehnhoff, P. H. Baumann, B. Bretthauer, G. E. A. Meier, “Computer aided evaluation of interferograms,” Exp. Fluids 16, 46–53 (1993).
[CrossRef]

Baumann, P. H.

H.-H. Bartels-Lehnhoff, P. H. Baumann, B. Bretthauer, G. E. A. Meier, “Computer aided evaluation of interferograms,” Exp. Fluids 16, 46–53 (1993).
[CrossRef]

Brangaccio, D. J.

Bretthauer, B.

H.-H. Bartels-Lehnhoff, P. H. Baumann, B. Bretthauer, G. E. A. Meier, “Computer aided evaluation of interferograms,” Exp. Fluids 16, 46–53 (1993).
[CrossRef]

Bruning, J. H.

Burow, R.

Carré, P.

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

Cheng, Y. Y.

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” SPIE Short Course Notes SC37 (Society of Photo-Optical Instrumentation Engineers, Bellingham, Washington, 1993).

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis; Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, 1993), Chap. 4, pp. 94–140.

Eiju, T.

Elssner, K.-E.

Gallagher, J. E.

Grzanna, J.

Hariharan, P.

Herriott, D. R.

Joenathan, C.

Kinnstaetter, K.

Lanen, T. A. W. M.

T. A. W. M. Lanen, C. Nebbeling, J. L. van Ingen, “Phase-stepping holographic interferometry in studying transparent density fields around 2-D objects of arbitrary shape,” Opt. Commun. 76, 268–276 (1990).
[CrossRef]

T. A. W. M. Lanen, “Digital holographic interferometry in compressible flow research,” Ph.D. dissertation (Delft University of Technology, Delft, The Netherlands, 1992).

Larkin, K. G.

K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
[CrossRef]

K. G. Larkin, B. F. Oreb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” in Proc. SPIE Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, eds., 1755, 219–227 (1993).

Lohman, A. W.

Meier, G. E. A.

H.-H. Bartels-Lehnhoff, P. H. Baumann, B. Bretthauer, G. E. A. Meier, “Computer aided evaluation of interferograms,” Exp. Fluids 16, 46–53 (1993).
[CrossRef]

Merkel, K.

Nebbeling, C.

T. A. W. M. Lanen, C. Nebbeling, J. L. van Ingen, “Phase-stepping holographic interferometry in studying transparent density fields around 2-D objects of arbitrary shape,” Opt. Commun. 76, 268–276 (1990).
[CrossRef]

Oreb, B. F.

K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
[CrossRef]

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. G. Larkin, B. F. Oreb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” in Proc. SPIE Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, eds., 1755, 219–227 (1993).

Rosenfeld, D. P.

Schwider, J.

Spolaczyk, R.

Streibl, N.

Timmerman, B. H.

B. H. Timmerman, D. W. Watt, “Tomographic high-speed digital holographic interferometry,” Meas. Sci. Technol. 6, 1270–1277 (1995).
[CrossRef]

van Ingen, J. L.

T. A. W. M. Lanen, C. Nebbeling, J. L. van Ingen, “Phase-stepping holographic interferometry in studying transparent density fields around 2-D objects of arbitrary shape,” Opt. Commun. 76, 268–276 (1990).
[CrossRef]

Watt, D. W.

B. H. Timmerman, D. W. Watt, “Tomographic high-speed digital holographic interferometry,” Meas. Sci. Technol. 6, 1270–1277 (1995).
[CrossRef]

White, A. D.

Wyant, J. C.

Appl. Opt. (7)

Exp. Fluids (1)

H.-H. Bartels-Lehnhoff, P. H. Baumann, B. Bretthauer, G. E. A. Meier, “Computer aided evaluation of interferograms,” Exp. Fluids 16, 46–53 (1993).
[CrossRef]

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

Meas. Sci. Technol. (1)

B. H. Timmerman, D. W. Watt, “Tomographic high-speed digital holographic interferometry,” Meas. Sci. Technol. 6, 1270–1277 (1995).
[CrossRef]

Metrologia (1)

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

Opt. Commun. (1)

T. A. W. M. Lanen, C. Nebbeling, J. L. van Ingen, “Phase-stepping holographic interferometry in studying transparent density fields around 2-D objects of arbitrary shape,” Opt. Commun. 76, 268–276 (1990).
[CrossRef]

Other (5)

P. Hariharan, Optical Interferometry (Academic, Sydney, Australia, 1985).

K. Creath, “Phase-measurement interferometry techniques,” SPIE Short Course Notes SC37 (Society of Photo-Optical Instrumentation Engineers, Bellingham, Washington, 1993).

T. A. W. M. Lanen, “Digital holographic interferometry in compressible flow research,” Ph.D. dissertation (Delft University of Technology, Delft, The Netherlands, 1992).

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis; Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, 1993), Chap. 4, pp. 94–140.

K. G. Larkin, B. F. Oreb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” in Proc. SPIE Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, eds., 1755, 219–227 (1993).

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Figures (4)

Fig. 1
Fig. 1

Rapid-switching digital holographic interferometer setup. Mi, mirror; Li, lens; BSC, beam-splitter cube; PBSC i , polarizing beam-splitter cube; PC, Pockels cell; H, holographic plate; PZT, piezoelectric transducer; BD, beam dump; S, shutter; SF, spatial filter; R4, (1/4)λ-retardation plate; R1,2,3, (1/2)λ-retardation plate; ↔, in-plane polarization; and ○, out-of-plane polarization.

Fig. 2
Fig. 2

Tangent and I as a function of the phase difference Δϕ for leakage ratios a 1 = b 1 = 0.01, a 2 = b 2 = 0.1, a 3 = b 3 = 0.4, a 4 = b 4= 0.6.

Fig. 3
Fig. 3

Error for calculating tangent instead of I as a function of the phase difference for leakage ratios a 1 = b 1 = 0.01, a 2 = b 2 = 0.1, a 3 = b 3 = 0.4, a 4 = b 4 = 0.6.

Fig. 4
Fig. 4

L: Part of modulation intensity that is a function of the leakage ratio. The leakage ratio for both recording instances is taken to be the same and is given by a = b = l.

Equations (14)

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tan   Δ ψ = e   cos   ψ -   sin   ψ 1 +   cos   ψ + e   sin   ψ ,
I i x ,   y = I bias x ,   y + I mod x ,   y × cos Δ ϕ x ,   y - Ψ i x ,   y ,
tan   Δ ϕ = I 1 - I 4 + I 2 - I 3 3 I 2 - I 3 - I 1 - I 4 1 / 2 I 2 + I 3 - I 1 + I 4 .
E x ,   y = E t = 1 path 1 + E t = 1 path 2 + E t = 2 path 1 + E t = 2 path 2 .
E ξ p η = Re E ξ r p η exp - i ϕ ξ η exp i 2 π ν t ,
I r x ,   y = I 1 p 1 + I 2 p 2 + I 1 p 2 + I 2 p 1 + 2 I 1 p 1 I 2 p 1 1 / 2 cos ϕ 1 1 - ϕ 2 1 + I 1 p 1 I 1 p 2 1 / 2 cos ϕ 1 1 - ϕ 1 2 - Ψ r + I 1 p 1 I 2 p 2 1 / 2 cos ϕ 1 1 - ϕ 2 2 - Ψ r + I 1 p 2 I 2 p 1 1 / 2 cos ϕ 2 1 - ϕ 1 2 - Ψ r + I 2 p 2 I 2 p 1 1 / 2 cos ϕ 2 1 - ϕ 2 2 - Ψ r + I 2 p 2 I 1 p 2 1 / 2 cos ϕ 1 2 - ϕ 2 2 ] ,
ϕ 2 2 - ϕ 1 2 = ϕ 2 2 - ϕ 1 1 = ϕ 2 1 - ϕ 1 2 = ϕ 2 1 - ϕ 1 1 = Δ ϕ , ϕ 2 2 - ϕ 2 1 = ϕ 1 2 - ϕ 1 1 = 0 .
I r x ,   y = D + P A + B + 1 cos   Δ ϕ cos Ψ r + 1 - B sin   Δ ϕ   sin   Ψ r ,
I I 1 - I 4 + I 2 - I 3 3 I 2 - I 3 - I 1 - I 4 1 / 2 I 2 + I 3 - I 1 + I 4 = | 1 - B sin   Δ ϕ | A + B + 1 cos   Δ ϕ ,
cos   Δ ϕ = - 2 AI 2 B + 1 ± 2 I 2 B + 1 2 B - 1 2 - I 2 A 2 B - 1 2 + B - 1 4 1 / 2 2 I 2 B + 1 2 + B - 1 2 ,
I 1 - I 4 + I 2 - I 3 = 2 P 1 - B sin   Δ ϕ 4   sin   α   cos 2   α .
I mod = 8 I - mod   sin 2   α   cos   α ,
I mod = - A 1 - B M + B + 1 B + 1 2 - A 2 N 2 + 1 - B 2 M 2 1 / 2 1 - B B + 1 2 - A 2 .
I mod x ,   y = L x ,   y I mod x ,   y ,

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