Quantitative imaging of nanometric optical path length modulations by time-averaged heterodyne holography in coherent frequency-division multiplexing regime

1.1 MB·English

by  Francois Bruno

Quantitative imaging of nanometric optical path length modulations by time-averaged
heterodyne holography in coherent frequency-division multiplexing regime.
Francois Bruno,1 Jean-Baptiste Laudereau,1 Max Lesaffre,1 Nicolas Verrier,1 and Michael Atlan1
1
Institut Langevin. Fondation Pierre-Gilles de Gennes. Centre National de la Recherche Scientifique (CNRS) UMR 7587,
Institut National de la Sant´e et de la Recherche M´edicale (INSERM) U 979, Universit´e Pierre et Marie Curie (UPMC),
Universit´e Paris 7. E´cole Sup´erieure de Physique et de Chimie Industrielles - 1 rue Jussieu. 75005 Paris. France
(Dated: February 1, 2013)
We report a demonstration of amplitude and phase imaging of out-of-plane sinusoidal vibration
at nanometer scales with a heterodyne holographic interferometer. Time-averaged holograms of a
3 phase-modulatedopticalfieldarerecordedwithanexposuretimemuchlongerthanthemodulation
1 period. Optical heterodyning, a frequency-conversion process aimed at shifting a given radiofre-
0 quencyopticalsidebandinthesensorbandwidth,isperformedwithanoff-axisandfrequency-shifted
2 opticallocaloscillator. Theoriginalityoftheproposedmethodistomakeuseofamultiplexedlocal

n oscillatortoaddressseveralopticalsidebandsintothetemporalbandwidthofthesensorarray. This
a process is called coherent frequency-division multiplexing. It enables simultaneous recording and
J pixel-to-pixeldivisionoftwosidebandholograms, whichpermitsquantitativemappingofthemod-

1 ulationdepthoflocalopticalpathlengthsyieldingsmallopticalphasemodulations. Additionally,a
3 linearfrequencychirpensurestheretrievalofthelocal mechanical phaseshift ofthevibrationwith
respecttotheexcitation signal, when screeningagiven frequencyrange. Theproposedapproach is

] validatedbyquantitativemotioncharacterizationofthelamellophoneofamusicalbox,behavingas
s
agroupofharmonicoscillators,underweaksinusoidalexcitation. Imagesofthevibrationamplitude
c
i versusexcitationfrequencyshowtheresonanceofthenanometricflexuralresponseofoneindividual
t cantilever, at which a phase hop is measured.
p
o
.
s Imaging nanometric optical path length modulations RF mixer
c
i is of interest in non-destructive testing of micro electro- !C+î!01 !+î!02
s
mechanical systems [1]. Among common approaches
y
h used for single-point nanometric vibration spectra !C+î!03 RF combiner
p measurements, laser Doppler heterodyne schemes [2] beam !
[ are methods of choice, because of their high sensitivity laser splitter C lamellophone
+
1 and wide detection bandwidth, though imaging is mirror piezo-electric
v hindered because scanning of the probe is required. actuator
2 In wide-field imaging of sinusoidal optical path length !L acousto-optic reference
3 modulations, either performed with incoherent or modulators ELO beam (LO)
5
7 coherent light sources, in direct image or holographic mirror illumination beam
1. recording conditions, the mechanical phase is usually beam splitter E
retrievedfromphase-steppedstroboscopicschemes[1,3]. object !
0 ò
3 In the particular case of holographic measurements, sensor array 0 beam
1 this approach is extensively used in the sinusoidal
: excitation regime [4–8]. Yet, in holography, the optical !S
v
phase retardation of the recorded scattered field can be
i
X exploited for the determination of the local mechanical FIG. 1: Experimental set-up.
r phase of an object motion with respect to the excitation
a
signalwhen the readoutrate ofthe arraysensoris faster
than the modulation period [9, 10]. An intermediate
The proposed arrangement consists of a Mach-Zehnder
recording regime is quasi-time-averaging [3], for which
holographic interferometer in an off-axis and frequency-
the modulationperiodis ofthe same orderofmagnitude
shifting configuration,withamultiplexed localoscillator
as the frame rate. Nevertheless, because the vibration
(fig. 1). Optical local oscillator multiplexing is realized
frequencies of interest in non-destructive vibration
by the coherent addition of radiofrequency (RF) signals
screening are still much higher than camera frame rates,
driving an acousto-optic modulator in the reference
time-averaged interferograms [11–14] are recorded, even
beamchannel. Coherentfrequency-divisionmultiplexing
with high-speed cameras. Quantitative imaging of
is a technique by which the total available bandwidth
nanometer-scale optical path length modulations and
is divided into non-overlapping frequency sub-bands
phase retardation in stationary regimes, at the time
carrying separate signals. Addressing simultaneously
scale of the frame exposure, may thus find applications.
distinct domains ofthe spectralbandwidth ofholograms
was performed with success for angular [15], wavelength
In this letter, we report a heterodyne holographic [16], space-division [17], and side band [18] multiplexing
arrangement designed for amplitude and phase imag- schemes. Coherent frequency-division multiplexing in
ing of nanoscale out-of-plane sinusoidal vibrations in heterodyneholographyofferstheopportunitytomeasure
time-averagedrecording conditions without strobe light. complex-valued maps of the RF spectral components of
2
the optical radiation field with accuracy and sensitivity,
3mm
as long as they can be isolated in sub bands of the
sensor’s temporal bandwidth. In the reported exper-
iment, high-quality factor flexural responses at audio
frequencies of a lamellophone (the metallic comb of
cantileversfrom a musicalbox) excited in a sympathetic
resonance regime are sought. Sympathetic vibration
is a harmonic phenomenon wherein a vibratory body
responds to external excitation in the vicinity of a
resonance frequency.
(a) (b)
Theopticalsetupusedtoperformnon-destructivetest-
ing of the cantilevers’ out-of-plane flexural responses at
nanometric scales is sketched in fig.1. It is designed to
screen quantitatively steady-state vibration amplitudes
andtodetectmechanicalphasejumpswithrespecttothe
excitationsignal,by interferometricdetectionofanopti-
calfieldscatteredbythe lamellophone,referredtoasthe
objectfield,beatingagainstareferenceopticalfield. The
out-of-planemotionofthecantileversprovokesanoptical 10-1010-9 10-8 10-7 m
pathlengthmodulationofthebackscatteredobjectfield.
(c) (d)
HencethetemporalpartoftheobjectfieldEundergoesa
sinusoidalphase modulationφ(t)=4πz(t)/λ. The equa-
tionofthe out-of-planemotionz(t)ofa cantilever,mod-
eled as a damped harmonic oscillator of spring constant
k, mass m, damping coefficient c, driven sinusoidally by
the force F(t) = F sin(ωt) provided by a piezo-electric
0
actuator is
∂2z ∂z
m +c +kz =F(t) (1)
∂t2 ∂t −π 0 π −π 0 π
We seek a steady-state solution at the excitation fre-
(e) (f)
quency with an induced phase change of ψ with respect
to the excitation F(t), as a result of viscous damping
FIG. 2: Magnitude of a statically-scattered light hologram
|H˜0| (a). Magnitude of side band holograms |H˜−1| (b) and
z(t)=z sin(ωt+ψ) (2)
0 |H˜1| (c). Amplitude map of the flexural response z0 of the
first cantilever for the first resonance at ω/(2π) = 541Hz
Its amplitude, z , is proportional to the driving force
0 (d). Phase images ψ calculated from 128 raw interferograms
−1/2 around540.9Hz(e)and541.8 Hz(f),intheneighborhoodof
z = F0 (2ω ζ)2+ 1 ω2−ω2 2 (3) thefirst resonance. Moviesof theamplitudeandphasemaps
0 mω (cid:20) 0 ω2 0 (cid:21) of the out-of-planevibration are reported in media 1 and 2.
(cid:0) (cid:1)
Twointermediatevariableswereintroduced: theangular
resonance frequency ω of the undamped oscillator, and The quantity
0
the dimensionless damping ratio ζ. They satisfy the re-
E =EJ (φ )einψ (6)
lationships k =mω2 and c=2mω ζ. The phase shift of n n 0
0 0
the sinusoidal response with respect to the driving force is the complex weight of the optical side band of order
is n, where E is the complex amplitude of the optical field,
and φ = 4πz /λ is the modulation depth of the optical
2ωω ζ 0 0
0
ψ =arctan (4) phase of the radiation backscattered by the cantilever.
(cid:18)ω2−ω2(cid:19)
0 The localamplitude z of the out-of-plane vibrationand
0
phase retardation ψ with respect to the sinusoidal driv-
Theresonancefrequenciesω =ω 1−2ζ2 ofeachcan-
r 0
ing force can be derived from the complex weight of two
tilever are almost equal to the repsonant frequencies ω
0
optical side bands with eq. 6 and the first-order Tay-
of the undamped system. The optical phase modulation
lor developments of J (φ ) and J (φ ), near φ = 0, for
φ(t)=φ sin(ωt+ψ) of depth φ =4πz /λ at the angu- 0 0 1 0 0
0 0 0
z ≪λ
lar frequency ω, can be decomposed on a basis of Bessel 0
functions of the first kind J (φ ), via the Jacobi-Anger E 2π
n 0 1 ≈ z eiψ (7)
identity E λ 0
0
E =EeiωLteiφ(t) =EeiωLt J (φ )ein(ωt+ψ) (5) The holographic interferometer is used for the detec-
n 0
tion of RF phase modulation side bands of an object
Xn
3
where the quantities δω satisfy −ω /2 < δω < ω /2.
n S n S
1012 The FFT-spectrum of N consecutive off-axis holograms
u.) H˜ (ω )= N H(2πp/ω )exp(−2inpπ/N) (11)
a. jHà j jHà j jHà j n p=1 S
( 1 0 à1
)jn 1011 exhibitsthreePpeaksatthefrequenciesδω , δω ,andδω
! 1 2 3
àH( (fig. 3). Their complex values yield holograms H˜m, of
j
rank m = −1,0,1, whose magnitude is proportional the
1010 modulation depth at the frequency δω
-10 -5 0 5 10 m+2
Frequency ! = ( 2 ù ) (Hz)
n H˜ =H˜ (δω )=AE (12)
m m+2 m
FIG. 3: Magnitude of theFFT-spectrum (eq.11 and eq. 12),
where A∝E∗ α is a complex constant. Thus, H˜ is a
where three modulation side bandsare addressed. LO m m
measure of the complex weight E . For small-amplitude
m
vibrations, a quantitative map of the local out-of-plane
field E in reflective geometry, beating againsta LO field vibration amplitude z can be assessed from the ratio
0
E , as reported in [18]. Its purpose is to enable a ro- of the first-order sideband H˜ and the non-shifted light
LO 1
bust quantitative measurement of the nanometric mod- component H˜
0
ulation amplitudes of cantilever resonances, as well as
λ
the associated phase shift with respect to the sinusoidal z ≈ H˜ /H˜ (13)
0 1 0
excitation signal. To do so, the adopted strategy is to 2π (cid:16)(cid:12)D E(cid:12)(cid:17)
(cid:12) (cid:12)
separate the sensor bandwidth [−ωS/2,ωS/2[ into three where hi corresponds to sp(cid:12)atial aver(cid:12)aging over the ex-
non-overlapping frequency sub-bands, centered at δω1, tent of the resonant cantilever, arrowed in fig. 2(b). In
δω2, and δω3, each of which is used to carry a separate the same manner, a map of the local mechanical phase
opticalmodulationsideband. TheLOsignaliscomposed retardation ψ with respect to the excitation signal can
of the sum of three phase-locked RF signals, to yield an be assessed from H˜ , and H˜
1 0
optical LO field of the form
ψ−ψ =Arg H˜ /H˜ (14)
3 0 1 0
(cid:16)D E(cid:17)
E =E eiωLt α ei∆ωnt (8)
LO LO n
Relationships 13 and 14 are derived from eqs. 7, 12.
nX=1
One can also make use of the third recorded band
where ∆ωn = (n−2)ω−δωn is the frequency shift and H˜−1 and the property J−1(φ0) = −J1(φ0) to bene-
α E is the complex weight of the LO component of fit from a small SNR improvement, since we also have
n LO
rank n = 1,2,3. The positive parameters αn are the z0 ≈ 2λπ H˜−1/H˜0 andψ−ψ0 =Arg H˜0/H˜−1 .
normalized weights of each LO component; in this ex- (cid:16)(cid:12)D E(cid:12)(cid:17) (cid:16)D E(cid:17)
(cid:12) (cid:12)
periment α1 = α2 =α3 = 1/3. The LO components are To perf(cid:12)orm the me(cid:12)asurement, an optical LO consist-
tuned aroundthe opticalmodulation bands of order −1, ing of the combination of three frequency components
0, and +1. Precisely,the LO component of rank n beats shifted by ∆ω , ∆ω , and ∆ω (eq.8) was realized.
1 2 3
againstthe opticalmodulation bandofrankn−2atthe Those shifts were chosen to yield non-symmetrical
apparent frequency δωn, which is set within the camera (non-opposed) algebraic modulation frequencies δω1,
bandwidth. Imagerenderingfromtherawinterferograms δω ,andδω inthedetectorbandwidthinordertoavoid
2 3
I(t) = |E(t)+ELO(t)|2 impinging on the sensor array spuriousaliasingorcross-talkeffects. Todoso,wemade
involves a diffraction calculation performed with a nu- use of phase-locked RF signals generated by frequency-
mericalFresneltransform. Itsimplementationinoff-axis synthesizers at the carrierfrequency ω /(2π)=80MHz,
C
holographic conditions [19] yields complex-valued time- andω +δω′,ω+δω′,ω +δω′,wheretheshiftsusedto
C 1 2 C 3
averaged [11–14] holograms I(t) from which the off-axis generate the three RF bands were δω′/(2π)=−1.25Hz,
1
region δω′/(2π) = 6.25Hz, δω′/(2π) = −2.5Hz. As sketched
2 3
infig. 1,thosesignalsweremixedandcombinedtodrive
∗
H(t)=E(t)ELO(t) (9) a set of acousto-optic modulators designed to operate
around 80 MHz, from which opposite diffraction orders
is the support of the object-reference fields cross-term.
were selected. The resulting modulation frequencies of
Thedemodulationprocess(eq. 11)fromN ofconsecutive ′ ′ ′
the holograms(eq.12) were δω =δω +δω , δω =δω ,
recordings of finite exposure times acts as a band-pass ′ ′ 1 1 2 2 3
δω = δω − δω . The three side band components
filter of width ωS/N in the temporal frequency domain. H˜−31, H˜0, a1nd H˜12, encoded at 5 Hz, -2.5 Hz, and -7.5
Onlythecontributionswithinthesensortemporalband-
Hz respectively, are visible on the magnitude of the
width, between the Nyquist frequencies ±ω /2, have to
S discrete Fourier spectrum (eq. 12) reported in fig. 3.
be taken into account in H(t), which can hence be ex-
Spectral maps of the vibration amplitude z (ω) were
0
pressed as
calculated from a set of sequential measurements at
each frequency. We took 3000 sequences of N = 8 raw
3
∗ interferograms I at a readout rate of ω /(2π) = 20Hz,
H(t)= En−2ELOαnexp(iδωnt) (10) S
nX=1 for excitation frequencies ω/(2π) rangingfrom 0 Hz to 3
4
kHz, in 1 Hz steps (supply voltage of the piezo electric (a) m)
actuator : 10 mV). The first resonance frequencies of e (n100
the lamellophone’s cantilevers are observed between d 80
u
500 Hz and 2500 Hz (Media 1). In particular, a reso- plit 60
nanceofthefirstcantileverwasidentifiedaround541Hz. m
n a 40
o
Afinerscreeningaroundthisresonancewasperformed, ati 20
r
b
from 248 sequences of N = 8 raw interferograms, for Vi 0
536 538 540 542 544 546
excitationfrequenciesrangingfrom536Hzto546Hz(the
Frequency (Hz)
values of z (ω) are reported in fig.4, points). But ψ is
0 0
rthanedpohmasizeerdetfraormdaotinoenfψreqcuanenncoytpboeinretttroietvheed.nTexot.prHeevnencet (b) rad) ù2
reference phase randomizationbetweenmeasurements,a se (
a
h
chirp of the frequency was performed. It consisted of a p 0
n
linear drift with time t of the excitation frequency from o
ωI/(2π)=536Hz to ωF/(2π)=546Hz in T =99.2s bratiàù
Vi
2
ω(t)=ω +(ω −ω )t/T (15)
I F I 536 538 540 542 544 546
Frequency (Hz)
Two out of the three components of the optical LO
(eq. 8) were chirped as well via ω(t) throughout the FIG.4: Vibrationamplitudez0 (a)andphaseψ (b)averaged
acquisition. During the time lapse T, a sequence of overthefirstcantilever,arrowedinfig. 2(b),versusexcitation
frequencyω/(2π). Insets: retrievedvibrationamplitudeand
248 × 8 = 1984 raw interferograms I was steadily
phase maps in the neighborhood of the resonance, reported
recorded. Amplitude z (ω) and phase ψ(ω) maps at
0 in figs. 2(b,e,f). The points were obtained from a sequential
eachfrequencyω werecalculatedfromthe demodulation
measurement, the lines were obtained with the linear chirp
of N =8 raw frame packets, reported in media 2. These (eq. 15). The theoretical resonance lines in gray were deter-
maps were coherently averaged over the first cantilever mined from eq3and eq.4.
according to eq. 13 and eq. 14 and reported in fig. 4
(blue lines). In agreement with eq.4, for non-zero vis-
cosity, a phase shift of ∼ π is observed at the resonance
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r
tricconstantsusinganopticalheterodyneinterferometer.
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