Abstract

Quantitative phase information from a single interferogram can be obtained using the Hilbert transform (HT). We have applied the HT method for quantitative evaluation of Bessel fringes obtained in time average TV holography. The method requires only one fringe pattern for the extraction of vibration amplitude and reduces the complexity in quantifying the data experienced in the time average reference bias modulation method, which uses multiple fringe frames. The technique is demonstrated for the measurement of out-of-plane vibration amplitude on a small scale specimen using a time average microscopic TV holography system.

© 2010 Optical Society of America

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References

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  1. P. K. Rastogi, Digital Speckle Pattern Interferometry and Related Techniques (Wiley, 2001).
  2. W.Osten, Ed., Optical Inspection of Microsystems (CRC, 2007).
  3. R. J. Pryputniewicz and K. A. Stetson, “Measurement of vibration pattern using electro-optic holography,” Proc. SPIE 1162, 456–468 (1989).
  4. S.-H. Baik, S.-K. Park, C.-J. Kimi, and S.-Y. Kim, “Analysis of phase measurement errors in electro-optic holography,” Opt. Rev. 8, 26–31 (2001).
    [CrossRef]
  5. U. Paul Kumar, Y. Kalyani, N. Krishna Mohan, and M. P. Kothiyal, “Time average TV holography for vibration fringe analysis,” Appl. Opt. 48, 3094–3101 (2009).
    [CrossRef] [PubMed]
  6. L. X. Yang, M. Schuth, D. Thomas, Y. H. Wang, and F. Voessing, “Stroboscopic digital speckle pattern interferometry for vibration analysis of microsystems,” Opt. Lasers Eng. 47, 252–258 (2009).
    [CrossRef]
  7. L. X. Yang and A. L. Bhangaonka, “Investigation of natural frequencies under free-free conditions on objects by digital holographic speckle pattern interferometry,” ISEM, Charlotte, USA, 2–4 June, 2003.
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    [PubMed]
  16. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

2009 (3)

U. Paul Kumar, Y. Kalyani, N. Krishna Mohan, and M. P. Kothiyal, “Time average TV holography for vibration fringe analysis,” Appl. Opt. 48, 3094–3101 (2009).
[CrossRef] [PubMed]

L. X. Yang, M. Schuth, D. Thomas, Y. H. Wang, and F. Voessing, “Stroboscopic digital speckle pattern interferometry for vibration analysis of microsystems,” Opt. Lasers Eng. 47, 252–258 (2009).
[CrossRef]

X. Guanlei, W. Xiaotong, and X. Xiaogang, “Improved bi-dimensional EMD and Hilbert spectrum for the analysis of textures,” Pattern Recogn. 42, 718–734 (2009).
[CrossRef]

2007 (1)

F. A. M. Rodriguez, A. Federico, and G. H. Kaufmann, “Phase measurement improvement in temporal spackle pattern interferometry using empirical mode decomposition,” Opt. Commun. 275, 38–41 (2007).
[CrossRef]

2006 (1)

B. Bhaduri, M. P. Kothiyal, and N. Krishna Mohan, “Vibration mode shape visualization with dual function DSPI system,” Proc. SPIE 6292, 629217 (2006).

2003 (1)

2001 (3)

1997 (1)

1989 (1)

R. J. Pryputniewicz and K. A. Stetson, “Measurement of vibration pattern using electro-optic holography,” Proc. SPIE 1162, 456–468 (1989).

Baik, S.-H.

S.-H. Baik, S.-K. Park, C.-J. Kimi, and S.-Y. Kim, “Analysis of phase measurement errors in electro-optic holography,” Opt. Rev. 8, 26–31 (2001).
[CrossRef]

Bhaduri, B.

B. Bhaduri, M. P. Kothiyal, and N. Krishna Mohan, “Vibration mode shape visualization with dual function DSPI system,” Proc. SPIE 6292, 629217 (2006).

Bhangaonka, A. L.

L. X. Yang and A. L. Bhangaonka, “Investigation of natural frequencies under free-free conditions on objects by digital holographic speckle pattern interferometry,” ISEM, Charlotte, USA, 2–4 June, 2003.

Bone, D. J.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th extended ed. (Cambridge, , 1999).
[PubMed]

Federico, A.

F. A. M. Rodriguez, A. Federico, and G. H. Kaufmann, “Phase measurement improvement in temporal spackle pattern interferometry using empirical mode decomposition,” Opt. Commun. 275, 38–41 (2007).
[CrossRef]

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

Guanlei, X.

X. Guanlei, W. Xiaotong, and X. Xiaogang, “Improved bi-dimensional EMD and Hilbert spectrum for the analysis of textures,” Pattern Recogn. 42, 718–734 (2009).
[CrossRef]

Kadono, H.

Kalyani, Y.

Kaufmann, G. H.

F. A. M. Rodriguez, A. Federico, and G. H. Kaufmann, “Phase measurement improvement in temporal spackle pattern interferometry using empirical mode decomposition,” Opt. Commun. 275, 38–41 (2007).
[CrossRef]

Kim, S.-Y.

S.-H. Baik, S.-K. Park, C.-J. Kimi, and S.-Y. Kim, “Analysis of phase measurement errors in electro-optic holography,” Opt. Rev. 8, 26–31 (2001).
[CrossRef]

Kimi, C.-J.

S.-H. Baik, S.-K. Park, C.-J. Kimi, and S.-Y. Kim, “Analysis of phase measurement errors in electro-optic holography,” Opt. Rev. 8, 26–31 (2001).
[CrossRef]

Kothiyal, M. P.

U. Paul Kumar, Y. Kalyani, N. Krishna Mohan, and M. P. Kothiyal, “Time average TV holography for vibration fringe analysis,” Appl. Opt. 48, 3094–3101 (2009).
[CrossRef] [PubMed]

B. Bhaduri, M. P. Kothiyal, and N. Krishna Mohan, “Vibration mode shape visualization with dual function DSPI system,” Proc. SPIE 6292, 629217 (2006).

Kumar, U. Paul

Larkin, K. G.

Lohmann, W.

Madjarova, V. D.

Mohan, N. Krishna

U. Paul Kumar, Y. Kalyani, N. Krishna Mohan, and M. P. Kothiyal, “Time average TV holography for vibration fringe analysis,” Appl. Opt. 48, 3094–3101 (2009).
[CrossRef] [PubMed]

B. Bhaduri, M. P. Kothiyal, and N. Krishna Mohan, “Vibration mode shape visualization with dual function DSPI system,” Proc. SPIE 6292, 629217 (2006).

Oldfield, M. A.

Park, S.-K.

S.-H. Baik, S.-K. Park, C.-J. Kimi, and S.-Y. Kim, “Analysis of phase measurement errors in electro-optic holography,” Opt. Rev. 8, 26–31 (2001).
[CrossRef]

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

Pryputniewicz, R. J.

R. J. Pryputniewicz and K. A. Stetson, “Measurement of vibration pattern using electro-optic holography,” Proc. SPIE 1162, 456–468 (1989).

Ramirez, J. G.

Rastogi, P. K.

P. K. Rastogi, Digital Speckle Pattern Interferometry and Related Techniques (Wiley, 2001).

Rodriguez, F. A. M.

F. A. M. Rodriguez, A. Federico, and G. H. Kaufmann, “Phase measurement improvement in temporal spackle pattern interferometry using empirical mode decomposition,” Opt. Commun. 275, 38–41 (2007).
[CrossRef]

Schuth, M.

L. X. Yang, M. Schuth, D. Thomas, Y. H. Wang, and F. Voessing, “Stroboscopic digital speckle pattern interferometry for vibration analysis of microsystems,” Opt. Lasers Eng. 47, 252–258 (2009).
[CrossRef]

Stetson, K. A.

R. J. Pryputniewicz and K. A. Stetson, “Measurement of vibration pattern using electro-optic holography,” Proc. SPIE 1162, 456–468 (1989).

Tepichin, E.

Thomas, D.

L. X. Yang, M. Schuth, D. Thomas, Y. H. Wang, and F. Voessing, “Stroboscopic digital speckle pattern interferometry for vibration analysis of microsystems,” Opt. Lasers Eng. 47, 252–258 (2009).
[CrossRef]

Voessing, F.

L. X. Yang, M. Schuth, D. Thomas, Y. H. Wang, and F. Voessing, “Stroboscopic digital speckle pattern interferometry for vibration analysis of microsystems,” Opt. Lasers Eng. 47, 252–258 (2009).
[CrossRef]

Wang, Y. H.

L. X. Yang, M. Schuth, D. Thomas, Y. H. Wang, and F. Voessing, “Stroboscopic digital speckle pattern interferometry for vibration analysis of microsystems,” Opt. Lasers Eng. 47, 252–258 (2009).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th extended ed. (Cambridge, , 1999).
[PubMed]

Xiaogang, X.

X. Guanlei, W. Xiaotong, and X. Xiaogang, “Improved bi-dimensional EMD and Hilbert spectrum for the analysis of textures,” Pattern Recogn. 42, 718–734 (2009).
[CrossRef]

Xiaotong, W.

X. Guanlei, W. Xiaotong, and X. Xiaogang, “Improved bi-dimensional EMD and Hilbert spectrum for the analysis of textures,” Pattern Recogn. 42, 718–734 (2009).
[CrossRef]

Yang, L. X.

L. X. Yang, M. Schuth, D. Thomas, Y. H. Wang, and F. Voessing, “Stroboscopic digital speckle pattern interferometry for vibration analysis of microsystems,” Opt. Lasers Eng. 47, 252–258 (2009).
[CrossRef]

L. X. Yang and A. L. Bhangaonka, “Investigation of natural frequencies under free-free conditions on objects by digital holographic speckle pattern interferometry,” ISEM, Charlotte, USA, 2–4 June, 2003.

Appl. Opt. (2)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

F. A. M. Rodriguez, A. Federico, and G. H. Kaufmann, “Phase measurement improvement in temporal spackle pattern interferometry using empirical mode decomposition,” Opt. Commun. 275, 38–41 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (1)

L. X. Yang, M. Schuth, D. Thomas, Y. H. Wang, and F. Voessing, “Stroboscopic digital speckle pattern interferometry for vibration analysis of microsystems,” Opt. Lasers Eng. 47, 252–258 (2009).
[CrossRef]

Opt. Rev. (1)

S.-H. Baik, S.-K. Park, C.-J. Kimi, and S.-Y. Kim, “Analysis of phase measurement errors in electro-optic holography,” Opt. Rev. 8, 26–31 (2001).
[CrossRef]

Pattern Recogn. (1)

X. Guanlei, W. Xiaotong, and X. Xiaogang, “Improved bi-dimensional EMD and Hilbert spectrum for the analysis of textures,” Pattern Recogn. 42, 718–734 (2009).
[CrossRef]

Proc. SPIE (2)

R. J. Pryputniewicz and K. A. Stetson, “Measurement of vibration pattern using electro-optic holography,” Proc. SPIE 1162, 456–468 (1989).

B. Bhaduri, M. P. Kothiyal, and N. Krishna Mohan, “Vibration mode shape visualization with dual function DSPI system,” Proc. SPIE 6292, 629217 (2006).

Other (5)

L. X. Yang and A. L. Bhangaonka, “Investigation of natural frequencies under free-free conditions on objects by digital holographic speckle pattern interferometry,” ISEM, Charlotte, USA, 2–4 June, 2003.

P. K. Rastogi, Digital Speckle Pattern Interferometry and Related Techniques (Wiley, 2001).

W.Osten, Ed., Optical Inspection of Microsystems (CRC, 2007).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th extended ed. (Cambridge, , 1999).
[PubMed]

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

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Figures (10)

Fig. 1
Fig. 1

(a) Line scan of a simulated closed fringe pattern, (b) π / 2 phase shifted signal from 1(a), and (c) HT signal of the pattern shown in Fig. 1a.

Fig. 2
Fig. 2

(a) Simulated J 0 ( Ω ) fringes, (b) fringe after applying HT to Fig. 2a, and (c) line scan profile of the patterns shown in Figs. 2a, 2b.

Fig. 3
Fig. 3

(a) Wrapped phase map calculated using the patterns shown in Figs. 2a, 2b, and (b) line scan profile.

Fig. 4
Fig. 4

(a) Calibration curve (lookup table) between the input Ω and calculated Ω , and (b) phase error ε v plot.

Fig. 5
Fig. 5

(a) Simulated J 0 fringe pattern, (b) fringe after applying HT to (a), (c) wrapped phase map with the π shift, (d) mask generated by identifying the parts where π shift is happening, and (e) wrapped phase map after removing the sign ambiguity.

Fig. 6
Fig. 6

Schematic of the time average microscopic TV holographic system: SF, spatial filter; BS1, beam splitter; CL, collimating lens; BS2, cube beam splitter; NDF, neutral density filter; M, mirror; A 1 , amplifier; A 2 , amplifier for object excitation; PZTM, piezoelectric transducer mirror; FG, function generator; DAQ, digital to analog converter.

Fig. 7
Fig. 7

Fringes obtained with a vibrating PZT cantilever at resonance frequency ( 5.72 kHz ) using Eqs. (3, 4), respectively.

Fig. 8
Fig. 8

PZT cantilever beam vibrating at fundamental frequency 5.72 kHz : (a) time average fringes, (b) central line scan profile, (c) filtered fringe pattern, (d) fringe after applying HT to (c), and (e) central line scan profile of the patterns shown in Figs. 8c, 8d.

Fig. 9
Fig. 9

(a) Wrapped phase map obtained using the HT method, (b) 3D view of the shape made, (c) wrapped phase map obtained using bias phase modulation method, and (d) Profiles A and B are the central line scan profiles of the phase maps shown in Figs. 9a, 9c after unwrapping and correcting the error.

Fig. 10
Fig. 10

Thin circular aluminum membrane vibrating at the fundamental frequency of 4 KHz : (a) fringe pattern, (b) wrapped phase map obtained using the HT method, (c) 3D view of the mode shape, (d) wrapped phase map obtained using bias phase modulation method, and (e) central line scan profiles (A and B) of the phase maps shown in Figs. 10b, 10d after unwrapping and correcting phase. A shift is introduced between Profiles A and B for clarity.

Equations (8)

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I avg ( 0 ) ( x , y ) = I o ( x , y ) ( 1 + V ( x , y ) cos ( ϕ ( x , y ) ) J 0 ( Ω ( x , y ) ) ) ,
I avg ( π ) ( x , y ) = I o ( x , y ) ( 1 V ( x , y ) cos ( ϕ ( x , y ) ) J 0 ( Ω ( x , y ) ) ) .
P = | I avg ( 0 ) I avg ( π ) | = | 2 V I o cos ( ϕ ) J 0 ( Ω ) | .
R = | I avg ( 0 ) I avg ( π ) I avg ( 0 ) + I avg ( π ) | = | V cos ( ϕ ) J 0 ( Ω ) | .
v ( x ) = H i { u ( x ) } = 1 π u ( x ) x x d x .
H i { u ( x ) } = 1 π x u ( x ) ,
Ω = arctan ( H i { f ( ϕ ( x ) ) } f ( ϕ ( x ) ) .
Ω = Ω + ε v ,

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