Abstract

The velocity measurement limit in dynamic interferometry is vNyq, the velocity at which the interferogram is sampled at the Nyquist limit. We show that vNyq can be exceeded by assuming continuity of the surface motion and unwrapping the velocity modulo 2vNyq. The technique was demonstrated in a high-speed speckle pattern interferometer with spatial phase stepping. Surface velocities of 4vNyq were measured experimentally. With a reduced exposure, high-speed sub-Nyquist interferometry could be implemented up to a maximum acceleration of vNyq/ts, where ts is the detector frame period.

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References

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  1. J. M. Kilpatrick, A. J. Moore, J. S. Barton, J. D. C. Jones, M. Reeves, and C. Buckberry, “Measurement of complex surface deformation by high-speed dynamic phase-stepped digital speckle pattern interferometry,” Opt. Lett. 25(15), 1068–1070 (2000).
    [Crossref]
  2. W. N. MacPherson, M. Reeves, D. P. Towers, A. J. Moore, J. D. C. Jones, M. Dale, and C. Edwards, “Multipoint laser vibrometer for modal analysis,” Appl. Opt. 46(16), 3126–3132 (2007).
    [Crossref] [PubMed]
  3. J. M. Huntley, G. H. Kaufmann, and D. Kerr, “Phase-shifted dynamic speckle pattern interferometry at 1 kHz,” Appl. Opt. 38(31), 6556–6563 (1999).
    [Crossref]
  4. X. Colonna de Lega and P. Jacquot, “Deformation measurement with object-induced dynamic phase shifting,” Appl. Opt. 35(25), 5115–5121 (1996).
    [Crossref] [PubMed]
  5. P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, and G. H. Kaufmann, “Phase errors in low-frequency vibration measurement with high-speed phase-shifting speckle pattern interferometry,” Opt. Eng. 40(9), 1984–1992 (2001).
    [Crossref]
  6. T. Wu, J. D. Jones, and A. J. Moore, “High-speed phase-stepped digital speckle pattern interferometry using a complementary metal-oxide semiconductor camera,” Appl. Opt. 45(23), 5845–5855 (2006).
    [Crossref] [PubMed]
  7. A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “Transient deformation measurement with electronic speckle pattern interferometry and a high-speed camera,” Appl. Opt. 38(7), 1159–1162 (1999).
    [Crossref]
  8. A. Ettemeyer and Z. Wang, “Verfahren und Vorrichtung zur Bestimmung von Phasen und Phasendifferenzen,” Patent DE 195 13 234 (1995).
  9. H. van Brug, “Temporal phase unwrapping and its application in shearography systems,” Appl. Opt. 37(28), 6701–6706 (1998).
    [Crossref]
  10. J. E. Greivenkamp, “Sub-Nyquist interferometry,” Appl. Opt. 26(24), 5245–5258 (1987).
    [Crossref] [PubMed]
  11. M. Reeves, A. J. Moore, D. P. Hand, and J. D. C. Jones, “Dynamic shape measurement system for laser materials processing,” Opt. Eng. 42(10), 2923–2929 (2003).
    [Crossref]
  12. A. J. P. Haasteren and H. J. Frankena, “Real-time displacement measurement using a multicamera phase-stepping speckle interferometer,” Appl. Opt. 33(19), 4137–4142 (1994).
    [Crossref] [PubMed]
  13. A. L. Weijers, H. van Brug, and H. J. Frankena, “Polarization phase stepping with a savart element,” Appl. Opt. 37(22), 5150–5155 (1998).
    [Crossref]
  14. T. D. Upton and D. W. Watt, “Optical and electronic design of a calibrated multichannel electronic interferometer for quantitative flow visualization,” Appl. Opt. 34(25), 5602–5610 (1995).
    [Crossref] [PubMed]
  15. B. B. García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, “Transient deformation measurement with electronic speckle pattern interferometry by use of a holographic optical element for spatial phase stepping,” Appl. Opt. 38(28), 5944–5947 (1999).
    [Crossref]
  16. B. Barrientos García, A. J. Moore, C. Perez-Lopez, L. Wang, and T. Tschudi, “Spatial phase-stepped interferometry using a holographic optical element,” Opt. Eng. 38(12), 2069–2074 (1999).
    [Crossref]
  17. J. Kranz, J. Lamprecht, A. Hettwer, and J. Schwider, “Fiber optical single frame speckle interferometer for measuring industrial surfaces,” Proc. SPIE 3407, 328–331 (1998).
    [Crossref]
  18. A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39(4), 960–966 (2000).
    [Crossref]
  19. J. E. Greivenkamp, A. E. Lowman, and R. J. Palum, “Sub-Nyquist interferometry: Implementation and measurement capability,” Opt. Eng. 35(10), 2962–2969 (1996).
    [Crossref]
  20. M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005).
    [Crossref] [PubMed]

2007 (1)

2006 (1)

2005 (1)

2003 (1)

M. Reeves, A. J. Moore, D. P. Hand, and J. D. C. Jones, “Dynamic shape measurement system for laser materials processing,” Opt. Eng. 42(10), 2923–2929 (2003).
[Crossref]

2001 (1)

P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, and G. H. Kaufmann, “Phase errors in low-frequency vibration measurement with high-speed phase-shifting speckle pattern interferometry,” Opt. Eng. 40(9), 1984–1992 (2001).
[Crossref]

2000 (2)

1999 (4)

1998 (3)

1996 (2)

J. E. Greivenkamp, A. E. Lowman, and R. J. Palum, “Sub-Nyquist interferometry: Implementation and measurement capability,” Opt. Eng. 35(10), 2962–2969 (1996).
[Crossref]

X. Colonna de Lega and P. Jacquot, “Deformation measurement with object-induced dynamic phase shifting,” Appl. Opt. 35(25), 5115–5121 (1996).
[Crossref] [PubMed]

1995 (1)

1994 (1)

1987 (1)

Barrientos García, B.

B. Barrientos García, A. J. Moore, C. Perez-Lopez, L. Wang, and T. Tschudi, “Spatial phase-stepped interferometry using a holographic optical element,” Opt. Eng. 38(12), 2069–2074 (1999).
[Crossref]

Barton, J. S.

Brock, N.

Buckberry, C.

Coggrave, C. R.

P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, and G. H. Kaufmann, “Phase errors in low-frequency vibration measurement with high-speed phase-shifting speckle pattern interferometry,” Opt. Eng. 40(9), 1984–1992 (2001).
[Crossref]

Colonna de Lega, X.

Dale, M.

Edwards, C.

Frankena, H. J.

García, B. B.

Greivenkamp, J. E.

J. E. Greivenkamp, A. E. Lowman, and R. J. Palum, “Sub-Nyquist interferometry: Implementation and measurement capability,” Opt. Eng. 35(10), 2962–2969 (1996).
[Crossref]

J. E. Greivenkamp, “Sub-Nyquist interferometry,” Appl. Opt. 26(24), 5245–5258 (1987).
[Crossref] [PubMed]

Haasteren, A. J. P.

Hand, D. P.

M. Reeves, A. J. Moore, D. P. Hand, and J. D. C. Jones, “Dynamic shape measurement system for laser materials processing,” Opt. Eng. 42(10), 2923–2929 (2003).
[Crossref]

A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “Transient deformation measurement with electronic speckle pattern interferometry and a high-speed camera,” Appl. Opt. 38(7), 1159–1162 (1999).
[Crossref]

Hayes, J.

Hettwer, A.

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39(4), 960–966 (2000).
[Crossref]

J. Kranz, J. Lamprecht, A. Hettwer, and J. Schwider, “Fiber optical single frame speckle interferometer for measuring industrial surfaces,” Proc. SPIE 3407, 328–331 (1998).
[Crossref]

Huntley, J. M.

P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, and G. H. Kaufmann, “Phase errors in low-frequency vibration measurement with high-speed phase-shifting speckle pattern interferometry,” Opt. Eng. 40(9), 1984–1992 (2001).
[Crossref]

J. M. Huntley, G. H. Kaufmann, and D. Kerr, “Phase-shifted dynamic speckle pattern interferometry at 1 kHz,” Appl. Opt. 38(31), 6556–6563 (1999).
[Crossref]

Jacquot, P.

Jones, J. D.

Jones, J. D. C.

Kaufmann, G. H.

P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, and G. H. Kaufmann, “Phase errors in low-frequency vibration measurement with high-speed phase-shifting speckle pattern interferometry,” Opt. Eng. 40(9), 1984–1992 (2001).
[Crossref]

J. M. Huntley, G. H. Kaufmann, and D. Kerr, “Phase-shifted dynamic speckle pattern interferometry at 1 kHz,” Appl. Opt. 38(31), 6556–6563 (1999).
[Crossref]

Kerr, D.

Kilpatrick, J. M.

Kranz, J.

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39(4), 960–966 (2000).
[Crossref]

J. Kranz, J. Lamprecht, A. Hettwer, and J. Schwider, “Fiber optical single frame speckle interferometer for measuring industrial surfaces,” Proc. SPIE 3407, 328–331 (1998).
[Crossref]

Lamprecht, J.

J. Kranz, J. Lamprecht, A. Hettwer, and J. Schwider, “Fiber optical single frame speckle interferometer for measuring industrial surfaces,” Proc. SPIE 3407, 328–331 (1998).
[Crossref]

Lowman, A. E.

J. E. Greivenkamp, A. E. Lowman, and R. J. Palum, “Sub-Nyquist interferometry: Implementation and measurement capability,” Opt. Eng. 35(10), 2962–2969 (1996).
[Crossref]

MacPherson, W. N.

Millerd, J.

Moore, A. J.

W. N. MacPherson, M. Reeves, D. P. Towers, A. J. Moore, J. D. C. Jones, M. Dale, and C. Edwards, “Multipoint laser vibrometer for modal analysis,” Appl. Opt. 46(16), 3126–3132 (2007).
[Crossref] [PubMed]

T. Wu, J. D. Jones, and A. J. Moore, “High-speed phase-stepped digital speckle pattern interferometry using a complementary metal-oxide semiconductor camera,” Appl. Opt. 45(23), 5845–5855 (2006).
[Crossref] [PubMed]

M. Reeves, A. J. Moore, D. P. Hand, and J. D. C. Jones, “Dynamic shape measurement system for laser materials processing,” Opt. Eng. 42(10), 2923–2929 (2003).
[Crossref]

J. M. Kilpatrick, A. J. Moore, J. S. Barton, J. D. C. Jones, M. Reeves, and C. Buckberry, “Measurement of complex surface deformation by high-speed dynamic phase-stepped digital speckle pattern interferometry,” Opt. Lett. 25(15), 1068–1070 (2000).
[Crossref]

A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “Transient deformation measurement with electronic speckle pattern interferometry and a high-speed camera,” Appl. Opt. 38(7), 1159–1162 (1999).
[Crossref]

B. B. García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, “Transient deformation measurement with electronic speckle pattern interferometry by use of a holographic optical element for spatial phase stepping,” Appl. Opt. 38(28), 5944–5947 (1999).
[Crossref]

B. Barrientos García, A. J. Moore, C. Perez-Lopez, L. Wang, and T. Tschudi, “Spatial phase-stepped interferometry using a holographic optical element,” Opt. Eng. 38(12), 2069–2074 (1999).
[Crossref]

North-Morris, M.

Novak, M.

Palum, R. J.

J. E. Greivenkamp, A. E. Lowman, and R. J. Palum, “Sub-Nyquist interferometry: Implementation and measurement capability,” Opt. Eng. 35(10), 2962–2969 (1996).
[Crossref]

Perez-Lopez, C.

B. Barrientos García, A. J. Moore, C. Perez-Lopez, L. Wang, and T. Tschudi, “Spatial phase-stepped interferometry using a holographic optical element,” Opt. Eng. 38(12), 2069–2074 (1999).
[Crossref]

Pérez-López, C.

Reeves, M.

Ruiz, P. D.

P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, and G. H. Kaufmann, “Phase errors in low-frequency vibration measurement with high-speed phase-shifting speckle pattern interferometry,” Opt. Eng. 40(9), 1984–1992 (2001).
[Crossref]

Schwider, J.

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39(4), 960–966 (2000).
[Crossref]

J. Kranz, J. Lamprecht, A. Hettwer, and J. Schwider, “Fiber optical single frame speckle interferometer for measuring industrial surfaces,” Proc. SPIE 3407, 328–331 (1998).
[Crossref]

Shen, Y.

P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, and G. H. Kaufmann, “Phase errors in low-frequency vibration measurement with high-speed phase-shifting speckle pattern interferometry,” Opt. Eng. 40(9), 1984–1992 (2001).
[Crossref]

Towers, D. P.

Tschudi, T.

B. B. García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, “Transient deformation measurement with electronic speckle pattern interferometry by use of a holographic optical element for spatial phase stepping,” Appl. Opt. 38(28), 5944–5947 (1999).
[Crossref]

B. Barrientos García, A. J. Moore, C. Perez-Lopez, L. Wang, and T. Tschudi, “Spatial phase-stepped interferometry using a holographic optical element,” Opt. Eng. 38(12), 2069–2074 (1999).
[Crossref]

Upton, T. D.

van Brug, H.

Wang, L.

B. Barrientos García, A. J. Moore, C. Perez-Lopez, L. Wang, and T. Tschudi, “Spatial phase-stepped interferometry using a holographic optical element,” Opt. Eng. 38(12), 2069–2074 (1999).
[Crossref]

B. B. García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, “Transient deformation measurement with electronic speckle pattern interferometry by use of a holographic optical element for spatial phase stepping,” Appl. Opt. 38(28), 5944–5947 (1999).
[Crossref]

Watt, D. W.

Weijers, A. L.

Wu, T.

Wyant, J.

Appl. Opt. (12)

W. N. MacPherson, M. Reeves, D. P. Towers, A. J. Moore, J. D. C. Jones, M. Dale, and C. Edwards, “Multipoint laser vibrometer for modal analysis,” Appl. Opt. 46(16), 3126–3132 (2007).
[Crossref] [PubMed]

J. M. Huntley, G. H. Kaufmann, and D. Kerr, “Phase-shifted dynamic speckle pattern interferometry at 1 kHz,” Appl. Opt. 38(31), 6556–6563 (1999).
[Crossref]

X. Colonna de Lega and P. Jacquot, “Deformation measurement with object-induced dynamic phase shifting,” Appl. Opt. 35(25), 5115–5121 (1996).
[Crossref] [PubMed]

T. Wu, J. D. Jones, and A. J. Moore, “High-speed phase-stepped digital speckle pattern interferometry using a complementary metal-oxide semiconductor camera,” Appl. Opt. 45(23), 5845–5855 (2006).
[Crossref] [PubMed]

A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “Transient deformation measurement with electronic speckle pattern interferometry and a high-speed camera,” Appl. Opt. 38(7), 1159–1162 (1999).
[Crossref]

H. van Brug, “Temporal phase unwrapping and its application in shearography systems,” Appl. Opt. 37(28), 6701–6706 (1998).
[Crossref]

J. E. Greivenkamp, “Sub-Nyquist interferometry,” Appl. Opt. 26(24), 5245–5258 (1987).
[Crossref] [PubMed]

A. J. P. Haasteren and H. J. Frankena, “Real-time displacement measurement using a multicamera phase-stepping speckle interferometer,” Appl. Opt. 33(19), 4137–4142 (1994).
[Crossref] [PubMed]

A. L. Weijers, H. van Brug, and H. J. Frankena, “Polarization phase stepping with a savart element,” Appl. Opt. 37(22), 5150–5155 (1998).
[Crossref]

T. D. Upton and D. W. Watt, “Optical and electronic design of a calibrated multichannel electronic interferometer for quantitative flow visualization,” Appl. Opt. 34(25), 5602–5610 (1995).
[Crossref] [PubMed]

B. B. García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, “Transient deformation measurement with electronic speckle pattern interferometry by use of a holographic optical element for spatial phase stepping,” Appl. Opt. 38(28), 5944–5947 (1999).
[Crossref]

M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005).
[Crossref] [PubMed]

Opt. Eng. (5)

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39(4), 960–966 (2000).
[Crossref]

J. E. Greivenkamp, A. E. Lowman, and R. J. Palum, “Sub-Nyquist interferometry: Implementation and measurement capability,” Opt. Eng. 35(10), 2962–2969 (1996).
[Crossref]

B. Barrientos García, A. J. Moore, C. Perez-Lopez, L. Wang, and T. Tschudi, “Spatial phase-stepped interferometry using a holographic optical element,” Opt. Eng. 38(12), 2069–2074 (1999).
[Crossref]

M. Reeves, A. J. Moore, D. P. Hand, and J. D. C. Jones, “Dynamic shape measurement system for laser materials processing,” Opt. Eng. 42(10), 2923–2929 (2003).
[Crossref]

P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, and G. H. Kaufmann, “Phase errors in low-frequency vibration measurement with high-speed phase-shifting speckle pattern interferometry,” Opt. Eng. 40(9), 1984–1992 (2001).
[Crossref]

Opt. Lett. (1)

Proc. SPIE (1)

J. Kranz, J. Lamprecht, A. Hettwer, and J. Schwider, “Fiber optical single frame speckle interferometer for measuring industrial surfaces,” Proc. SPIE 3407, 328–331 (1998).
[Crossref]

Other (1)

A. Ettemeyer and Z. Wang, “Verfahren und Vorrichtung zur Bestimmung von Phasen und Phasendifferenzen,” Patent DE 195 13 234 (1995).

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Figures (7)

Fig. 1
Fig. 1

Schematic of spatial phase-stepped speckle interferometer. L, lens; A, aperture and P, polariser. Other components described in text.

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Fig. 2
Fig. 2

Mean phase difference between corresponding pixels in the two diffracted orders plotted against lateral translation of grating G2 in its plane. Error bars show ± 3 standard deviations.

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Fig. 3
Fig. 3

Spatial phase-stepped results for out-of-plane deformation. Speckle interferograms recorded (a) at tn-1 and (b) tn. (c) Direct subtraction of images, e.g. In,N-1 - In-1,N-1. (d) Cross-subtraction of images, e.g. In,N-1 - In-1,N.

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Fig. 4
Fig. 4

(a) Spatiotemporal speckle interferogram for a horizontal region of interest recorded at 20,000 fps for object vibrating harmonically at 518 Hz with nominal maximum velocity 0.1vNyq . (b) Normalized velocity, ΔΦ/π, calculated from temporal phase stepped data. Comparison of normalized velocity for a single column calculated from temporal and spatial phase stepped data at (c) 0.1vNyq and (d) 0.5vNyq .

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Fig. 5
Fig. 5

Normalized velocity, ΔΦ/π, calculated from spatial phase stepped data at nominal maximum velocities of (a) 0.5 v N y q and (c) 4 v N y q . Images recorded at 20,000 fps for object vibrating harmonically at 250 Hz. Unwrapped normalized velocity for one pixel (column) is shown in (b) and (d).

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Fig. 6
Fig. 6

Maximum normalized velocity error plotted against (a) normalized velocity and (b) number of frames per vibration period.

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Fig. 7
Fig. 7

Maximum normalized velocity error plotted against number of frames per vibration period, for v/vNyq = 4. (a) Influence of exposure time. (b) Influence of zero-mean, Gaussian-distributed intensity noise. (c) Influence of phase step error.

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Equations (13)

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v N y q = λ 2 η t s
v N y q , P S =   v N y q − v P S   = λ 2 η t s ( 1 − 2 N )
I n ( x , y , t n ) = I ( 1 + V sinc [ ϕ 2 ]   cos [ Φ O − Φ R ] )
Φ O ( t n ) − Φ R ( t n ) = tan − 1 [ 3 ( I n − I n + 1 ) − ( I n − 1 − I n + 2 ) ]   [ ( I n − I n + 1 ) + ( I n − 1 − I n + 2 ) ] ( I n + I n + 1 ) − ( I n − 1 + I n + 2 )
w n ′ ( x , y , t n ) = λ 2 π η t s Δ Φ O ( x , y , t n ) o r w n ′ ( x , y , t n ) v N y q = Δ Φ O ( x , y , t n ) π
I n , N ( x , y , t n ) = I ( 1 + V sinc [ ϕ 2 ]   cos [ Φ O − Φ R ] )
Δ Φ O ( t n ) − π 2   = 2 tan − 1 I n − 1 , N − I n , N − 1 I n − 1 , N − 1 − I n , N
Δ Φ O ( t n ) π ¯ = Δ Φ O ( t n ) π ± k 2 v N y q
w n ″   ( x , y , t n ) = λ 2 π η t s 2 Δ   2 Φ O ( x , y , t n ) o r w n ″ ( x , y , t n ) v N y q / t s = Δ   2 Φ O ( x , y , t n ) π
a   = v N y q t s
Φ O ( x , t n ) = 2 π η λ   w 0   sin ( 2 π f t n )   sin ( 2 π x )   +   2 π x
w 0 = ( v v N y q )   v N y q 2 π f
1 25 ( 2 π f t s ) 2 ( v v N y q )

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