Abstract

Time-average TV holography is widely used method for vibration measurement. The method generates speckle correlation time-averaged J0 fringes that can be used for full-field qualitative visualization of mode shapes at resonant frequencies of an object under harmonic excitation. In order to map the amplitudes of vibration, quantitative evaluation of the time-averaged fringe pattern is desired. A quantitative evaluation procedure based on the phase-shifting technique used in two beam interferometry has also been adopted for this application with some modification. The existing procedure requires a large number of frames to be recorded for implementation. We propose a procedure that will reduce the number of frames required for the analysis. The TV holographic system used and the experimental results obtained with it on an edge-clamped, sinusoidally excited square aluminium plate sample are discussed.

© 2009 Optical Society of America

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References

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  1. P. K. Rastogi, ed., Digital Speckle Pattern Interferometry and Related Techniques (Wiley, 2001).
  2. A. F. Doval, “A systematic approach to TV holography,” Meas. Sci. Technol. 11, R1-R36 (2000).
    [CrossRef]
  3. N. K. Mohan, “Speckle techniques for optical metrology,” in Progress in Laser and Electro-Optics Research, W. T. Arkin, ed. (Nova Science, 2007), Chap. 9, pp. 173-222.
  4. L. X. Yang, W. Steinchen, G. Kupfer, P. Maeckel, and F. Voessing, “Vibration analysis by digital shearography,” Opt. Lasers Eng. 30, 199-212 (1998).
    [CrossRef]
  5. 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]
  6. R. J. Pryputniewicz and K. A. Stetson, “Measurement of vibration pattern using electro-optic holography,” Proc. SPIE 1162, 456-468 (1989).
  7. H. O. Saldner, N. K. Mohan, and N.-E. Molin, “Comparative TV holography for vibration analysis,” Opt. Eng. 34, 486-492 (1995).
    [CrossRef]
  8. 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]
  9. O. J. Løkberg and K. Høgmoen, “Vibration phase mapping using electronic speckle pattern interferometry,” Appl. Opt. 15, 2701-2704 (1976).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  12. W. O. Wong, “Vibration mode shape visualization with time average TV holography system,” Int. J. Eng. Sci. 14, 241-247 (1998).
  13. D. N. Borza, “High-resolution time-average electronic holography for vibration measurement,” Opt. Lasers Eng. 41, 515-527 (2004).
    [CrossRef]
  14. C. H. Huang and Y.-Y. Chen, “Transverse vibration analysis for piezoceramic rectangular plates using Ritz's method with equivalent constants,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 53, 265-273 (2006).
    [CrossRef] [PubMed]
  15. G. S. Schajer and M. Steinzig, “Sawblade vibration mode shape measurement using ESPI,” J. Test. Eval. 36, 1-5(2008).
  16. C.-C. Ma and H.-Y. Lin, “Experimental measurements on transverse vibration characteristics of piezoceramic rectangular plates by optical methods,” J. Sound Vib. 286, 587-600(2005).
    [CrossRef]
  17. K. A. Stetson, “Effect of phase mismatch on pseudo-phase-step analysis of time-average hologram recording of vibration modes,” Appl. Opt. 45, 6473-6476 (2006).
    [CrossRef] [PubMed]
  18. L. Yang and P. Colbourne, “Digital laser micro-interferometer and its applications,” Opt Eng. 42, 1417-1426 (2003).
    [CrossRef]
  19. U. P. Kumar, B. Bhaduri, N. K. Mohan, M. P. Kothiyal, and A. K. Asundi “Microscopic TV holography for MEMS deflection and 3-D surface profile characterization,” Opt. Lasers Eng. 46, 687-694 (2008).
    [CrossRef]
  20. L. X. Yang and A. L. Bhangaonka, “Investigation of natural frequencies under free-free conditions on objects by digital holographic speckle pattern interferometry,” presented at Annual Conference of the International Society of Experimental Mechanics, Charlotte, N.C., USA, 2-4 June, 2003.
  21. B. Bhaduri, M. P. Kothiyal, and N. K. Mohan, “Vibration mode shape visualization with dual function DSPI system,” Proc. SPIE 6292, 629217-629223 (2006).
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    [CrossRef] [PubMed]
  23. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

2009 (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]

2008 (2)

G. S. Schajer and M. Steinzig, “Sawblade vibration mode shape measurement using ESPI,” J. Test. Eval. 36, 1-5(2008).

U. P. Kumar, B. Bhaduri, N. K. Mohan, M. P. Kothiyal, and A. K. Asundi “Microscopic TV holography for MEMS deflection and 3-D surface profile characterization,” Opt. Lasers Eng. 46, 687-694 (2008).
[CrossRef]

2006 (3)

B. Bhaduri, M. P. Kothiyal, and N. K. Mohan, “Vibration mode shape visualization with dual function DSPI system,” Proc. SPIE 6292, 629217-629223 (2006).

C. H. Huang and Y.-Y. Chen, “Transverse vibration analysis for piezoceramic rectangular plates using Ritz's method with equivalent constants,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 53, 265-273 (2006).
[CrossRef] [PubMed]

K. A. Stetson, “Effect of phase mismatch on pseudo-phase-step analysis of time-average hologram recording of vibration modes,” Appl. Opt. 45, 6473-6476 (2006).
[CrossRef] [PubMed]

2005 (1)

C.-C. Ma and H.-Y. Lin, “Experimental measurements on transverse vibration characteristics of piezoceramic rectangular plates by optical methods,” J. Sound Vib. 286, 587-600(2005).
[CrossRef]

2004 (1)

D. N. Borza, “High-resolution time-average electronic holography for vibration measurement,” Opt. Lasers Eng. 41, 515-527 (2004).
[CrossRef]

2003 (1)

L. Yang and P. Colbourne, “Digital laser micro-interferometer and its applications,” Opt Eng. 42, 1417-1426 (2003).
[CrossRef]

2001 (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]

2000 (1)

A. F. Doval, “A systematic approach to TV holography,” Meas. Sci. Technol. 11, R1-R36 (2000).
[CrossRef]

1998 (2)

L. X. Yang, W. Steinchen, G. Kupfer, P. Maeckel, and F. Voessing, “Vibration analysis by digital shearography,” Opt. Lasers Eng. 30, 199-212 (1998).
[CrossRef]

W. O. Wong, “Vibration mode shape visualization with time average TV holography system,” Int. J. Eng. Sci. 14, 241-247 (1998).

1996 (1)

1995 (2)

1993 (1)

1989 (1)

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

1976 (1)

Asundi, A. K.

U. P. Kumar, B. Bhaduri, N. K. Mohan, M. P. Kothiyal, and A. K. Asundi “Microscopic TV holography for MEMS deflection and 3-D surface profile characterization,” Opt. Lasers Eng. 46, 687-694 (2008).
[CrossRef]

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.

U. P. Kumar, B. Bhaduri, N. K. Mohan, M. P. Kothiyal, and A. K. Asundi “Microscopic TV holography for MEMS deflection and 3-D surface profile characterization,” Opt. Lasers Eng. 46, 687-694 (2008).
[CrossRef]

B. Bhaduri, M. P. Kothiyal, and N. K. Mohan, “Vibration mode shape visualization with dual function DSPI system,” Proc. SPIE 6292, 629217-629223 (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,” presented at Annual Conference of the International Society of Experimental Mechanics, Charlotte, N.C., USA, 2-4 June, 2003.

Borza, D. N.

D. N. Borza, “High-resolution time-average electronic holography for vibration measurement,” Opt. Lasers Eng. 41, 515-527 (2004).
[CrossRef]

Chen, Y.-Y.

C. H. Huang and Y.-Y. Chen, “Transverse vibration analysis for piezoceramic rectangular plates using Ritz's method with equivalent constants,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 53, 265-273 (2006).
[CrossRef] [PubMed]

Colbourne, P.

L. Yang and P. Colbourne, “Digital laser micro-interferometer and its applications,” Opt Eng. 42, 1417-1426 (2003).
[CrossRef]

Creath, K.

Doval, A. F.

A. F. Doval, “A systematic approach to TV holography,” Meas. Sci. Technol. 11, R1-R36 (2000).
[CrossRef]

Ghiglia, D. C.

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

Høgmoen, K.

Huang, C. H.

C. H. Huang and Y.-Y. Chen, “Transverse vibration analysis for piezoceramic rectangular plates using Ritz's method with equivalent constants,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 53, 265-273 (2006).
[CrossRef] [PubMed]

Hwang, C.-H.

Kato, J.

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. P. Kumar, B. Bhaduri, N. K. Mohan, M. P. Kothiyal, and A. K. Asundi “Microscopic TV holography for MEMS deflection and 3-D surface profile characterization,” Opt. Lasers Eng. 46, 687-694 (2008).
[CrossRef]

B. Bhaduri, M. P. Kothiyal, and N. K. Mohan, “Vibration mode shape visualization with dual function DSPI system,” Proc. SPIE 6292, 629217-629223 (2006).

Kumar, U. P.

U. P. Kumar, B. Bhaduri, N. K. Mohan, M. P. Kothiyal, and A. K. Asundi “Microscopic TV holography for MEMS deflection and 3-D surface profile characterization,” Opt. Lasers Eng. 46, 687-694 (2008).
[CrossRef]

Kupfer, G.

L. X. Yang, W. Steinchen, G. Kupfer, P. Maeckel, and F. Voessing, “Vibration analysis by digital shearography,” Opt. Lasers Eng. 30, 199-212 (1998).
[CrossRef]

Lin, H.-Y.

C.-C. Ma and H.-Y. Lin, “Experimental measurements on transverse vibration characteristics of piezoceramic rectangular plates by optical methods,” J. Sound Vib. 286, 587-600(2005).
[CrossRef]

Lin, S.-Y.

Løkberg, O. J.

Ma, C.-C.

C.-C. Ma and H.-Y. Lin, “Experimental measurements on transverse vibration characteristics of piezoceramic rectangular plates by optical methods,” J. Sound Vib. 286, 587-600(2005).
[CrossRef]

Maeckel, P.

L. X. Yang, W. Steinchen, G. Kupfer, P. Maeckel, and F. Voessing, “Vibration analysis by digital shearography,” Opt. Lasers Eng. 30, 199-212 (1998).
[CrossRef]

Mohan, N. K.

U. P. Kumar, B. Bhaduri, N. K. Mohan, M. P. Kothiyal, and A. K. Asundi “Microscopic TV holography for MEMS deflection and 3-D surface profile characterization,” Opt. Lasers Eng. 46, 687-694 (2008).
[CrossRef]

B. Bhaduri, M. P. Kothiyal, and N. K. Mohan, “Vibration mode shape visualization with dual function DSPI system,” Proc. SPIE 6292, 629217-629223 (2006).

H. O. Saldner, N. K. Mohan, and N.-E. Molin, “Comparative TV holography for vibration analysis,” Opt. Eng. 34, 486-492 (1995).
[CrossRef]

N. K. Mohan, “Speckle techniques for optical metrology,” in Progress in Laser and Electro-Optics Research, W. T. Arkin, ed. (Nova Science, 2007), Chap. 9, pp. 173-222.

Molin, N.-E.

H. O. Saldner, N. K. Mohan, and N.-E. Molin, “Comparative TV holography for vibration analysis,” Opt. Eng. 34, 486-492 (1995).
[CrossRef]

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]

Ping, Q.

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).

Saldner, H. O.

H. O. Saldner, N. K. Mohan, and N.-E. Molin, “Comparative TV holography for vibration analysis,” Opt. Eng. 34, 486-492 (1995).
[CrossRef]

Schajer, G. S.

G. S. Schajer and M. Steinzig, “Sawblade vibration mode shape measurement using ESPI,” J. Test. Eval. 36, 1-5(2008).

Schmit, J.

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]

Steinchen, W.

L. X. Yang, W. Steinchen, G. Kupfer, P. Maeckel, and F. Voessing, “Vibration analysis by digital shearography,” Opt. Lasers Eng. 30, 199-212 (1998).
[CrossRef]

Steinzig, M.

G. S. Schajer and M. Steinzig, “Sawblade vibration mode shape measurement using ESPI,” J. Test. Eval. 36, 1-5(2008).

Stetson, K. A.

K. A. Stetson, “Effect of phase mismatch on pseudo-phase-step analysis of time-average hologram recording of vibration modes,” Appl. Opt. 45, 6473-6476 (2006).
[CrossRef] [PubMed]

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

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]

L. X. Yang, W. Steinchen, G. Kupfer, P. Maeckel, and F. Voessing, “Vibration analysis by digital shearography,” Opt. Lasers Eng. 30, 199-212 (1998).
[CrossRef]

Wang, W.-C.

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]

Wong, W. O.

W. O. Wong, “Vibration mode shape visualization with time average TV holography system,” Int. J. Eng. Sci. 14, 241-247 (1998).

Yamaguchi, I.

Yang, L.

L. Yang and P. Colbourne, “Digital laser micro-interferometer and its applications,” Opt Eng. 42, 1417-1426 (2003).
[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, W. Steinchen, G. Kupfer, P. Maeckel, and F. Voessing, “Vibration analysis by digital shearography,” Opt. Lasers Eng. 30, 199-212 (1998).
[CrossRef]

L. X. Yang and A. L. Bhangaonka, “Investigation of natural frequencies under free-free conditions on objects by digital holographic speckle pattern interferometry,” presented at Annual Conference of the International Society of Experimental Mechanics, Charlotte, N.C., USA, 2-4 June, 2003.

Appl. Opt. (5)

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

C. H. Huang and Y.-Y. Chen, “Transverse vibration analysis for piezoceramic rectangular plates using Ritz's method with equivalent constants,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 53, 265-273 (2006).
[CrossRef] [PubMed]

Int. J. Eng. Sci. (1)

W. O. Wong, “Vibration mode shape visualization with time average TV holography system,” Int. J. Eng. Sci. 14, 241-247 (1998).

J. Sound Vib. (1)

C.-C. Ma and H.-Y. Lin, “Experimental measurements on transverse vibration characteristics of piezoceramic rectangular plates by optical methods,” J. Sound Vib. 286, 587-600(2005).
[CrossRef]

J. Test. Eval. (1)

G. S. Schajer and M. Steinzig, “Sawblade vibration mode shape measurement using ESPI,” J. Test. Eval. 36, 1-5(2008).

Meas. Sci. Technol. (1)

A. F. Doval, “A systematic approach to TV holography,” Meas. Sci. Technol. 11, R1-R36 (2000).
[CrossRef]

Opt Eng. (1)

L. Yang and P. Colbourne, “Digital laser micro-interferometer and its applications,” Opt Eng. 42, 1417-1426 (2003).
[CrossRef]

Opt. Eng. (1)

H. O. Saldner, N. K. Mohan, and N.-E. Molin, “Comparative TV holography for vibration analysis,” Opt. Eng. 34, 486-492 (1995).
[CrossRef]

Opt. Lasers Eng. (4)

D. N. Borza, “High-resolution time-average electronic holography for vibration measurement,” Opt. Lasers Eng. 41, 515-527 (2004).
[CrossRef]

L. X. Yang, W. Steinchen, G. Kupfer, P. Maeckel, and F. Voessing, “Vibration analysis by digital shearography,” Opt. Lasers Eng. 30, 199-212 (1998).
[CrossRef]

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]

U. P. Kumar, B. Bhaduri, N. K. Mohan, M. P. Kothiyal, and A. K. Asundi “Microscopic TV holography for MEMS deflection and 3-D surface profile characterization,” Opt. Lasers Eng. 46, 687-694 (2008).
[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]

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. K. Mohan, “Vibration mode shape visualization with dual function DSPI system,” Proc. SPIE 6292, 629217-629223 (2006).

Other (4)

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

L. X. Yang and A. L. Bhangaonka, “Investigation of natural frequencies under free-free conditions on objects by digital holographic speckle pattern interferometry,” presented at Annual Conference of the International Society of Experimental Mechanics, Charlotte, N.C., USA, 2-4 June, 2003.

N. K. Mohan, “Speckle techniques for optical metrology,” in Progress in Laser and Electro-Optics Research, W. T. Arkin, ed. (Nova Science, 2007), Chap. 9, pp. 173-222.

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

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Figures (9)

Fig. 1
Fig. 1

Fringes obtained with a vibrating sample at resonance, using Eqs (3, 4, 5).

Fig. 2
Fig. 2

Phase error as a function of percentage bias error for the three-step [Eq. (8)], 4-step A [Eq. (10)], and 4-step B [Eq. (11)] phase-shifting algorithms.

Fig. 3
Fig. 3

(a) Lookup table in graphical form for the error ε between Ω and Ω ( B = π / 2 ); (b) phase error introduced (ε) when the Bessel function assumed to be a cosine function.

Fig. 4
Fig. 4

Schematic of a time-average TV holography arrangement.

Fig. 5
Fig. 5

Mode shapes at different resonant frequencies of an aluminum plate ( 80  mm × 80   mm × 1   mm ) rigidly clamped at the bottom.

Fig. 6
Fig. 6

Phase shift calibration using the phase bias modulation method with the PMRM reference mirror. The aluminium plate ( 80  mm × 80   mm × 1   mm ) is rigidly clamped at the bottom and is vibrating at a frequency of 1772 Hz . (a) J 0 ( Ω ) fringes without the bias modulation; (b) π phase-shifted J 0 ( Ω ) fringes with the phase bias modulation and an arbitrary phase relation between the object and the reference beam excitations; (c) π phase-shifted J 0 ( Ω ) fringes with the phase bias modulation and an in-phase relation between the object and the reference beam excitations; (d) π phase-shifted J 0 ( Ω ) fringes with the phase bias modulation and another in-phase relation by addition of 180 ° to the current phase value between the object and the reference beam excitations.

Fig. 7
Fig. 7

Phase-shifted J 0 ( Ω ) fringe patterns on the sample at the vibration frequency of 1772 Hz . (a) Initial fringes, (b)  π / 2 -phase-shifted fringes, (c) π-phase-shifted fringes, (d)  3 π / 2 -phase-shifted fringes.

Fig. 8
Fig. 8

Results of analysis of phase-shifted J 0 ( Ω ) fringe patterns shown in Fig. 7. (a) Raw wrapped phase map, (b) filtered wrapped phase map using median filtering with 3 × 3 window, and (c) 3D plot of out-of-plane vibration amplitude.

Fig. 9
Fig. 9

Results of analysis when the object is vibrating at a frequency 2921 Hz . (a) Raw wrapped phase map, (b) filtered wrapped phase map, and (c) 3D plot of out-of-plane vibration amplitude.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

I avg = I o [ 1 + V cos ( ϕ ) J 0 ( Ω ) ] ,
I 1 = I o [ 1 + V cos ( ϕ ) J 0 ( Ω ) ] , I 2 = I o [ 1 V sin ( ϕ ) J 0 ( Ω ) ] , I 3 = I o [ 1 V cos ( ϕ ) J 0 ( Ω ) ] , I 4 = I o [ 1 + V sin ( ϕ ) J 0 ( Ω ) ] .
T = ( I 1 I 3 ) 2 + ( I 2 I 4 ) 2 = 4 I o 2 V 2 J 0 2 ( Ω ) .
I s = I r I avg = I o ( 1 + V cos ( ϕ ) ) I o ( 1 + V cos ( ϕ ) J o ( Ω ) ) = I o V cos ( ϕ ) ( 1 J o ( Ω ) ) .
R = | I 1 I 3 | = | 2 I o V cos ( ϕ ) J 0 ( Ω ) | .
T 1 = 4 I o 2 V 2 J 0 2 ( Ω ) , T 2 = 4 I o 2 V 2 J 0 2 ( Ω + B ) , T 3 = 4 I o 2 V 2 J 0 2 ( Ω + 2 B ) .
T ( n + 1 ) = 4 I o 2 V 2 J 0 2 ( Ω + n B ) ,
Ω = arctan ( T 1 2 T 2 + T 3 T 1 T 3 ) .
R ( n + 1 ) = | 2 I o V cos ( ϕ ) J 0 ( Ω + n B ) | .
Ω = arctan ( R 1 3 R 2 + R 3 + R 4 R 1 + R 2 3 R 3 + R 4 ) .
Ω = arctan ( R 4 R 2 R 1 R 3 ) .
Ω = Ω + ε ,
Ω = 4 π λ A .

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