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  • Editor: Gregory W. Faris
  • Vol. 2, Iss. 7 — Jul. 16, 2007
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Tracking high amplitude auto-oscillations with digital Fresnel holograms

Pascal Picart, Julien Leval, Francis Piquet, Jean Pierre Boileau, Thomas Guimezanes, and Jean-Pierre Dalmont  »View Author Affiliations


Optics Express, Vol. 15, Issue 13, pp. 8263-8274 (2007)
http://dx.doi.org/10.1364/OE.15.008263


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Abstract

Method for tracking vibrations with high amplitude of several hundreds of micrometers is presented. It is demonstrated that it is possible to reconstruct a synthetic high amplitude deformation of auto-oscillations encoded with digital Fresnel holograms. The setup is applied to the auto-oscillation of a clarinet reed in a synthetic mouth. Tracking of the vibration is performed by using the pressure signal delivered by the mouth. Experimental results show the four steps of the reed movement and especially emphasize the shocks of the reed on the mouthpiece.

© 2007 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

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

]. Digital Fresnel holography is a powerful tool for metrological applications such as biological imaging, polarization imaging, heterogeneous material investigation, surface shape measurement or also fluid mechanics investigation. Some demonstrative examples can be found in references [2–6

2. P. Ferraro, D. Alferi, S. De Nicola, L. De Petrocellis, A. Finizio, and G. Pierattini, “Quantitative phase-contrast microscopy by a lateral shear approach to digital holographic image reconstruction,” Opt. Lett. 31, 1405–1407 (2006). [CrossRef] [PubMed]

]. Lately the suitability of digital Fresnel holography for vibration analysis and full field vibrometry was demonstrated [7–10

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

]. We have also proposed an extension of the principle described in [10

10. 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]

] into simultaneous 2D full field vibration analysis by combining digital holography and spatial multiplexing of holograms [11

11. P. Picart, J. Leval, M. Grill, J.-P. Boileau, J.C. Pascal, J.-M. Breteau, B. Gautier, and S. Gillet, “2D full field vibration analysis with multiplexed digital holograms,” Opt. Express 13, 8882–8892 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-22-8882. [CrossRef] [PubMed]

]. One particularity of such setup and strategies is that the vibration is under a controlled monochromatic excitation. This means that the maximum amplitude of oscillation to be measured is chosen so that the number of fringes in measured phases is low. Therefore a few holograms to extract amplitude and phase of the vibration are needed. Typically three holograms are necessary in the method described in [10

10. 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]

]. In practice, acoustical engineers are also confronted with free oscillations which are multi-chromatic and have high amplitude as regards the optical wavelength. This kind of vibration is generally not controlled since it is often initiated with non linear phenomena leading to an auto oscillation phenomenon.

2. Theory

When a rough object is under a free oscillation regime and illuminated by a coherent beam, it appears as a spatiotemporal optical phase modulator and the complex amplitude at its surface can simply be written

Axyt=A0xyexp[iψ0xy]exp[iΔφxyt]
(1)

where Δφ(x,y,t) = 2π S(x,y).U(x,y,t)/λ is the spatio-temporal phase induced by the dynamic displacement of the object, S is the sensitivity vector of the experimental set-up and ψ 0(x,y) is the random phase to the rough object surface. In the case of an object vibrating such a cantilevered beam, we have U(t) = uz(t)k, {i,j,k} being a set of reference vectors attached to the object plane, whose coordinate are {x,y,z} and k is perpendicular to the object surface. Note that in the case of free oscillations, displacement U(t) is often a multi harmonic function. In digital Fresnel holography, encoding of the object wave is performed through a Fresnel diffraction and interferences with a uniform reference plane wave [7

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

]. In this paper, the reference wave is written as R(x’,y’) = a 0exp[-2{u 0 x’+v 0 y’)], where {u 0,v 0} are the spatial frequencies of the reference wave. Here {x’,y’,z} is a set of coordinate attached to the interference plane, also the recording plane. The field diffracted along a distance d 0 from the object plane to the recording plane is mathematically written in the Fresnel approximations, so that we have [16

16. J.W. Goodman, Introduction to Fourier Optics (Mc Graw Hill Editions, New York, 2nd Edition, 1996).

]

Oxyd0t=iexp(i2πd0λ)λd0exp[iπλd0(x2+y2)]
×++Axytexp[iπλd0(x2+y2)]exp[2iπλd0(xx+yy)]dxdy
(2)

Omitting the effect of the pixel surface which spatially integrates the signal, the recorded hologram is temporally varying and can be written as

H(x,y,d0,t)=O(x,y,d0,t)2+R(x,y)2
+R*(x,y)O(x,y,d0,t)+R(x,y)O*(x,y,d0,t)
(3)

If the recording exposure is sufficiently short for the instantaneous hologram to be considered as frozen during the recording, then at time tn corresponding to the recording H(x’,y’,d 0,tn) can be used for a digital reconstruction of the initial object wave [1

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

]. Note that if the exposure time or light pulse is too large compared to both amplitude and vibration period, then distortion will appear in the numerically computed optical phase [10

10. 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]

]. Here we consider that there is no such distortion and that the recorded hologram is the frozen instantaneous one. This will be achieved experimentally by using a pulsed laser with nanosecond pulse width. Now, with an ideal recording, the numerical process is performed according to diffraction theory considered in the Fresnel approximation [1

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

, 7

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

]. The reconstructed object is then computed according to Eq. (4) by considering a diffraction distance equal to –d 0 and a discrete version of diffraction

AR(x,y,d0,tn)=iexp(2iπd0λ)λd0exp[iπλd0(x2+y2)]
×k=0k=K1l=0l=L1Hlpxkpyd0tnexp[iπλd0(l2px2+k2py2)]exp[2iπλd0(lxpx+kypy)]
(4)

In Eq. (4), {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. (4) and reference [10

10. 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]

], the reconstructed object in the +1 order takes the form of

A+1R(x,y,d0,tn)MNλ4d04R*xyexp[iπλd0(u02+v02)]
×A0xyexp[iψ0xy]exp[iΔφxytn]*δ(xλu0d0,yλv0d0).
(5)

where {M,N} are the number of effective pixels of the CCD area, * means convolution and δ(x,y) is the Dirac distribution. At any time tn, the optical phase extracted from the reconstructed +1 order is then given by

ψnxy=2π(u0x+v0y)πλd0(u02+v02)+ψ0xy+Δφxytn.
(6)

Interferometric metrology is a comparative process in the sense that with only one optical wavelength, absolute values cannot always be determined. If the vibrating structure is clearly clamped then it is easy to impose its zero displacement. In any other case, if the static part of the structure is not included in the field of view, it will only be possible to impose a relative zero to the displacement map. This means that movement like piston will remain undetermined. Furthermore, in digital holography the instantaneous optical phase has a random nature which is due to the presence of phase ψ 0(x,y) in Eq. (6). However, this random nature can be overcome by considering the first instantaneous result as the reference one for the set of temporal recordings. In this strategy, it is now possible to construct the deformation of the object between two instants tn and tm sampled by a set of recordings by summing the set of intermediate results between the two instants, i.e. summing quantities Δψnm = ψn - ψm. Physical deformation will be computed by inverting the sensitivity relation between optical phase and true displacement.

3. Artificial mouth

As pointed out in the introduction, the aim of this work is to study the vibration of the reed under quasi playing conditions, i.e. when the clarinet reed is in the auto-oscillation regime. In order to produce suitable oscillation conditions, J.-P. Dalmont et al [17

17. J.P. Dalmont, J. Gilbert, J. Kergomard, and S. Ollivier, “An analytical prediction of the oscillation and extinction thresholds of a clarinet,” J. Acoust. Soc. Am. 118, 3294–3305 (2005). [CrossRef] [PubMed]

] built an artificial mouth modeling that of a musician playing clarinet. This artificial mouth is constituted mainly with an air-proof caisson and an artificial lip modeling the tight-lipped going of the musician. The schematic diagram of the artificial mouth is depicted in Fig. 1.

Fig. 1. Artificial mouth

The artificial lip is made with a rubber membrane filled with water such that it has quite the same consistency as the real musician’s lip. Note that practically the humidity of the reed has a damping role, however it will not be considered in this setup since the mouth is not coupled with a dry pipe. The mouth includes an optical window which allows illumination of the reed. To initiate free oscillation of the reed, flow is blown in the mouth. The over pressure in the mouth is of the order of 60 hPa. It induces a flexure of the reed and introduces an air flow in the instrument. Hereafter a threshold pressure, the equilibrium position of the reed becomes unstable and the reed oscillates at a frequency inversely proportional to the effective length of the clarinet. The sound generated by this oscillation is periodic but multi-chromatic. In our set-up, the instrument is a simple cylindrical tube which produces a sound whose fundamental frequency is about 162 Hz. The particularity of the observed auto-oscillation regime, classically called beating reed regime, is that the amplitude is imposed by the characteristics of the mouthpiece. This amplitude can vary from a few tenths of millimetre to one millimetre.

A pressure sensor made with piezoresistive gages (microphone) is placed in the mouthpiece of the clarinet, as indicated in Fig. 1 (top view). This sensor measures the pressure fluctuations induced by the mechanical vibration of the reed. The signal issued from the microphone is correlated with the displacement of the reed. Figure 2 shows two periods of such a signal.

Fig. 2. Pressure signal issued form the artificial mouth

The signal is composed of numerous harmonics and remains stationary during about 40 minutes after the reed is destroyed by the shocks on the mouthpiece. Figure 2 exhibits four important parts during one period of oscillation. The first part is called the opening reed since the reed is in a high level position regarding to the mouthpiece. Then, the second part corresponds to the closing phase during which the reed goes tight to the mouthpiece. The third part corresponds to the closing of the reed : the reed closes the reed channel. After that, the last part of the cycle corresponds to the opening phase during which the reed goes to its high level position. So, the movement of the reed is bi-stable.

In order to illustrate the geometry of the different phases, Fig. 3 shows the position of the reed on the mouthpiece for the static case and the close case.

Fig. 3. Different phases of the movement of the reed

4. Optical set-up

The optical setup is described in Fig. 4. The laser beam is issued from a double frequency pulsed NdYAG laser (λ = 532 nm) with a pulse width of 20 ns and pulse energy about 7 mJ. The laser is pumped with a flash lamp and the cavity is switched by means of a Pockels cell. The mouth is placed in such a way that the reed is at distance d 0 = 312 mm from the detector area. The illumination of the reed is realized through lenses L3 and L4 and the optical window of the mouth. In the set-up, illuminating angle θ is set to 22°.

The off-axis holographic recording is carried out using lens L2 which is displaced out of its optical axis by means of two micrometric transducers [4

4. 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]

]. Therefore, spatial frequencies of reference wave are adjusted to {u 0,v 0} ≈ {60mm-1,-60mm-1}. The detector is a 12-bit digital CCD with (M×N) = (1024×1360) pixels of pitch px = py = 4.65μm (PCO Pixel Fly) and maximum acquisition rate is 7 Hz. Digital reconstruction was performed with K = L = 2048 data points by use of zero-padding.

Fig. 4. Optical set-up

The setup includes a microprocessor unit with electronics devoted to the management of the laser source. A graphical interface allows users to program a sequence for the laser fire. The pressure signal issued from the microphone is used as an input for the microprocessor. Timers at 1 MHz allowing time resolution of 1 μs exploit the zero-crossing of the reference signal to generate useful signals for the laser flash, Pockels cell and trigger signal of CCD. The signals are generated taking into account the delay programmed between pulses in the fire sequence. Figure 5 shows the amplitude image obtained with a digital Fresnel hologram recorded with one laser pulse. Note that the artificial lip is not in focus and that the structure of the natural cane of the reed can be seen.

Fig. 5. Full field reconstruction of diffracted field

For each unwrapped phase difference Δψnm extracted from recording at each two consecutive laser pulses emitted at instants tn and tm, the deformation can be found by computing Δuznm = λΔψnm/2π(1+cosθ).

5. Experimental results

Considering the fundamental frequency of the reference signal (162 Hz), the laser fire must be designed by taking into account the available rate of the flash lamp (20 Hz to 60 Hz), the necessary thermal equilibrium of the Nd YAG rode and the maximum camera rate (7 Hz). So the trigger frequency was chosen to be 5.4 Hz, leading to a laser pulse fired every three pressure period (mean laser rate 54 Hz) and a digital hologram acquired each 10 laser pulses. This strategy imposes that the reed movement is sampled over a large number of oscillation periods, typically one digital hologram every 30 periods. The deformation of the reed during one period is reconstructed into a synthetic movement built up with sampling point temporally spaced from 185.19 ms. Acquisition of digital holograms is performed during a unique sequence which is programmed for the generation of 3150 laser pulses with a synthetic temporal sampling of 2 μs. This ensures a covering of slightly more than one period of the reference signal. At the end of the full process, 3150 digital holograms including the relative phase of the reed displacement are obtained. After digital reconstruction, phase extraction and phase unwrapping, phase maps are suitable for the deformation reconstruction. Results presented in this paper are organized into 4 segments describing the 4 parts of the oscillation period, i.e. opened reed, closing reed, closed reed and opening reed. Note that the arbitrary zero of the deformation was set at half reed near the lip boundary. Movies of Fig. 6 to 9 show the dynamic reed deformation relatively to this reference region. Colour maps give the displacement in micrometers. The graphic at the top of the image represents the dynamic deformation whereas the one at the bottom represents a period of the pressure signal. Red bars indicate the window for the full laser fire and blue bar corresponds to the time evolution correlated with the top movie.

Fig. 6. 2506 Ko Movie 1 - Dynamic deformation for the opened reed [Media 1]
Fig. 7. 914 Ko Movie 2 - Dynamic deformation for the closing reed [Media 2]
Fig. 8. 2988 Ko Movie 3 - Dynamic deformation for the closed reed [Media 3]
Fig. 9. 844 Ko Movie 4 - Dynamic deformation for the opening reed [Media 4]

In Fig. 6, the reed is in the opened position. Just after its opening, its stabilizes; then displacement fluctuates around an average position. During this time interval, the reed is submitted to combined torsion and flexure modes.

Figure 7 shows the beginning of the closing of the reed when it is going tight to the mouthpiece. The reed sinks into the mouthpiece with a significant amplitude of about 350 μm for a time interval of 594 μs.

Closed reed is illustrated in Fig. 8. When the reed slams the mouthpiece, it is strongly deformed. The reed exhibits shocks with a bounce of about 50 μm amplitude. After the shock, the reed closes on the mouthpiece and its movement becomes negligible (less than 3 μm). Then, the reed exhibits small oscillations indicating the future opening phase.

In Fig. 9, the reed is leaving its closed position and goes to the opened one with a deformation relative to the reference point of about 355.58 μm in a time interval of 702 μs. Approximately, this corresponds to the deformation of the closing phase. Note that the global piston movement of the reed is not represented in these experimental results.

As seen on Fig. 6 to 9, the experimental results are quite coherent with the reference signal used to manage the laser fire which is only an “image” of the reed movement.

6. Conclusion

This paper presents a tracking method devoted to high amplitude free oscillations. The strategy is based on pulsed digital Fresnel holography. Deformation of a clarinet reed inserted in an artificial mouth can be extracted with the recording of 3150 digital holograms virtually temporally spaced of 2 μs. The method allows high spatial resolution on the reconstructed image and needs only cheap CCD. Experimental results show the vibration behavior of the clarinet reed under auto-oscillation regime, exhibiting high amplitude shocks during the closed phase. To the best of our knowledge, these results constitute a first attempt to reconstruct a synthetic reed deformation on a pseudo-period of its auto-oscillation regime in an artificial mouth. Thanks to digital Fresnel holography, information is full field allowing to observe for the first time the shocks on the mouthpiece. These works give opportunity to acoustician researchers to better understand sound production and musical quality of a clarinet. Future works will focus on hybrid optical methods to simultaneously measure deformation and piston movement of the reed in the artificial mouth.

Acknowledgments

The authors wish to express their appreciation to Emmanuel Brasseur from Laboratoire d’Acoustique de l’Université du Maine for useful help in electronic setup.

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.

P. Ferraro, D. Alferi, S. De Nicola, L. De Petrocellis, A. Finizio, and G. Pierattini, “Quantitative phase-contrast microscopy by a lateral shear approach to digital holographic image reconstruction,” Opt. Lett. 31, 1405–1407 (2006). [CrossRef] [PubMed]

3.

T. Nomura, B. Javidi, S. Murata, E. Nitanai, and T. Numata, “Polarization imaging of a 3D object by use of on-axis phase-shifting digital holography,” Opt. Lett. 32, 481–483 (2007). [CrossRef] [PubMed]

4.

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]

5.

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

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.

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

8.

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]

9.

A. Asundi and V.R. Singh, “Time-averaged in-line digital holographic interferometry for vibration analysis,” App. Opt. 45, 2391–2395 (2006). [CrossRef]

10.

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]

11.

P. Picart, J. Leval, M. Grill, J.-P. Boileau, J.C. Pascal, J.-M. Breteau, B. Gautier, and S. Gillet, “2D full field vibration analysis with multiplexed digital holograms,” Opt. Express 13, 8882–8892 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-22-8882. [CrossRef] [PubMed]

12.

S. C. Thompson, “The effect of the reed resonance on woodwind tone production,” J. Acoust. Soc. Am. 66, 1299–1307 (1979). [CrossRef]

13.

P. Hoekje and G. Roberts, “Observed vibration patterns of clarinet reeds,” J. Acoust. Soc. Am. 99, 2462 (A) (1996). [CrossRef]

14.

I.M. Lindevald and J. Gower, “Vibrational modes of clarinet reeds,” J. Acoust. Soc. Am. 102, 3085 (A) (1997). [CrossRef]

15.

F. Pinard, B. Laine, and H. Vach, “Musical quality assessment of clarinet reeds using optical holography,” J. Acoust. Soc. Am. 113, 1736–1742 (2003). [CrossRef] [PubMed]

16.

J.W. Goodman, Introduction to Fourier Optics (Mc Graw Hill Editions, New York, 2nd Edition, 1996).

17.

J.P. Dalmont, J. Gilbert, J. Kergomard, and S. Ollivier, “An analytical prediction of the oscillation and extinction thresholds of a clarinet,” J. Acoust. Soc. Am. 118, 3294–3305 (2005). [CrossRef] [PubMed]

OCIS Codes
(090.0090) Holography : Holography
(090.2880) Holography : Holographic interferometry
(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.4630) Instrumentation, measurement, and metrology : Optical inspection

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: April 2, 2007
Revised Manuscript: May 22, 2007
Manuscript Accepted: May 22, 2007
Published: June 18, 2007

Virtual Issues
Vol. 2, Iss. 7 Virtual Journal for Biomedical Optics

Citation
Pascal Picart, Julien Leval, Francis Piquet, Jean P. Boileau, Thomas Guimezanes, and Jean-Pierre Dalmont, "Tracking high amplitude auto-oscillations with digital Fresnel holograms," Opt. Express 15, 8263-8274 (2007)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-13-8263


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References

  1. U. Schnars, W. Jüptner, "Direct recording of holograms by a CCD target and numerical reconstruction,’’App. Opt. 33, 179-181 (1994). [CrossRef]
  2. P. Ferraro, D. Alferi, S. De Nicola, L. De Petrocellis, A. Finizio, G. Pierattini, "Quantitative phase-contrast microscopy by a lateral shear approach to digital holographic image reconstruction,’’Opt. Lett. 31, 1405-1407 (2006). [CrossRef] [PubMed]
  3. T. Nomura, B. Javidi, S. Murata, E. Nitanai, T. Numata, "Polarization imaging of a 3D object by use of on-axis phase-shifting digital holography,’’Opt. Lett. 32, 481-483 (2007). [CrossRef] [PubMed]
  4. P. Picart, B. Diouf, E. Lolive, J.-M. Berthelot, "Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms," Opt. Eng. 43, 1169-1176 (2004). [CrossRef]
  5. I. Yamaguchi, J. Kato, S. Ohta, "Surface shape measurement by phase shifting digital holography,’’Opt. Rev. 8, 85-89 (2001). [CrossRef]
  6. N. Demoli, D. Vukicevic, 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. P. Picart, J. Leval, D. Mounier, S. Gougeon, "Time averaged digital holography," Opt. Lett. 28, 1900-1902 (2003). [CrossRef] [PubMed]
  8. N. Demoli, 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]
  9. A. Asundi, V.R. Singh, "Time-averaged in-line digital holographic interferometry for vibration analysis," App. Opt. 45, 2391-2395 (2006). [CrossRef]
  10. J. Leval, P. Picart, J.-P. Boileau, J.-C. Pascal, "Full field vibrometry with digital Fresnel holography," Appl. Opt. 44, 5763-5772 (2005). [CrossRef] [PubMed]
  11. P. Picart, J. Leval, M. Grill, J.-P. Boileau, J.C. Pascal, J.-M. Breteau, B. Gautier, S. Gillet, "2D full field vibration analysis with multiplexed digital holograms,’’ Opt. Express 13, 8882-8892 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-22-8882. [CrossRef] [PubMed]
  12. S. C. Thompson, "The effect of the reed resonance on woodwind tone production,’’ J. Acoust. Soc. Am. 66, 1299-1307 (1979). [CrossRef]
  13. P. Hoekje, G. Roberts, "Observed vibration patterns of clarinet reeds," J. Acoust. Soc. Am. 99, 2462 (1996). [CrossRef]
  14. I.M. Lindevald, J. Gower, "Vibrational modes of clarinet reeds," J. Acoust. Soc. Am. 102, 3085 (1997). [CrossRef]
  15. F. Pinard, B. Laine, H. Vach, "Musical quality assessment of clarinet reeds using optical holography," J. Acoust. Soc. Am. 113, 1736-1742 (2003). [CrossRef] [PubMed]
  16. J.W. Goodman, Introduction to Fourier Optics (Mc Graw Hill Editions, New York, 2nd Edition, 1996).
  17. J.P. Dalmont, J. Gilbert, J. Kergomard, S. Ollivier, "An analytical prediction of the oscillation and extinction thresholds of a clarinet," J. Acoust. Soc. Am. 118, 3294-3305 (2005). [CrossRef] [PubMed]

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