Abstract

opportunities for full field 2D amplitude and phase vibration analysis are presented. It is demonstrated that it is possible to simultaneously encode-decode 2D the amplitude and phase of harmonic mechanical vibrations. The process allows the determination of in plane and out of plane vibration components when the object is under a pure sinusoidal excitation. The principle is based on spatial multiplexing in digital Fresnel holography. Experimental results are presented in the case of an industrial application.

©2005 Optical Society of America

1. Introduction

Digital holography appeared in the last decade with cheap high resolution CCD cameras and the increasing power of computers [1]. Digital Fresnel holography is a powerful tool for metrological applications such as tomographic imaging, biological imaging, surface measurement and characterization, fluid mechanics investigation or vibration analysis. Some demonstrative examples can be found in references [2,3,4,5,6,7,8]. In a recent publication [4], vibrations are analyzed by use of the time averaging principle [9,10] with quasi-Fourier holograms. But the phase of the vibration signal is lost with time-averaging. However, for vibration analysis, amplitude and phase retrieving is a challenge for full field optical metrology because it is necessary to give insight of the mechanical vibrating behavior. Lately was demonstrated the possibility of using digital Fresnel holography for full field vibrometry [11]. This paper proposes an extension of this principle to 2D full field vibration analysis by combining digital holography and spatial multiplexing of holograms. The paper is organized as follows : Section 2 presents the theoretical analysis and briefly describes the setup which was used. Section 3 focuses on simultaneous 2D amplitude and phase retrieving. Section 4 presents experimental results in the case of an industrial application with an automotive car joint made of elastomer; in particular, mean quadratic velocities along two vibration directions are estimated and a kinetic representation of the velocity field is shown through a number of films.

2. Theory and set-up

When a rough object is submitted to a pure sinusoidal excitation, it induces a spatio-temporal displacement vector which can be written as

U(t)=uxsin(ω0t+φx)i+uysin(ω0t+φy)j+uzsin(ω0t+φz)k,

where {ux ,uy ,uz } are the maximum amplitudes at pulsation ω 0 = 2π/T 0, and {φx , φy , φz } are the phases of the mechanical vibration along the three directions of a set of reference axis (x,y,z) attached to the object. When illuminated by a coherent laser beam, the object induces a spatio-temporal phase modulation given by

A(t)=A0exp(iψ0)exp[2S.U(t)/λ],

where S is the sensitivity vector of the set-up, λ is the wavelength of the laser and ψ 0 is the random phase due to the roughness of the object. The field diffracted along a distance d 0 from the object plane can be encoded by use of interferences with a smooth plane reference beam R(x’,y’). The recorded hologram H(x’,y’,d 0,t) can be used for a digital reconstruction of the initial object wave [1]. The numerical process is performed according to diffraction theory considered in the Fresnel approximation [1]. The reconstructed object is then computed following Eq. (3)

AR(x,y,d0,t)=iexp(2iπd0/λ)λd0exp[iπλd0(x2+y2)]
×k=0k=K1l=0l=L1H(lpx,kpy,d0,t)exp[iπλd0(l2px2+k2py2)]exp[2iπλd0(lxpx+kypy)].

In Eq. (3), {K,L} are the number of pixels of the reconstructed field and {px ,py } are the pixel pitches of the CCD area used for the recording. According to Eq. (3) and references [9,10], the reconstructed object in the +1 order takes the form of

A+1R(x,y,d0,t)MNλ4d04R*(x,y)exp[iπλd0(u02+v02)]
×A0(x,y)exp[iψ0(x,y)]exp[2S.U(t)/λ]*δ(xλu0d0,yλv0d0).

In Eq. (4), {u 0,v 0} are the spatial frequencies of the reference wave R(x,y), {M,N} are the number of effective pixels of the CCD area, * means convolution and δ(x,y) is the bidimensionnal Dirac distribution. If we consider now that the object is illuminated by two beams and that each diffracted beam interferes with its appropriated reference wave, then Eq. (4) will contain two reconstructed objects whose localizations are governed by spatial frequencies {ui ,vi }, i = 1,2. Related phase terms depend on each associated sensitivity. A possible set-up enabling this is described in Fig. 1.

 figure: Fig. 1.

Fig. 1. Experimental set-up for simultaneous 2D vibration analysis

Download Full Size | PPT Slide | PDF

In the case of the results presented in this paper, the incoherent addition of the two multiplexed holograms is performed by use of a delay line between the two Mach-Zehnder interferometers; indeed, some depolarization occurs due to the nature of the material composing the object (see Section 4). Note that pure polarizing interferometers can be used in other cases [13]. The spatial frequencies of the reference waves are adjusted such that there is no overlapping between the five diffracted orders when the field is reconstructed [12]. So, at any time tj when a recording is performed and followed by a numerical reconstruction, the phase of the i th (i = 1,2) reconstructed object is given by

ψJi=ψ0±Δφxsin(θ)sin(ω0tj+φx)Δφz[1+cos(θ)]sin(ω0tj+φz),

where Δφ x,z = 2πu x,z /λ and irrelevant phase terms are omitted. The two spatially multiplexed holograms include amplitude and phase of the vibration along x and z directions. Demultiplexing consists in determining the point to point relation between the two holograms. It is achieved according to reference [12] and it allows numerical computation of sum and difference of phase terms resulting from the two multiplexed holograms.

3. 2D amplitude and phase retrieving

Equation (5) indicates that phases ψji carry information on in plane and out of plane vibrations. In a pure sinusoidal regime, vibrating terms can be retrieved by use of a synchronic excitation-acquisition scheme. For example, if one records holograms at three different synchronous times t 1, t 2, t 3 such that ω 0(t 2-t 1) = π/2 and ω 0(t 3-t 1) = π, the reconstructed phase terms ψji are then used to extract phase quantities Δψkli = ψki - ψli . Synchronisation can be carried out with a stroboscopic set-up. The device is exhaustively described in reference [11]. Quantities Δψ13i, Δψ21i and Δψ23i carry only information on in plane and out of plane vibrations and there is no dependence with the random phase ψ 0. Note that these quantities are determined modulo 2π and that they must be unwrapped. According to Eq. (5), pure in plane and out of plane vibration terms are determined by computing continuous quantities Δψkl_x = Δψkl1- Δψkl2 and Δψkl_z = Δψkl1 + Δψkl2. With these quantities it is now possible to extract the amplitude and the phase of the vibration along x and z directions according to the following algorithms [11]

ΔφA=12[Δψ13_A]2+[Δψ23_A+Δψ21_A]2,

and

φA=arctan[Δψ13_AΔψ23_A+Δψ21_A],

where A = x or A = z.

4. Experimental results

The set-up of Fig. 1 and the measurement principle were applied to a multifaceted industrial automotive car joint made of elastomer. The unit is clamped along the y direction such that it can be considered as infinite in this direction, so vibrations exist mainly in z and x directions. The car joint is excited at its rear face by a louspeaker and laser beams illuminate its front face which is placed at a distance of 1348 mm from the CCD (see Fig. 1). The zone under inspection on the unit is 58.4 mm by 15.4 mm. The stroboscopic set-up generates excitation of the loudspeaker in synchronisation with the illumination of the part with light pulses from a mechanical chopper. Acoustic waves emitted by the loudspeaker cause the front face of the car joint to vibrate by vibroacoustic coupling inside the mechanical structure. The vibration of the front face is of interest in this study, particularly its behavior in relation to the excitation frequency. The off-axis holographic recording is carried out using lenses L1 and L3 which are displaced out of their optical axis by means of two micrometric transducers [12]. Therefore, frequencies of carrier waves are {u 1,v 1} = {76mm-1,54mm-1} and {u 2,v 2} = {71mm-1, -50mm-1}. The detector is a 12-bit digital CCD with (M×N) = (1024×1360) pixels of pitch px = py = 4.65μm (PCO Pixel Fly). Digital reconstruction was performed with K = L = 2048 data points by use of zero-padding. In the set-up, illuminating angle θ is set to 45°. Figure 2 shows the multiplexed holograms of the unit.

 figure: Fig. 2.

Fig. 2. Multiplexed holograms of the car joint piece

Download Full Size | PPT Slide | PDF

The evaluation of the vibration amplitude and phase according to algorithms 6,7 can be used to determine the mean quadratic velocity of the front face of the piece. This parameter is currently used as criterion by acoustic engineers for qualifying vibrations. It is defined for pulsation ω 0 by

vA2=ω02πSs0T0vA(x,y,t)2dtdxdy,

where S is the surface of the object under consideration and A = x or A = z. Considering the sensitivity vectors, we have

vz(x,y,t)=λω04π(1+cosθ)Δφz(x,y)cos(ω0+φz),

and

vz(x,y,t)=λω04πsin(θ)Δφx(x,y)cos(ω0t+φx).

As an example of the results that were obtained by this technique, Fig. 3 shows in plane and out of plane vibration amplitudes and phases for a frequency of 680Hz.

 figure: Fig. 3.

Fig. 3. 2D vibration amplitude and phase at a frequency of 680 Hz

Download Full Size | PPT Slide | PDF

The region of interest in each map contains approximately 143550 data points. Compared to classical vibrometers which use a point wise optical probe [14], the digital holographic set-up leads to 2D full field information on vibration. As pointed out previously, the determination of amplitude and phase allows the computation of the mean quadratic velocity along the two sensitivities. Figure 4 shows the mean quadratic velocities extracted from the set of data for an excitation frequency varying from 200Hz to 1000Hz.

 figure: Fig. 4.

Fig. 4. Mean quadratic velocities extracted from the experimental results

Download Full Size | PPT Slide | PDF

Qualitative information on how the surface of the piece vibrates can be appreciated also by observing the kinetic representation of the vibration in the two measured directions. The parameter of interests is now the combined velocity of each point of the surface by taking into account vibration amplitudes and phases. Considering Eq. (1) in which uy component is set equal to zero, then the velocity of a surface point is v(t) = ω 0 ux cos(ω 0 t+(ψx ) + ω 0 uz cos(ω 0 t+(ωz ). Figures 5 to 12 show the kinetic representation of v(t) for 27×5 points on the surface of the piece for excitation frequencies 340Hz, 430Hz, 460Hz, 680Hz, 730Hz, 880Hz, 900Hz and 930Hz. This set of frequencies corresponds to peaks in the mean quadratic velocity curves and are indicated by vertical arrows in Fig. 4. In Fig. 5 to 12, the representation is chosen as follows : a horizontal arrow means a pure in plan vibration whereas a vertical arrow means a pure out of plane vibration. From films of Fig. 5 to 12, an elliptic behavior is observed for some points of the surface.

 figure: Fig. 5.

Fig. 5. 1064 Ko Movie 1 - Kinetic representation of Fresnel for 2D vibration at 340Hz [Media 9]

Download Full Size | PPT Slide | PDF

 figure: Fig. 6.

Fig. 6. 912 Ko Movie 2 - Kinetic representation of Fresnel for 2D vibration at 430Hz

Download Full Size | PPT Slide | PDF

 figure: Fig. 7.

Fig. 7. 1403 Ko Movie 3 - Kinetic representation of Fresnel for 2D vibration at 460Hz

Download Full Size | PPT Slide | PDF

 figure: Fig. 8.

Fig. 8. 1609 Ko Movie 4 - Kinetic representation of Fresnel for 2D vibration at 680Hz

Download Full Size | PPT Slide | PDF

 figure: Fig. 9.

Fig. 9. 1586 Ko Movie 5 - Kinetic representation of Fresnel for 2D vibration at 730Hz

Download Full Size | PPT Slide | PDF

 figure: Fig. 10.

Fig. 10. 1251 Ko Movie 6 - Kinetic representation of Fresnel for 2D vibration at 880Hz

Download Full Size | PPT Slide | PDF

 figure: Fig. 11.

Fig. 11. 1093 Ko Movie 7 - Kinetic representation of Fresnel for 2D vibration at 900Hz

Download Full Size | PPT Slide | PDF

 figure: Fig. 12.

Fig. 12. 1321 Ko Movie 8 - Kinetic representation of Fresnel for 2D vibration at 930Hz

Download Full Size | PPT Slide | PDF

5. Conclusion

This paper has presented a 2D full field vibrometer based on digital Fresnel holography and spatial multiplexing of holograms. Amplitude and phase of the vibration along z and x directions can be extracted with the recording of only three synchronous holograms. The velocity of surface points is easy to determine and can be used for analysis of the mechanical behavior of the object being tested. Experimental results were presented and exhibit the relevance of digital Fresnel holography in 2D vibrometry.

References and Links

1 . U. Schnars and W. Jüptner , “ Direct recording of holograms by a CCD target and numerical reconstruction ,” App. Opt. 33 , 179 – 181 ( 1994 ). [CrossRef]  

2 . L. Yu and M.K. Kim , “ Wavelength scanning digital interference holography for variable tomographic scanning ,” Opt. Express 13 , 5621 – 5627 ( 2005 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-15-5621. [CrossRef]   [PubMed]  

3 . M. Paturzo , P. Ferraro , S. Grilli , D. Alfieri , P. De Natale , M. de Angelis , A. Finizio , S. De Nicola , G. Pierattini , F. Caccavale , D. Callejo , and A. Morbiato , “ On the origin of internal field in Lithium Niobate crystals directly observed by digital holography ,” Opt. Express 13 , 5416 – 5423 ( 2005 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-14- 5416. [CrossRef]   [PubMed]  

4 . N. Demoli and I. Demoli , “ Dynamic modal characterization of musical instruments using digital holography ,” Opt. Express 13 , 4812 – 4817 ( 2005 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-13-4812. [CrossRef]   [PubMed]  

5 . B. Javidi , I. Moon , S. Yeom , and E. Carapezza , “ Three-dimensional imaging and recognition of microorganism using single-exposure on-line (SEOL) digital holography ,” Opt. Express 13 , 4492 – 4506 ( 2005 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-12-4492. [CrossRef]   [PubMed]  

6 . N. Demoli , D. Vukicevic , and M. Torzynski , “ Dynamic digital holographic interferometry with three wavelengths ,” Opt. Express 11 , 767 – 774 ( 2003 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-767. [CrossRef]   [PubMed]  

7 . I. Yamaguchi , J. Kato , and S. Ohta , “ Surface shape measurement by phase shifting digital holography ,” Opt. Rev. 8 , 85 – 89 ( 2001 ). [CrossRef]  

8 . F. Dubois , L. Joannes , and J.C. Legros , “ Improved three-dimensional imaging with a digital holographic microscope with a source of partial spatial coherence ,” Appl. Opt. 38 , 7085 – 7094 ( 1999 ). [CrossRef]  

9 . P. Picart , J. Leval , D. Mounier , and S. Gougeon , “ Time averaged digital holography ,” Opt. Lett. 28 , 1900 – 1902 ( 2003 ). [CrossRef]   [PubMed]  

10 . P. Picart , J. Leval , D. Mounier , and S. Gougeon , “ Some opportunities for vibration analysis with time-averaging in digital Fresnel holography ,” Appl. Opt. 44 , 337 – 343 ( 2005 ). [CrossRef]   [PubMed]  

11 . J. Leval , P. Picart , J.-P. Boileau , and J.-C. Pascal , “ Full field vibrometry with digital Fresnel holography ,” Appl. Opt. 44 , 5763 – 5772 ( 2005 ). [CrossRef]   [PubMed]  

12 . P. Picart , E. Moisson , and D. Mounier , “ Twin sensitivity measurement by spatial multiplexing of digitally recorded holograms ,” Appl. Opt. 42 , 1947 – 1957 ( 2003 ). [CrossRef]   [PubMed]  

13 . P. Picart , B. Diouf , E. Lolive , and J.-M. Berthelot , “ Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms ,” Opt. Eng. 43 , 1169 – 1176 ( 2004 ). [CrossRef]  

14 . Technical data sheet available for example on http://www.polytec.com.

References

  • View by:

  1. U. Schnars and W. Jüptner , “ Direct recording of holograms by a CCD target and numerical reconstruction ,” App. Opt.   33 , 179 – 181 ( 1994 ).
    [Crossref]
  2. L. Yu and M.K. Kim , “ Wavelength scanning digital interference holography for variable tomographic scanning ,” Opt. Express   13 , 5621 – 5627 ( 2005 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-15-5621.
    [Crossref] [PubMed]
  3. M. Paturzo , P. Ferraro , S. Grilli , D. Alfieri , P. De Natale , M. de Angelis , A. Finizio , S. De Nicola , G. Pierattini , F. Caccavale , D. Callejo , and A. Morbiato , “ On the origin of internal field in Lithium Niobate crystals directly observed by digital holography ,” Opt. Express   13 , 5416 – 5423 ( 2005 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-14- 5416.
    [Crossref] [PubMed]
  4. N. Demoli and I. Demoli , “ Dynamic modal characterization of musical instruments using digital holography ,” Opt. Express   13 , 4812 – 4817 ( 2005 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-13-4812.
    [Crossref] [PubMed]
  5. B. Javidi , I. Moon , S. Yeom , and E. Carapezza , “ Three-dimensional imaging and recognition of microorganism using single-exposure on-line (SEOL) digital holography ,” Opt. Express   13 , 4492 – 4506 ( 2005 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-12-4492.
    [Crossref] [PubMed]
  6. N. Demoli , D. Vukicevic , and M. Torzynski , “ Dynamic digital holographic interferometry with three wavelengths ,” Opt. Express   11 , 767 – 774 ( 2003 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-767.
    [Crossref] [PubMed]
  7. I. Yamaguchi , J. Kato , and S. Ohta , “ Surface shape measurement by phase shifting digital holography ,” Opt. Rev.   8 , 85 – 89 ( 2001 ).
    [Crossref]
  8. F. Dubois , L. Joannes , and J.C. Legros , “ Improved three-dimensional imaging with a digital holographic microscope with a source of partial spatial coherence ,” Appl. Opt.   38 , 7085 – 7094 ( 1999 ).
    [Crossref]
  9. P. Picart , J. Leval , D. Mounier , and S. Gougeon , “ Time averaged digital holography ,” Opt. Lett.   28 , 1900 – 1902 ( 2003 ).
    [Crossref] [PubMed]
  10. P. Picart , J. Leval , D. Mounier , and S. Gougeon , “ Some opportunities for vibration analysis with time-averaging in digital Fresnel holography ,” Appl. Opt.   44 , 337 – 343 ( 2005 ).
    [Crossref] [PubMed]
  11. J. Leval , P. Picart , J.-P. Boileau , and J.-C. Pascal , “ Full field vibrometry with digital Fresnel holography ,” Appl. Opt.   44 , 5763 – 5772 ( 2005 ).
    [Crossref] [PubMed]
  12. P. Picart , E. Moisson , and D. Mounier , “ Twin sensitivity measurement by spatial multiplexing of digitally recorded holograms ,” Appl. Opt.   42 , 1947 – 1957 ( 2003 ).
    [Crossref] [PubMed]
  13. P. Picart , B. Diouf , E. Lolive , and J.-M. Berthelot , “ Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms ,” Opt. Eng.   43 , 1169 – 1176 ( 2004 ).
    [Crossref]
  14. Technical data sheet available for example on http://www.polytec.com.

2005 (6)

L. Yu and M.K. Kim , “ Wavelength scanning digital interference holography for variable tomographic scanning ,” Opt. Express   13 , 5621 – 5627 ( 2005 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-15-5621.
[Crossref] [PubMed]

M. Paturzo , P. Ferraro , S. Grilli , D. Alfieri , P. De Natale , M. de Angelis , A. Finizio , S. De Nicola , G. Pierattini , F. Caccavale , D. Callejo , and A. Morbiato , “ On the origin of internal field in Lithium Niobate crystals directly observed by digital holography ,” Opt. Express   13 , 5416 – 5423 ( 2005 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-14- 5416.
[Crossref] [PubMed]

N. Demoli and I. Demoli , “ Dynamic modal characterization of musical instruments using digital holography ,” Opt. Express   13 , 4812 – 4817 ( 2005 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-13-4812.
[Crossref] [PubMed]

B. Javidi , I. Moon , S. Yeom , and E. Carapezza , “ Three-dimensional imaging and recognition of microorganism using single-exposure on-line (SEOL) digital holography ,” Opt. Express   13 , 4492 – 4506 ( 2005 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-12-4492.
[Crossref] [PubMed]

P. Picart , J. Leval , D. Mounier , and S. Gougeon , “ Some opportunities for vibration analysis with time-averaging in digital Fresnel holography ,” Appl. Opt.   44 , 337 – 343 ( 2005 ).
[Crossref] [PubMed]

J. Leval , P. Picart , J.-P. Boileau , and J.-C. Pascal , “ Full field vibrometry with digital Fresnel holography ,” Appl. Opt.   44 , 5763 – 5772 ( 2005 ).
[Crossref] [PubMed]

2004 (1)

P. Picart , B. Diouf , E. Lolive , and J.-M. Berthelot , “ Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms ,” Opt. Eng.   43 , 1169 – 1176 ( 2004 ).
[Crossref]

2003 (3)

2001 (1)

I. Yamaguchi , J. Kato , and S. Ohta , “ Surface shape measurement by phase shifting digital holography ,” Opt. Rev.   8 , 85 – 89 ( 2001 ).
[Crossref]

1999 (1)

1994 (1)

U. Schnars and W. Jüptner , “ Direct recording of holograms by a CCD target and numerical reconstruction ,” App. Opt.   33 , 179 – 181 ( 1994 ).
[Crossref]

Alfieri, D.

Angelis, M. de

Berthelot, J.-M.

P. Picart , B. Diouf , E. Lolive , and J.-M. Berthelot , “ Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms ,” Opt. Eng.   43 , 1169 – 1176 ( 2004 ).
[Crossref]

Boileau, J.-P.

Caccavale, F.

Callejo, D.

Carapezza, E.

Demoli, I.

Demoli, N.

Diouf, B.

P. Picart , B. Diouf , E. Lolive , and J.-M. Berthelot , “ Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms ,” Opt. Eng.   43 , 1169 – 1176 ( 2004 ).
[Crossref]

Dubois, F.

Ferraro, P.

Finizio, A.

Gougeon, S.

Grilli, S.

Javidi, B.

Joannes, L.

Jüptner, W.

U. Schnars and W. Jüptner , “ Direct recording of holograms by a CCD target and numerical reconstruction ,” App. Opt.   33 , 179 – 181 ( 1994 ).
[Crossref]

Kato, J.

I. Yamaguchi , J. Kato , and S. Ohta , “ Surface shape measurement by phase shifting digital holography ,” Opt. Rev.   8 , 85 – 89 ( 2001 ).
[Crossref]

Kim, M.K.

Legros, J.C.

Leval, J.

Lolive, E.

P. Picart , B. Diouf , E. Lolive , and J.-M. Berthelot , “ Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms ,” Opt. Eng.   43 , 1169 – 1176 ( 2004 ).
[Crossref]

Moisson, E.

Moon, I.

Morbiato, A.

Mounier, D.

Natale, P. De

Nicola, S. De

Ohta, S.

I. Yamaguchi , J. Kato , and S. Ohta , “ Surface shape measurement by phase shifting digital holography ,” Opt. Rev.   8 , 85 – 89 ( 2001 ).
[Crossref]

Pascal, J.-C.

Paturzo, M.

Picart, P.

Pierattini, G.

Schnars, U.

U. Schnars and W. Jüptner , “ Direct recording of holograms by a CCD target and numerical reconstruction ,” App. Opt.   33 , 179 – 181 ( 1994 ).
[Crossref]

Torzynski, M.

Vukicevic, D.

Yamaguchi, I.

I. Yamaguchi , J. Kato , and S. Ohta , “ Surface shape measurement by phase shifting digital holography ,” Opt. Rev.   8 , 85 – 89 ( 2001 ).
[Crossref]

Yeom, S.

Yu, L.

App. Opt. (1)

U. Schnars and W. Jüptner , “ Direct recording of holograms by a CCD target and numerical reconstruction ,” App. Opt.   33 , 179 – 181 ( 1994 ).
[Crossref]

Appl. Opt. (4)

Opt. Eng. (1)

P. Picart , B. Diouf , E. Lolive , and J.-M. Berthelot , “ Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms ,” Opt. Eng.   43 , 1169 – 1176 ( 2004 ).
[Crossref]

Opt. Express (5)

Opt. Lett. (1)

Opt. Rev. (1)

I. Yamaguchi , J. Kato , and S. Ohta , “ Surface shape measurement by phase shifting digital holography ,” Opt. Rev.   8 , 85 – 89 ( 2001 ).
[Crossref]

Other (1)

Technical data sheet available for example on http://www.polytec.com.

Supplementary Material (9)

Media 1: MPG (1064 KB)     
Media 2: MPG (911 KB)     
Media 3: MPG (1402 KB)     
Media 4: MPG (1608 KB)     
Media 5: MPG (1586 KB)     
Media 6: MPG (1251 KB)     
Media 7: MPG (1092 KB)     
Media 8: MPG (1321 KB)     
Media 9: MPG (811 KB)     

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1.
Fig. 1. Experimental set-up for simultaneous 2D vibration analysis
Fig. 2.
Fig. 2. Multiplexed holograms of the car joint piece
Fig. 3.
Fig. 3. 2D vibration amplitude and phase at a frequency of 680 Hz
Fig. 4.
Fig. 4. Mean quadratic velocities extracted from the experimental results
Fig. 5.
Fig. 5. 1064 Ko Movie 1 - Kinetic representation of Fresnel for 2D vibration at 340Hz [Media 9]
Fig. 6.
Fig. 6. 912 Ko Movie 2 - Kinetic representation of Fresnel for 2D vibration at 430Hz
Fig. 7.
Fig. 7. 1403 Ko Movie 3 - Kinetic representation of Fresnel for 2D vibration at 460Hz
Fig. 8.
Fig. 8. 1609 Ko Movie 4 - Kinetic representation of Fresnel for 2D vibration at 680Hz
Fig. 9.
Fig. 9. 1586 Ko Movie 5 - Kinetic representation of Fresnel for 2D vibration at 730Hz
Fig. 10.
Fig. 10. 1251 Ko Movie 6 - Kinetic representation of Fresnel for 2D vibration at 880Hz
Fig. 11.
Fig. 11. 1093 Ko Movie 7 - Kinetic representation of Fresnel for 2D vibration at 900Hz
Fig. 12.
Fig. 12. 1321 Ko Movie 8 - Kinetic representation of Fresnel for 2D vibration at 930Hz

Equations (12)

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

U ( t ) = u x sin ( ω 0 t + φ x ) i + u y sin ( ω 0 t + φ y ) j + u z sin ( ω 0 t + φ z ) k ,
A ( t ) = A 0 exp ( i ψ 0 ) exp [ 2 S . U ( t ) / λ ] ,
A R ( x , y , d 0 , t ) = i exp ( 2 i π d 0 / λ ) λ d 0 exp [ i π λ d 0 ( x 2 + y 2 ) ]
× k = 0 k = K 1 l = 0 l = L 1 H ( l p x , k p y , d 0 , t ) exp [ i π λ d 0 ( l 2 p x 2 + k 2 p y 2 ) ] exp [ 2 i π λ d 0 ( lx p x + ky p y ) ] .
A + 1 R ( x , y , d 0 , t ) MN λ 4 d 0 4 R * ( x , y ) exp [ iπλ d 0 ( u 0 2 + v 0 2 ) ]
× A 0 ( x , y ) exp [ i ψ 0 ( x , y ) ] exp [ 2 S . U ( t ) / λ ] * δ ( x λ u 0 d 0 , y λ v 0 d 0 ) .
ψ J i = ψ 0 ± Δ φ x sin ( θ ) sin ( ω 0 t j + φ x ) Δ φ z [ 1 + cos ( θ ) ] sin ( ω 0 t j + φ z ) ,
Δ φ A = 1 2 [ Δ ψ 13 _ A ] 2 + [ Δ ψ 23 _ A + Δ ψ 21 _ A ] 2 ,
φ A = arctan [ Δ ψ 13 _ A Δ ψ 23 _ A + Δ ψ 21 _ A ] ,
v A 2 = ω 0 2 πS s 0 T 0 v A ( x , y , t ) 2 dtdxdy ,
v z ( x , y , t ) = λ ω 0 4 π ( 1 + cos θ ) Δ φ z ( x , y ) cos ( ω 0 + φ z ) ,
v z ( x , y ,t ) = λ ω 0 4 π sin ( θ ) Δ φ x ( x , y ) cos ( ω 0 t + φ x ) .

Metrics