Vibration of low amplitude imaged in amplitude and phase by sideband versus carrier correlation digital holography

0.42 MB·English

by  N Verrier

Vibration of low amplitude imaged in amplitude and phase by sideband versus carrier
correlation digital holography
N. Verrier1,2, L. Alloul1 and M. Gross1
1Laboratoire Charles Coulomb - UMR 5221 CNRS-UM2 Universit´e
Montpellier II Bat 11. Place Eug`ene Bataillon 34095 Montpellier
2LaboratoireHubertCurienUMR5516CNRS-Universit´eJeanMonnet18RueduProfesseurBenoˆıtLauras42000Saint-Etienne
Sideband holography can beused toget fieldsimages (E0 and E1) of avibrating object for both
the carrier (E0) and the sideband (E1) frequency with respect to vibration. We propose here to
∗
recordE0 andE1 sequentially,andtoimagethecorrelation E1E0. Weshowthatthiscorrelation is
5
insensitivethephaserelatedtotheobjectroughnessanddirectlyreflectthephaseofthemechanical
1
motion. The signal to noise can be improved by averaging the correlation over neighbor pixel.
0
Experimental validation is made with vibrating cube of wood and with a clarinet reed. At 2 kHz,
2
vibrations of amplitude down to0.01 nm are detected.

n
a PACSnumbers: 120.7280,090.1995,040.2840,120.2880
J

9
2 Citation: N. Verrier, L. Alloul, and M. Gross, Opt. al. [14]haveshownthatonecansimultaneouslymeasure
Lett. 40, 411-414(2015) E and E by using a localoscillatorwith two frequency
0 1
] http://dx.doi.org/10.1364/OL.40.000411 components. One can thus infer the mechanical phase
s
c of the vibration [15]. However, Bruno et al [16] shows
i There is a big demandfor full field vibrationmeasure- that this simultaneous detection of E and E can lead
t 0 1
p ments, in particular in industry. Different holographic to crosstalk, andto a lossof detection sensitivity,which
o techniquesareabletoimageandanalyzesuchvibrations. becomesannoyingwhenthevibrationamplitudeissmall.
.
s Double pulse [1, 2] or multi pulse holography [3] records In this letter, we show that simultaneous detection of
c severalhologramswithtimeseparationinthe1...1000µs E and E is not necessary,and that equivalent or supe-
i 0 1
s range, getting the instantaneous velocity from the phase rior performances can be obtained by detecting E and
0
y
difference. Ifthevibrationfrequencyisnottoohigh,one E sequentially. Indeed, the cross talk effects seen by
h 1
can also directly track the vibration of the object with Bruno et al. [16] disappear in that case. We also show
p
[ fast CMOS cameras [4, 5]. The analysis of the motion that the random phase variations caused by the rough-
canbe done by phasedifference or byFourieranalysisin ness of the object can be eliminated by calculating the
1 the time domain. For periodic vibration motions, mea- correlation E E∗. It is then possible to increase the sig-
v 1 0
surementscanbedonewithslowcamera. Indeed,anhar- nal to noise ratio (SNR) by averaging correlation over
5
monically vibrating object illuminated by a laser yields neighboring pixels. The sequential measurement of E
5 0
5 alternatedarkandbrightfringes[6],thatcanbeanalyzed and E1, and the calculation of the correlation E1E0∗,
7 by time averaged holography [7]. Although the time av- make possible to image the vibration “full field”, and to
0 eraged method gives a way to determine the amplitude measure quantitatively its amplitude and phase. Maxi-
1. of vibration [8] quantitative measurement remains quite mum sensitivity is achieved by focusing the illumination
0 difficult for low and high vibration amplitudes. in the studied point and by averaging the correlation in
5 We have developed heterodyne holography [9, 10], that region. Finally, we prove that the sensitivity is lim-
1 which is a variant of phase shifting holography,in which ited by a sideband signal of one photo-electron per de-
v: thefrequency,phaseandamplitudeofbothreferenceand modulatedimagesequence. Thedeviceandmethodwere
i signal beam are controlled by acousto optic modulators validated experimentally by studying a cube of wood vi-
X
(AOM). Heterodyne holographyis thus extremely versa- brating at ≃ 20 kHz, and a clarinet reed at ≃ 2 kHz.
ar tile. By shifting the frequencyofthe localoscillatorωLO Measurement sensitivity of 0.01 nm for a vibration at 2
withrespecttoilluminationω0,itisforexamplepossible kHz, comparable to the sensitivity obtained by Bruno et
todetecttheholographicsignalatafrequencyωdifferent al. [16] at 40 kHz, is demonstrated.
than illumination ω0. This ability is extremely useful to Consideranobjectilluminated bya laseratfrequency
analyze vibration, since heterodyne holography can de- ω that vibrate at frequency ω with an out of plane vi-
0 A
tect selectively the signal that is scattered by object on bration amplitude z . The out of plane coordinate is
max
vibration sideband of frequency ωm = ω0+mωA, where z(t) = zmaxsin(ωAt). The field scattered by the object
ωA is the vibration frequency and m and integer index. is E =E(t)eiω0t+ c.c., where c.c. is the complex conju-
AswasreportedbyUedaetal,[11]thedetectionofthe gate and E(t) the field complex amplitude. In reflection
sidebandm=1isadvantageouswhenthevibrationampli- geometry, we have
tude is small. Nanometric vibration amplitude measure-
ments were achieved with sideband digital holography E(t)=E e|Φ|sin(ωAt+argΦ)
wo
on the m =1 sideband [12], and comparison with single
pointlaserinterferometryhas been made [13]. Verrieret where E is the complex field without movement,
wo
2
FIG. 2. (a,b) Reconstructed images of a cube of wood vi-
∗
FIG. 1. Heterodyne holography setup applied to analyse vi- brating at ωA/2π= 21.43 kHz. (a) E∗1E0 correlation im-
blartaotriso;nM. L: :mmiraroinr;lBasSe:r;bAeaOmMs1p,liAttOerM; C2:CaDc:oucastmoe-orap.tic modu- aagrge:Eb1rEig0∗h).tn(ebs)s3isDadmisppliltauydoef(ti.hee. p|Eh1aEse0a|)r,gcEol1oEr0∗is. phase(i.e.
Forsmallvibrationamplitude(|Φ|≪1),correlationsim-
and Φ is a complex quantity that describes the
plifies to E E∗ ≃ |E |2Φ/2. Correlation E E∗ is a
phase modulation. The phase of this modulation is 1 0 wo 1 0
powerful tool since gives directly the phase of mechani-
argΦ, while the amplitude is |Φ| = 4πz /λ. Be-
max calmotionargΦ. Nevertheless,problemscanbe encoun-
cause of the Jacobi-Anger expansion, we have E(t) =
teredwhenthe signal|E |2 scatteredwithoutvibration
E J (|Φ|) ejm(ωAt+argΦ) where J is the mth- wo
wo Pm m m vanish.
order Bessel function of the first kind. The scattered
Figure 1 shows the holographic experimental setup
field E is then the sum of fields components E =
m used to measure successively E and E in order to get
E eiωmt + c.c. of frequency ω = ω + mω , where 0 1
m m 0 A E E∗. The main laser L is a Sanyo DL-7140-201 diode
m is the sideband index (m = 0 for the carrier) with 1 0
laser(λ=785nm,50mW).Itissplitintoanillumination
E = E J (|Φ|) ejmargΦ. When the vibration am-
m wo m beam(frequencyω ,complexfieldE ),andinaLObeam
plitude Φ becomes small, the energy within sidebands I I
(ω ,E ). The illumination light scattered by the ob-
decrease very rapidly with the sideband indexes m, and LO LO
ject interferes with the reference beam on the camera
one has only to consider the carrier m=0, and the first
(Lumenera2-2: 1616×1216pixelsof4.4×4.4µm)whose
sideband m=±1. We have thus:
frame rate is ω =10 Hz. To simplify further Fourier
CCD
transformcalculations,the 1616×1216measuredmatrix
E (Φ)=E J (|Φ|) (1)
0 wo 0
is truncated to 1024×1024.
E (Φ)=E J (|Φ|) ejargΦ
1 wo 1 TheilluminationandLObeamfrequenciesω andω
LO I
are tuned by using two acousto-opticmodulators AOM1
Note that time averaged holography [7] that detects the
and AOM2 (Bragg cells), and we have ω = ω +
carrier field E is not efficient in detecting small ampli- LO L
0 ω andω =ω +ω ,whereω ≃80MHz
tude vibration |Φ|, because E varies quadratically with AOM1 I L AOM2 AOM1/2
0 are the frequencies of RF signals that drive the AOMs.
|Φ|. Ontheotherhand,sidebandholographythatisable
The RF signals are tuned to have ω −ω = ω /4
to detect selectively the sideband field E is much more LO I CCD
1 to get E , and to have ω −ω =ω +ω /4 to get
sensitive, because E varies with linearly with |Φ|. 0 LO I A CCD
1 E . Successive sequences of n = 128 camera frames
1 max
One must notice that the field scattered by the sam-
(i.e. I ,I ...I ) are recorded by tuning the RF signals
0 1 127
ple without vibration E depends strongly on the x,y
wo firstonthe carrier(E ), thenonthe sideband(E ). The
0 1
position. In a typical experiment, the sample rugosity is
carrier and sideband complex hologram H are obtained
such that this field is a fully developed speckle. The re-
fromthesesequencesby4phasedemodulationwithn
max
constructed fields E and E are thus random in phase,
0 1 frames:
from one pixel (x,y) to the next (x +1,y). One can-
not thus simply extract the phase of the vibration from n=nmax−1
a measurement made on a single sideband. To remove H(x,y)= X jnIn(x,y) (3)
the pixel to pixel random phase of E , we propose to
wo n=0
record the hologram successively on the carrier m = 0
and the sideband m = 1, to reconstruct the correspond- The fields images of the object E0(x,y) and E1(x,y) are
ing field image of the object E (x,y) and E (x,y), and then reconstructed from H by the Schnars et. al [17]
0 1
to calculate and image the correlation E E∗, since this methodthat involves1 Fouriertransformation. The cor-
1 0
correlationdonotinvolvesEwo,but|Ewo|2,whichisreal relation E1E0∗ is then calculated.
and has no phase. Indeed, we have: Figure 2 (a) shows the reconstructed correlation im-
agesof a cube of wood(2 cm×2 cm) vibrating atits res-
E E∗ =|E |2 J (|Φ|) J (|Φ|) ejargΦ (2) onance frequency ω /2π= 21.43 kHz. Brightness is the
1 0 wo 1 0 A
3
correlation amplitude (i.e. |E E∗|) and color the corre-
1 0
lation phase (i.e. argE E∗). As seen, neighbor point
1 0
havethe same phase (same color). In orderto getbetter
SNR for the phase, the complex correlationsignal E E∗
1 0
isaveragedoverneighborx,y pointsbyusinga2DGaus-
sian blur filter of radius 4 pixels. Figure 2 (b) displays
the phase of the averaged correlation E E∗ in 3D. As
1 0
seen, the opposite corners (upper left and bottom right
for example) vibrate in phase, while the neighbor cor-
ners (upper left and upper right for example) vibrate in
phaseopposition. Note thatthe cube is excitedinoneof
itscornerbyaneedle. Thismayexplainwhytheopposite
corners are not perfectly in phase in Fig. 2 (c).
We can go further and use E and E to calculate the
0 1
vibration complex amplitude Φ. To increase the SNR FIG. 3. Ratio |hE1E0∗i|/h|E0|2i calculated by Monte Carlo
let us average, over neighbor reconstructed pixels, the by decreasing the sideband signal |E1|2. Horizontal axis is
correlation E1E0∗ and the carrier intensity |E0|2: the total sideband energy: nmaxNpixh|E1|2i in photo elec-
tron Units. Simulation is made with nmax=400, Npix =502,
|E0|2 =|Ewo|2 J02(Φ)≃|Ewo|2Φ/2 (4) |ELO|2 =104 and |E0|2 =102 photo electrons.
By averaging we get:
hE1E0∗ix,y =(1/Npix)XE1(x′,y′)E0∗(x′,y′) (5)
x′,y′
≃(Φ(x,y)/2) (1/Npix)X|Ewo(x′,y′)|2
x′,y′
h|E0|2ix,y ≃(1/Npix)X|Ewo(x′,y′)|2
x′,y′
where isthesummationovertheN pixelsofthe
Px′,y′ pix
averaging zone located around the point of coordinate
x,y. The vibration amplitude Φ is then
Φ(x,y)=2hE E∗i /h|E |2i (6)
1 0 x,y 0 x,y
FIG. 4. (a) Clarinet reed with illumination beam focused
We get here Φ that gives both the amplitude z =
max in x0,y0. (b) Sideband m = 1 reconstructed image of the
λ|Φ|/4π and the phase argΦ of the mechanical motion. vibratingreed. The display is made in arbitrary log scale for
Note that it is also possible to get Φ by calculating the thefield intensity |E1(x,y)|2.
ratio E /E ≃ Φ as done by Verrier et. al [14], but
1 0
the ratio calculationis unstable for the points x,y of the
object where E is close to zero.
wo
To evaluate the limits of sensibility of the correlation one frame n to another,and fromone sequence m to an-
+averagingmethod,wehavecalculatedby MonteCarlo other. By this way, we have obtained the Monte Carlo
the detection limit of |Φ|, for an ideal holographicdetec- frame signals I =I′ +s with whom we have performed
n n
tion that is only limited by shot noise. The calculation the 4 phase demodulation with n frames of Eq. (3).
max
is similar to one made by Lesaffre et al. [18]. For each WehavethencalculatedthereconstructedsignalE and
0
pixel (x,y), each frame (n) and each sequence (m=0 or E , the correlationsand intensities E E∗ and |E |2, and
1 1 0 0
m=1), we have calculated the ideal camera signal I′ in the means hE E∗i and h|E |2i. We have then calculated
n 1 0 0
theabsenceofshotnoise. WehaveIn′ =|ELO+jnE0′/1|2, the ratio hE1E0∗i/h|E0|2i that gives Φ using Eq. (6).
wherethefactorjn accountsforthephaseshiftofthelo- Figure3givestheresultoftheMonteCarlosimulation
cal oscillator field ELO with respect to object field E0/1 for the ratio |hE1E0∗i|/h|E0|2i. Each point correspond
for frame n. To accountfor the roughnessofthe sample, to simulation made with a double sequence m=0 and
E′ and E′ are taken proportional to a Gaussian speckle 1. The simulation is performed by decreasing the side-
0 1
E′ (x,y)thatisuncorrelatedfromonepixeltothenext, band averaged signal field intensity h|E |2i=1, 0.5, 0.25
wo 1
but remains the same for all frames n and all sequences ... photo electron per pixel and per frame. The other
m=0or1. Toaccountforshotnoise,wehaveaddedtoI′ parameters of the simulation are n = 400, N =502,
n max pix
a Gaussianrandomnoise s of variance of I′, where I′ |E |2=104 and h|E |2i=102 photo electrons. For each
p n n LO 0
is expressed in electron photo electron Units. The noise valueofh|E |2i,10simulationsareperformed. Ascanbe
1
s is uncorrelated from one pixels x,y to another, from seen in Fig. 3 the ratio |hE E∗i|/h|E |2i decreases with
1 0 0
4
the sideband signal E1 proportionally with ph|E1|2i. crease the sensitivity, the vibration amplitude is mea-
When E becomes very small, the ratio reaches a noise sured on a single point (x ,y ) (see Fig.4 (a)). The
1 0 0
floorrelatedtoshotnoise. Tosimplifythepresentdiscus- illumination has been focused on that point, and the
sion, the results are displayed as a function of the total calculations have been made with an averaging region
number of photos electrons on the sideband m =1. The centered on that point, whose size (radius 50 pixels for
x-axisisthusn N h|E |2i. AsseenofFig.3thenoise example) has been chosento include the whole illumina-
max pix 1
floor is reached for n N h|E |2i = 1. We have ver- tion zone. We have reconstructed the field image of the
max pix 1
ified by making simulation not displayed on Fig.3 that reed at the carrier frequency (i.e. E (x,y)) (see Fig.4
0
this result do not depend on |E |2 and h|E |2i, pro- (b)), and verified on the holographic data that most of
LO 0
vided that |E |2 ≫ h|E |2i ≫ h|E |2i. The noise floor the energy |E |2 is within the averaging zone x ,y . Se-
LO 0 1 0 0 0
corresponds thus to a minimal ratio |hE E∗i|/h|E |2i = quences of n =128 frames I have been recordedfor
1 0 0 max n
1/pnmaxNpixh|E0|2i(i.e. to10−4),andtoaminimalde- both carrier (E0) and sideband (E1), while the peak to
tectable vibration amplitude Φ = 2/pnmaxNpixh|E0|2i peak voltageVpp ofthe loudspeakersinusoidalsignalhas
(i.e. to 2 10−4). been decreased. We have then calculated H by Eq. (3),
reconstructed the fields images of the reed E and E ,
0 1
and calculated the correlation E E∗, the intensity|E |2,
1 0 0
andthe means hE E∗i and h|E |2i. We have then calcu-
1 0 0
latedthe ratio|hE E∗i|/h|E |2i. The latteris plottedon
1 0 0
Fig.5 as a function of the loudspeaker voltage V that
pp
is proportionalto vibration amplitude Φ. The measured
points follow |hE E∗i|/h|E |2i ≃ |Φ|/2 ∝ V down to
1 0 0 pp
a vibration amplitude noise floor of |Φ| ≃ 7 10−5 that
corresponds to z ≃ 10−2 nm i.e λ/78000. Note that
max
the noise floor measure here is about ×20 lower than in
previousholographicexperiments[12,14]andlowerthan
the limit λ/3500 predicted by Ueda [11]. Similar noise
floor has been detected by holography by Bruno et al.
[16], but at much higher vibration frequency (40 kHz).
In future work, it would be interesting to explore the
limits of sensitivity of the correlation technique for low
vibration amplitude. This can be done by using a laser
FIG. 5. Ratio |hE1E0∗i|/h|E0|2i as function of the reed with lower noise, by increasing the vibration frequency
excitation voltage converted in vibration amplitude zmax in ωA (and in all case by measuring the laser noise at ωA).
nm Units. Measurements: dark grey points, and theory One could also increase the illumination power, and the
J1(|Φ|)/J0(|Φ|)≃|Φ|/2 : light grey curve. numberofframeofthecoherentdetection(n >128).
max
Abettercontrolofthevibrationω versuscameraω
A CCD
frequencies, and or a proper choice of the demodulation
equation could be also important, in order to avoid leak
We have tested the ability of the correlation + av- detection of E , when detection is tuned on E .
0 1
eraging method to measure low vibration amplitude in The Authors wishes to acknowledge ANR-ICLM
an experiment made with a vibrating clarinet reed ex- (ANR-2011-BS04-017-04) for financial support, and M.
cited at ω =2 kHz with a loudspeaker. In order to in- Lesaffre for fruitful discussion and helpful comments.
A
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