Full-field vibrometry with digital Fresnel holography Julien Leval, Pascal Picart, Jean Pierre Boileau, and Jean Claude Pascal
A setup that permits full-field vibration amplitude and phase retrieval with digital Fresnel holography is presented. Full reconstruction of the vibration is achieved with a three-step stroboscopic holographic recording, and an extraction algorithm is proposed. The finite temporal width of the illuminating light is considered in an investigation of the distortion of the measured amplitude and phase. In particular, a theoretical analysis is proposed and compared with numerical simulations that show good agreement. Experimental results are presented for a loudspeaker under sinusoidal excitation; the mean quadratic velocity extracted from amplitude evaluation under two different measuring conditions is presented. Comparison with time averaging validates the full-field vibrometer. © 2005 Optical Society of America OCIS codes: 090.0090, 090.2880, 120.3180, 120.0120, 120.4630.
1. Introduction
Classic holographic interferometry has proved in past decades to be a useful industrial tool for the measurement of full-field surface deformations of naturally rough objects.1 Digital holography originated in the early 1970s (Ref. 2) and became properly available in the past decade3 with cheap high-speed CCD cameras and the increasing power of computers. Recently it was demonstrated that digital Fresnel holography offers new opportunities for metrological applications, examples of which include object deformation,4,5 surface shape measurement,6,7 phase-contrast microscopic imaging,8 and twin-sensitivity measurements.9 Vibration analysis with optical holographic techniques started with the works of Powell and Stetson,10 who established the principle of time averaging. The use of time averaging in digital Fresnel holography was described recently.11,12 However, the phase of the vibration signal is lost in time averaging. Using a sinusoidal modulation of the reference wave, it is also possible to determine the phase relation between points, and a full mapping of the vibration amplitude is accessible.13 Time averaging is a useful tool for studying vibrations, J. Leval, P. Picart (
[email protected]), and J. C. Pascal are with the Laboratoire d’Acoustique de l’Université du Maine, Unite Mixté de Recherche, Centre National de la Recherche Scientifique, 6613, Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France. P. Picart, J. P. Boileau, and J. C. Pascal are with the École Nationale Supérieure d’Ingénieurs du Mans, Rue Aristote, 72085 Le Mans Cedex 9, France. Received 4 January 2005; revised manuscript received 27 April 2005; accepted 28 April 2005. 0003-6935/05/275763-10$15.00/0 © 2005 Optical Society of America
and some interesting examples can be found in Refs. 14 –19. Nevertheless, vibration amplitude and phase retrieving is a challenge for full-field optical metrology because it is necessary for some applications. For example, modal analysis and structural intensity determination require vibration amplitude and phase determination.20,21 In the past, speckle interferometry was demonstrated to offer an efficient tool in combination with time averaging and digital image processing,22 fiber optics and acousto-optic stroboscopic illumination,23 laser diode modulation,24 and a pulsed laser coupled to a classic vibrometer.25 Furthermore, some authors26,27 were interested in studying the influence of exposure time during the recording process because it is one of the most important parameters of vibration analysis. However, the combined effect of amplitude and cyclic ratio has not been explained. In this paper we present an alternative approach to speckle interferometry that uses digital Fresnel holography. Reconstruction of the object is direct, and it can be achieved by use of a single hologram without any phase shifting. Thus vibration amplitude and phase extraction do not need too many images, and information is a full field, thus leading to direct full-field vibrometry. The paper is organized as follows: In Section 2 we present the theoretical background for reconstructions of the object, vibration amplitude, and vibration phase; a simple three-step algorithm is presented. In Section 3 we focus on the influence of the pulse width on the measurement, and a theoretical analysis is presented that is then validated by numerical simulations. In Sections 4 and 5 we discuss the stroboscopic and digital holographic setups, and in Section 6 we present experimental results; in particular, the mean quadratic velocity is estimated by use of 20 September 2005 兾 Vol. 44, No. 27 兾 APPLIED OPTICS
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the full field digital holographic vibrometer. In Section 7 we draw the conclusions of our study.
⫽ lpx and y⬘ ⫽ kpy. At any distance dR, the reconstructed ⫹1 order is
2. Theory
A rough object subjected to harmonic excitation and illuminated by a coherent laser beam induces a spatiotemporal optical phase modulation, which can be written as ⌬共t兲 ⫽ ⌬m sin共0t ⫹ 0兲, where ⌬m is the maximum amplitude at pulsation, 0 ⫽ 2兾T0, and 0 is the phase of the mechanical vibration. Thus, at any time t, the surface of the illuminated object diffuses an optical wave, written as
A⫹1R(X, Y, dR, tj) ⫽
冋
⫻
O(x⬘, y⬘, d0, t) ⫽
冋
册
i exp(i2d0兾) i exp (x⬘2 ⫹ y⬘2) d0 d0
冕 冕 ⫹⬁
⫻
⫺⬁
⫹⬁
⫺⬁
册 冋
⫹ y ) exp ⫺ 2
冋
A(x, y, t)exp
i 2 (x d0
册
2i (xx⬘ ⫹ yy⬘) dxdy. d0 (2)
Thus, in a recording plane at a distance d0 in front of the object, the instantaneous hologram generated by interference between the diffused wave and the plane reference wave, R共x⬘, y⬘兲 ⫽ aR exp关2i共u0x⬘ ⫹ v0y⬘兲兴, is expressed by H(x⬘, y⬘, d0, t) ⫽ ⱍO(x⬘, y⬘d0t)ⱍ2 ⫹ ⱍR(x⬘, y⬘)ⱍ2 ⫹ R*(x⬘, y⬘)O(x⬘, y⬘, d0, t) ⫹ R(x⬘, y⬘)O*(x⬘, y⬘, d0, t). (3) After digital reconstruction with the recorded hologram, the first two terms of Eq. (3) will give the so-called zero order, which is irrelevant in this paper; the third term will give the ⫹1 order and the last, the ⫺1 order. The instantaneous hologram will necessarily be time integrated by the solid-state sensor; if we denote by T the exposure time, then we have
H(x⬘, y⬘, d0) ⫽
冕
兺
兺
k⫽0
l⫽0
冋冕
册 R*(kpx, lpy)
⫻ O(kpx, lpy, d0, t)dt
冋 冋
册
册
i (k2px2 ⫹ l2py2) dR 2i ⫻ exp ⫺ (kXpx ⫹ lYpy) , (5) dR
⫻ exp
册
where 兵K, L其 may be equal to the number of pixels 兵M, N其 if we use raw data or may be greater and a multiple of a power of 2 if we apply zero padding. In Eq. (5), 兵px, py其 is the pixel pitch of the solid-state sensor. Note that Eq. (5) does not include the pixel width effect, but it will be considered a good approximation of what occurs in the digital reconstruction. The computation of the reconstructed field in ⫹1 order when dR ⫽ ⫺d0 leads to A⫹1R(x, y, ⫺d0, tj) ⬵ NM4d04R*(x, y) ⫻ exp[⫺id0(u02 ⫹ v02)] ⫻
冕
tj⫹T
A(x ⫺ u0d0, y
tj
⫺ v0d0, t)dt.
(6)
In Eq. (6) it is considered that the filtering function that is due to the two-dimensional Fourier transform ˜ 共x, y兲 ⬵ MN␦共x, y兲.11 The ⫹1 order is is reduced to W NM then localized at coordinates 共u0d0, v0d0兲 in the image plane. We can derive temporal integration by considering the harmonic excitation expressed in Eq. (1), so we have
冕
tj⫹T
tj
exp[i⌬m sin(0t ⫹ 0)]dt ⫽ T 兺 Jk(⌬m) k
冉 冊 冋冉
⫻ sinc k0
0T T exp ik 0tj ⫹ 0 ⫹ 2 2
冊册
. (7)
tj⫹T
H(x⬘, y⬘, d0, t)dt.
(4)
tj
The numerical reconstruction of the object plane is based on the diffraction integral when the hologram is considered transmittance and for a discrete spatial sampling of H共x⬘, y⬘, d0兲 by the detector with x⬘ 5764
k⫽K⫺1 l⫽L⫺1
tj⫹T
tj
A(x, y, t) ⫽ A0(x, y)exp[i0(x, y)]exp{i⌬m(x, y) ⫻ sin[0t ⫹ 0(x, y)]}. (1) In Eq. (1), A0 is the modulus of the diffused wave and 0 is a reference random phase uniformly distributed over interval 关⫺, ⫹兴. At any distance d0 where the object wave can be mixed with a reference wave of spatial frequencies 兵u0, v0其, the diffracted field is given in the Fresnel approximations by
i exp(2idR兾) dR i ⫻ exp (X2 ⫹ Y2) dR
APPLIED OPTICS 兾 Vol. 44, No. 27 兾 20 September 2005
Considering that the sinc function can be expanded as
冉 冊
sinc k with
冉 冊
T T ⫽ 1 ⫹ P k , T0 T0
(8)
P(x) ⫽
n⫽⬁
兺 n⫽1
x2n (⫺1) , (2n ⫹ 1)! n
(9)
we get
冕
tj⫹T
exp[i⌬m sin(0t ⫹ 0)]dt ⫽
tj
冋
冉
冊册
0T T exp i⌬m sin 0tj ⫹ 0 ⫹ 2 ⫹ Tqj exp(i⌰j),
(10)
where
再兺 冉 冊 冊册冎 冏再 冉 冊 冊册冏
⌰j ⫽ arg
k⫽⫹⬁
P k
k⫽⫺⬁
0T ⫹ 2
qj ⫽
,
k⫽⫹⬁
兺 P k
k⫽⫺⬁
⫹
0T 2
冋冉
T J (⌬m)exp ik 0tj ⫹ 0 T0 k (11)
冋冉
and it is simple to demonstrate that the amplitude and the phase of the vibration may be extracted by use of the two following equations: ⌬m(x, y) ⫽ ½ 兵⌬132(x, y) ⫹ 关⌬23(x, y) ⫹ ⌬21(x, y)]2其1兾2,
0(x, y) ⫽ arctan (12)
Note that complex term qj exp共i⌰j兲 is given by its complex Fourier expansion whose coefficients are P共kT兾T0兲Jk共⌬m兲 and its periodicity is related to variable tj and is equal to T0. Let us consider a phase term at any instant tj: ⌬j ⫽ ⌬m sin(0tj ⫹ 0 ⫹ 0T兾2);
arg[A⫹1R(tj)] ⫽ j ⫽ 0⬘ ⫹ ⌬m sin[0tj ⫹ 0 ⫹ (j ⫺ 1)兾2]mod(2), j ⫽ 1, 2, 3, (16)
冋
T J (⌬m)exp ik 0tj ⫹ 0 T0 k
.
It appears that because the first of Eqs. (15) contains three unknowns, only three equations are necessary for extracting the amplitude and phase of the vibration term. We can obtain the set of three equations with three unknowns by applying 兾2 phase shift between excitation and recording. Under these conditions, we get
(13)
then the phase of the digitally reconstructed object is expressed by
册
⌬13(x, y) , (18) ⌬23(x, y) ⫹ ⌬21(x, y)
where ⌬kl ⫽ k ⫺ l. Note that 0 is determined to an unknown constant and that Eq. (17) needs continuous absolute quantities. Because ⌬13, ⌬21, and ⌬23 are determined with a 2k uncertainty, k 僆 Z, it is necessary for order k to be the same for these three phase terms. Then phase unwrapping of ⌬kl must be started from the same region in each phase map, especially near a nodal region of the object. 3. Influence of Pulse Width
In practical situations, it is common for T to be not much smaller than T0, especially when it is not pos-
arg关A⫹1R(x, y, ⫺d0, tj)兴 ⫽ ⫺2i(u0x ⫹ v0y) ⫺ id0(u02 ⫹ v02) ⫹ 0(x, y) ⫹ ⌬j(x, y) qj(x, y)sin[⌬j(x, y) ⫺ ⌰j(x, y)] ⫺arctan . 1 ⫹ qj(x, y)cos[⌬j(x, y) ⫺ ⌰j(x, y)]
再
Because of the random nature of phase 0, the argument of the diffracted field is also a random variable, which encodes phase ⌬j, a reference phase term composed of determinist and random terms and a term whose limit is 0 when exposure time T is much smaller than T0. Now let us consider the ideal case in which the exposure time is infinitely small compared with the vibration period; we have lim arg关A⫹1R(tj)兴 ⫽ 0⬘ ⫹ ⌬msin(0tj ⫹ 0),
T兾T0→0
lim qj(tj) ⫽ 0. (15) T兾T0→0
(17)
冎
(14)
sible to work with a pulsed laser and a subnanosecond pulse width. This is the case when stroboscopic illumination is used with a continuous-wave laser and with acousto-optic23 or mechanical shuttering or with laser diode modulation.24 Indeed, if Rc ⫽ T兾T0 is the cyclic ratio of the illumination setup, the energy of each pulse is proportional to Rc; if the pulse width is too short compared with the photometric sensitivity of the sensor, then measurement is not possible or takes too long because of photon accumulation. Furthermore, the measurement is influenced by nonstationarity of the object. Thus it is important to get information on how the shuttering must be designed 20 September 2005 兾 Vol. 44, No. 27 兾 APPLIED OPTICS
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to prevent systematic distortions that may arise because of the nonzero last term of Eq. (14). Considering Eq. (17) and a linear approximation,28,29 a systematic error in the measurement of ⌬m is expressed as d⌬m ⫽
⭸⌬m ⭸⌬m ⭸⌬m d⌬13 ⫹ d⌬23 ⫹ d⌬21. ⭸⌬13 ⭸⌬23 ⭸⌬21 (19)
A similar relation holds for variation of phase 0. In Eq. (19), d⌬kl ⫽ dk ⫺ dl are variations on ⌬kl that are due to the nonzero pulse width. Variation terms dj are extracted from Eq. (14) and are approximated as dj ⬵ ⫺
qj sin(⌬j ⫺ ⌰j) . 1 ⫹ qj cos(⌬j ⫺ ⌰j)
(20)
From Eqs. (11), (12), and (20) it follows that d1, d2, and d3 are periodic functions with regard to variable t1 and their period is T0; d1 and d3 are odd functions; d2 is an even function; d1 is equal to d2 delayed by 兾2; and d3 ⫽ ⫺d1, it is then possible to expand the dj with Fourier series. Considering these properties, we have
Equations (24) and (25) give qualitative information on distortion. Indeed, Eq. (24) indicates that the error for measurement of vibration amplitude is an even periodic function with a nonzero mean value. This error has a period that is one quarter of the vibration period. Thus distortion that is due to the effect of a finite pulse width depends on the first instant t1 at which measurement is begun; the error has an offset value, so averaging of different measurements taken at different instants t1 does not remove it completely. The offset is found to be equal to the coefficient of the fundamental harmonic of the Fourier expansion of d1. Equation (25) indicates that error for the phase measurement is an odd periodic function with a zero mean value. This error has a period that is one quarter of the vibration period. To quantify the error it is useful to define a criterion that takes into account these characteristics. The following criteria are proposed:
再
Sd⌬m ⫽ sgn(␣11) ␣112 ⫹
k⫽⫹⬁
兺
k⫽0
␣1(2k⫹1)sin[(2k ⫹ 1)(0t1 ⫹ 0 ⫹ 0T兾2)], (21)
d2 ⫽
k⫽⫹⬁
兺
k⫽0
(⫺1)k␣1(2k⫹1)cos[(2k ⫹ 1)(0t1 ⫹ 0
⫹ 0T兾2)],
(22)
with 2 ␣1k ⫽ ⫺ T0
冕
T0
冉
0
q1(t1)sin[⌬1(t1) ⫺ ⌰1(t1)] 1 ⫹ q1(t1)cos[⌬1(t1) ⫺ ⌰1(t1)]
⫻ sin 2
冊
k t dt . T0 1 1
(23)
However, no analytical expression can be obtained from Eq. (23), and Fourier development must be numerically evaluated. According to Eqs. (19), (21), and (22), the phase error is d⌬m ⫽ ␣11 ⫹
k⫽⫹⬁
兺
k⫽1
[␣1(4k⫹1) ⫺ ␣1(4k⫺1)]cos(4k0t1 ⫹ 4k0
⫹ 2k0T).
(24)
In the same way, the phase error for phase 0 is d0 ⫽
5766
1 k⫽⫹⬁ 兺 [␣1(4k⫹3) ⫹ ␣1(4k⫹5)] ⌬m k⫽0 ⫻ sin[2(2k ⫹ 2)(0t1 ⫹ 0 ⫹ oT兾2)]. (25)
APPLIED OPTICS 兾 Vol. 44, No. 27 兾 20 September 2005
k⫽⫹⬁
兺 k⫽1
冎
1兾2
[␣1(4k⫹1) ⫺ ␣1(4k⫺1)]2
, (26)
Sd0 ⫽
再 兺 [␣ ⌬ 冑2 1
m
d1 ⫽
1 2
k⫽⫹⬁ k⫽0
1(4k⫹3)
冎
⫺ ␣1(4k⫹5)]2
1兾2
.
(27)
The first criterion corresponds to the square of the power of the harmonic expansion multiplied by the sign of the mean value. The signum of the fundamental indicates direction of the distortion. If it is positive, the estimated vibration amplitude is greater than the exact amplification, and if it is negative, the amplitude is smaller. To validate this theoretical investigation, simulations were carried out. One develops a numerical simulation by considering Eq. (7), computing associated arguments, evaluating amplitude and phase according to Eqs. (17) and (18), and then calculating errors. We evaluate the numerical equivalent criteria to Eqs. (26) and (27), using numerical error, by computing the square root of the average of the squared error and multiplying the result by the sign of the error offset. Analytical computation is performed with expressions (11), (12), (19), (20), and (23)–(25) and criteria Eqs. (26) and (27). Three simulations were performed with ⌬m ⫽ 20 rad and 0 ⫽ ⫺0.698 rad, the first with Rc ⫽ T兾T0 ⫽ 1兾50, the second with Rc ⫽ 1兾20, and the last with Rc ⫽ 1兾17. Figures 1–3 present errors for ⌬m and 0, expressed in percent, obtained with the three simulations. Numerical and analytical results are seen to match closely, which validates the theoretical analysis. It can be seen that there is little distortion in the two first results, whereas in the last one distortion is significant. This evidences the highly nonlinear behavior of distortion in relation to vibration amplitude and cyclic ratio. Comparisons of numerical and analytical statistics are given in Tables 1–3. The statistics are found to agree well. Furthermore, criteria proposed in Eqs. (26) and (27) appear to
Fig. 1. Error (%) for amplitude and phase for Rc ⫽ 1兾50 (circles, numerical; curves, analytical).
Fig. 3. Error (%) for amplitude and phase for Rc ⫽ 1兾17 (circles, numerical; curves, analytical).
be good indicators of the distortion. The slight difference in the numbers that one can observe in the rows of Table 3 is due to the fact that Eq. (19) is a linear approximation, which may be not sufficient to describe error when the cyclic ratio is high. Figure 4 presents three-dimensional plot criterion Sd⌬m as a function of ⌬m and Rc. Figure 5 presents threedimensional plot criterion Sd0 for 0 ⫽ 0 as a function of ⌬m and Rc. It is shown that for a maximum amplitude of approximately 30 rad the cyclic ratio must be smaller than 1兾30 for distortion smaller than 0.05% for ⌬m and than 0.1 rad for 0; for a maximum amplitude of ⬃20 rad, Rc can be 1兾20 for distortion less than 0.12% for amplitude and less than 0.1 rad for phase. As a conclusion to these results, it can be seen that, to prevent distortion, for a maximum vibration amplitude of ⌬m the cyclic ratio Rc must be smaller than 1兾⌬m. 4. Stroboscopic Setup
During a stroboscopic process, the recorded hologram results from an accumulation of instantaneous synchronic holograms, each of which is integrated during the pulse exposure. The temporal width of the accumulation defines the total exposure time of the sensor. Thus the mathematical expression for the recorded hologram becomes
Fig. 2. Error (%) for amplitude and phase for Rc ⫽ 1兾20 (circles, numerical; curves, analytical).
Ha(tj) ⫽
k⫽Kp
兺
k⫽0
冕
tj⫹T
H(t ⫺ kT0)dt,
(28)
tj
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Table 1. Statistics of Error in the First Simulation (%)
Amplitude ⌬m Type
Numerical
Analytical
⫺3
⫺3
⫺62.0 ⫻ 10 2.6 ⫻ 10⫺3 7.8 ⫻ 10⫺3 ⫺62.0 ⫻ 10⫺3
Offset Standard deviation Peak to valley Criterion
Phase 0
⫺62.0 ⫻ 10 2.6 ⫻ 10⫺3 7.8 ⫻ 10⫺3 ⫺62.0 ⫻ 10⫺3
where KpT0 ⫹ T is the total exposure time of the detector. Considering that A共x, y, t ⫺ kT0兲 ⫽ A共x, y, t兲 under sinusoidal excitation, the reconstructed object is given by A⫹1R(x, y, ⫺d0, tj) ⬵ (Kp ⫹ 1)NM 4d04 ⫻ R*(x, y) ⫻ exp[⫺id0(u02 ⫹ v02)]{A0(x, y) ⫻ exp[i0(x, y)] ⫻
冕
tj⫹T
exp[i⌬m(x, y, t)]dt}
tj
ⴱ ␦(x ⫺ u0d0, y ⫺ v0d0), (29) where ␦共x, y兲 is the bidimensional Dirac distribution and ⴱ means two-dimensional convolution. Note that the hologram accumulation does not change the properties of the reconstructed object if illumination is synchronized with excitation. Algorithms (17) and (18) are still valid. The setup to carry out the stroboscopic measurement with synchronization between illumination and excitation depends closely on the device used. In this section a system based on a mechanical shuttering with a rotating aperture is presented. The aperture is composed of a disk 102 mm in diameter with four slits 0.5 mm in length. This gives a natural cyclic
Numerical
Analytical
⫺19
⫺1 ⫻ 10⫺19 4.1 ⫻ 10⫺3 11.6 ⫻ 10⫺3 4.1 ⫻ 10⫺3
⫺1 ⫻ 10 4.1 ⫻ 10⫺3 11.6 ⫻ 10⫺3 4.1 ⫻ 10⫺3
ratio of 1兾144. The electronic setup must be designed with some important parameters taken into account: the maximum rotation speed of the engine is 125 Hz, the maximum frequency to be measured is 5 kHz, the maximum exposure time of the CCD sensor is 10 s, and continuous laser power is 30 mW. The foursector chopper permits an acceptable compromise between these magnitudes. The setup is depicted in Fig. 6. Setting ratio N establishes the illuminating frequency of the object. In the present case, N ⫽ 8 was chosen; i.e., there is one light pulse every eight vibration periods. Thus the cyclic ratio of the stroboscope is Rc ⬇ 1兾18, and this was experimentally verified. The square signal issued from the optocoupler has a frequency of f0兾N, and it controls a first phase look loop that generates a signal at frequency f0. This signal orders a second phase look loop, which generates a clock signal at a frequency of 256 ⫻ f0 (1280 kHz maximum) for incrementing a binary counter that carries out an exit encoded on 8 bits. This bus addresses a 64 K memory and a digital-to-analog converter. The signal at the DAC output is sinusoidal at frequency f0. A low-pass filter cuts parasitic high frequencies, and an amplifier regulates the amplitude of the signal applied to the object. The rising edge of signal VE and the synchronization signal (trigger) drive the binary counter to define the phase origin of signal VS. A five-bit bus controls the phase shift of VS. The phase-shift quantum is set to ⫽ 2兾25
Table 2. Statistics of Error in the Second Simulation (%)
Amplitude ⌬m
Phase 0
Type
Numerical
Analytical
Numerical
Analytical
Offset Standard deviation Peak to valley Criterion
0.0443 0.1047 0.5187 0.1136
0.0304 0.1176 0.5135 0.1214
⫺5 ⫻ 10⫺6 2.3689 10.2646 2.3677
1 ⫻ 10⫺19 2.3673 10.1664 2.3673
Table 3. Statistics of Error in the Last Simulation (%)
Phase 0
Amplitude ⌬m Type Offset Standard deviation Peak to valley Criterion
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Numerical ⫺17.26 2.11 5.84 ⫺17.39
APPLIED OPTICS 兾 Vol. 44, No. 27 兾 20 September 2005
Analytical ⫺17.30 2.08 5.72 ⫺17.42
Numerical
Analytical
⫺5
1 ⫻ 10⫺19 3.81 12.84 3.81
3.8 ⫻ 10 4.54 15.24 4.54
Fig. 4. Three-dimensional plot of criterion Sd⌬m for amplitude distortion versus Rc and ⌬m关%兴. Fig. 7. Experimental setup: L’s, lenses.
⫽ 兾16. Thus it is possible to phase shift illumination and excitation and then to produce the three phaseshifted digital holograms that are suitable for applying algorithms (17) and (18). Considering the results of Section 3, with Rc ⬇ 1兾18 the stroboscopic setup is designed for a vibration amplitude smaller than 18 rad for the amplitude distortion to be less than 0.15%. 5. Experimental Setup
Fig. 5. Three-dimensional plot of criterion Sd0 for amplitude distortion versus Rc and ⌬m 关rad兴.
The digital holographic setup is described in Fig. 7. The object under the sinusoidal excitation is a loudspeaker 60 mm in diameter, placed at d0 ⫽ 1440 mm from the detector area. Off-axis holographic recording is carried out with lens L2 displaced out of the afocal axis by means of two micrometric transducers.11 So the frequencies of the carrier wave are u0 ⬵ 76.5 mm⫺1 and v0 ⬵ ⫺69 mm⫺1. The detector is a
Fig. 6. Stroboscopic setup: DAC, digital-to-analog converter; VCO, voltage-controlled oscillator. 20 September 2005 兾 Vol. 44, No. 27 兾 APPLIED OPTICS
5769
Fig. 8. Vibration amplitude (left) and phase (right) at 2 kHz [rad].
12-bit digital CCD with 共M ⫻ N兲 ⫽ 共1024 ⫻ 1360兲 pixels of pitch px ⫽ py ⫽ 4.65 m (PCO Pixel Fly). Digital reconstruction was performed with K ⫽ L ⫽ 2048 data points. If the vibration of the membrane of the loudspeaker is considered to be a pure out-ofplane movement, i.e. U ⫽ uzk, the k amplitude is then related to ⌬m according to 1 uz(x, y) ⫽ ⌬m(x, y). 2 1 ⫹ cos
(30)
6. Results of the Experiments
The loudspeaker was excited in a sinusoidal regime from 1.36 to 4.32 kHz in steps of 40 Hz. The exposure time was set to 10 s. Figure 8 shows the vibration amplitude and phase at 2 kHz extracted from three 兾2 phase-shifted digital holograms as described in Section 2. In Fig. 8 the maximum amplitude is 17.1 rad, so distortion is less than 0.25% for ⌬m. For this frequency Kp ⫽ 20 ⫻ 103, so the acquired hologram corresponds to the accumulation of 20,000 instantaneous holograms. Figures 9 and 10 show the vibration amplitude and phase at 2.44 and 3.88 kHz. The region of interest in each map contains ⬃240,000 data points. Compared to classic vibrometers, which use a pointwise optical probe, the digital holographic setup leads to full-field information on vibration. The evaluation of the amplitude and the phase of the membrane of the loudspeaker determine its velocity along the z component, which is given by
1 具| v| 典 ⫽ ST0 2
冕冕 冕
T0
|v共x, y, t)|2dtdxdy,
0
(32) where S is the surface of the membrane. Figure 11 shows the mean quadratic velocity of the loudspeaker for a frequency that varies from 1.36 to 4.32 kHz. This curve is similar to a Bode diagram. Four resonance frequencies are clearly seen in the curve. This curve was computed after the recording of 225 holograms. Figure 12 shows the mean quadratic velocity of the loudspeaker after the surface of the membrane has been painted to enhance reflected light. It can be shown that the effect of painting modifies the resonance frequencies; this is due to a modification of the membrane stiffness. Note that high frequencies near 4 kHz have disappeared. To validate the stroboscopic measurement, Fig. 13 shows a comparison between time-averaged measurement11 and a simulation of time averaging from the stroboscopic result at 2.8 kHz; this corresponds to the second peak in Fig. 12. The experimental Bessel fringes and those that were simulated can be seen to be in close agreement; this confirms the suitability of the setup to perform accurate full-field amplitude and phase vibration measurements.
f0 ⌬m(x, y)cos(0t ⫹ 0). (31) 1 ⫹ cos
Fig. 9. Vibration amplitude (left) and phase (right) at 2.44 kHz [rad]. 5770
A criterion usually used by acoustic engineers for qualifying vibration is the mean quadratic velocity. It is defined for frequency f0 by
S
In the setup, angle is set to 45°.
vz(x, y, t) ⫽
Fig. 10. Vibration amplitude (left) and phase (right) at 3.88 Hz [rad].
APPLIED OPTICS 兾 Vol. 44, No. 27 兾 20 September 2005
Fig. 11. Mean quadratic velocity of the membrane (*, experimental; solid curve, fitting curve).
Fig. 12. Mean quadratic velocity after surface painting (*, experimental; solid curve, fitting curve).
Fig. 13. Comparison between time-averaging and stroboscopic measurement (left, experimental; right, simulation).
7. Conclusions
In this paper we have discussed a full-field vibrometer based on digital Fresnel holography. The influence of the pulse width was studied, and the analytical expressions of amplitude and phase distortion were proposed. Numerical simulation has validated the theoretical analysis. As a result, it was demonstrated that amplitude and phase extraction are possible with a cyclic ratio of ⬃1兾⌬m if the maximum vibration amplitude does not exceed ⌬m. Thus the use of a moderated cyclic ratio is possible in vibration analysis. Experimental results were presented and illustrated the relevance of digital Fresnel holography in vibrometry. A comparison between time-averaging and stroboscopic measurement confirmed the accuracy of the method. References 1. Th. Kreis, Holographic Interferometry—Principles and Methods, Vol. 1 of Akademie Verlag Series in Optical Metrology (Akademie Verlag, Berlin, 1996). 2. M. A. Kronrod, N. S. Merzlyakov, and L. P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333–334 (1972). 3. U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179 –181 (1994). 4. G. Pedrini and H. J. Tiziani, “Digital double pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 18, 251–260 (1995). 5. G. Pedrini, Y. L. Zou, and H. J. Tiziani, “Digital double pulse holographic interferometry for vibration analysis,” J. Mod. Opt. 42, 367–374 (1995).
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