Vibrations of microspheres probed
with ultrashort optical pulses
T. Dehoux,1T. A. Kelf,1M. Tomoda,1O. Matsuda,1O. B. Wright,1,* K. Ueno,2
Y. Nishijima,2S. Juodkazis,2H. Misawa,2V. Tournat,3and V. E. Gusev4
1Department of Applied Physics, Faculty of Engineering, Hokkaido University, Sapporo 060-8628, Japan
2Research Institute for Electronic Science, Hokkaido University, Sapporo 001-0021, Japan
3LAUM, CNRS, Université du Maine, F-72085 Le Mans, France
4Laboratoire de Physique de l’Etat Condensé, UMR CNRS 6087, Le Mans 72085, France
* Corresponding author: firstname.lastname@example.org
Received July 30, 2009; accepted October 14, 2009;
posted November 9, 2009 (Doc. ID 114981); published November 30, 2009
We use ultrashort optical pulses to excite and detect vibrations of single silica spheres with a diameter of
5 ?m placed at the surface of an acoustically mismatched substrate. In addition to the photoelastic detection
of picosecond longitudinal acoustic pulses propagating inside the bulk, we detect gigahertz acoustic reso-
nances of the sphere through probe beam defocusing. The mode frequencies are in close accord with those
calculated from the elastic vibrations of a free sphere. We also record a resonant enhancement in the am-
plitude of specific modes of two touching spheres. © 2009 Optical Society of America
OCIS codes: 320.5390, 110.5125, 110.7170.
Understanding vibrations in microscale systems is
important given the trends for the miniaturization of
mechanical resonators and actuators, with applica-
tions in filtering and mass sensing [1,2]. Having typi-
cal resonant frequencies in the gigahertz range, the
vibrations of microstructures can be conveniently
probed in the time domain using ultrashort optical
pulses for noncontact excitation and detection [3,4].
These experiments have been carried out in one- and
two-dimensional periodic microstructures. Single mi-
crostructures such as isolated spheres , disks ,
or cantilevers  are of particular interest for sens-
ing applications, because their vibrational spectra
are well known. Experiments on such structures
have generally been conducted in the frequency do-
main. It is, however, very useful to carry out mea-
surements in the time domain, because phase infor-
mation is easier to access and frequency domain data
are available by Fourier transforming. Despite the
attention paid to the vibrations of uniform isotropic
spheres—a useful model system for vibrational
analysis—from centimeter down to nanometer di-
mensions [5–10], there have been no time domain
studies of the vibrations of isolated spheres with a di-
ameter of ?1 ?m. In this Letter we present such
measurements on single microscopic silica spheres
using ultrashort laser pulse excitation and detection.
We also examine a system of two spheres in contact.
The samples are prepared by exploiting the adhe-
sion of silica spheres (medium 2) of diameter D
=5±0.05 ?m to a poly-dimethylsiloxane (PDMS,
medium 3) coated silicon substrate. During the con-
tacting process, the spheres penetrate the PDMS to a
depth of ?=0.34 ?m, measured by scanning electron
microscopy (SEM). A single-sphere site and a SEM
image are shown in Fig. 1(a). The small ?, combined
with the large mismatch between the acoustic imped-
ances of the PDMS  Z3=?3vL3=1.1 MPa s m−1
and that of silica  Z2=?2vL2=13 MPa s m−1(lon-
gitudinal velocities vL3=1.0 and vL2=6.0 km/s; densi-
ties ?3=1.1 and ?2=2.2 g/cm3), provides vibrational
Q factors ?10. (Q=11 is found for a Si particle in wa-
ter; see .) The low Young’s modulus of the PDMS,
?0.7 MPa , ensures ?1% shift from the free-
sphere resonance frequencies . Finally a metal film
(medium 1, 80% Pt, 20% Pd) with a thickness of
?160 nm is sputtered on the sample for photoelastic
transduction. We determine the stiffness coefficients
and the density ?1=19.6 g/cm3of the layer with an
effective medium theory . Using the values for Pt
and Pd in , the effective stiffness coefficients are
C11=345, C12=258, and C44=75.5 GPa. We calculate
the longitudinal sound velocity vL1=4.3 km/s using
We use the optical pump-probe technique shown
schematically in Fig. 1(a). Pump pulses from a
Ti:Sapphire mode-locked laser with a wavelength of
810 nm, a pulse duration of 200 fs, and a repetition
frequency of 80 MHz excite acoustic vibrations. This
light is chopped at 1.1 MHz for lock-in detection.
Each pump pulse, of energy E=25 pJ, is initially ab-
sorbed in the metal layer over a depth comparable to
the optical skin depth of 13 nm. Conduction band
electrons are excited and diffuse over a depth of
???/g?1/2=8 nm during their thermalization with the
lattice, where ?=72 W m−1K−1is the thermal con-
mirror; PBS, polarizing beam splitter). Inset, SEM. (b) Nor-
malized angular frequency ?D/2vT2versus l.
(Color online) (a) Sample and setup (DM, dichroic
OPTICS LETTERS / Vol. 34, No. 23 / December 1, 2009
0146-9592/09/233740-3/$15.00 © 2009 Optical Society of America
ductivity  and g=11?1017W m−3K−1is the
electron-phonon coupling constant in Pt . The
subsequent laser-induced heating yields a transient
temperature rise of ?90 K, producing a thermoelas-
tic expansion that couples to longitudinal acoustic
pulses and sphere vibrations, and a maximum steady
state temperature rise of ?2 K (avoiding damage to
the PDMS substrate). The reflectivity change ?R?t? is
measured with frequency-doubled circularly polar-
ized probe pulses with a wavelength of 405 nm and
energy of 3 pJ as a function of the pump-probe time
delay t using a balanced photodiode. (The unmodu-
lated acoustic oscillations excited by the probe pulse
are not detected by the lock-in detection system.) The
delay is achieved by multiple passes through a mo-
torized mechanical delay line. The coaxial pump and
probe beams are focused onto the top of a sphere at
normal incidence through a 100? objective lens with
an NA of 0.8 to Gaussian spots of diameters w1
?1 ?m and w2?0.5 ?m FWHM, respectively. As-
suming that the incident probe beam waist is exactly
at the surface of the sphere, the reflected Gaussian
probe beam has a radius of curvature of D/4 at this
position. We measured a 60% clipping of this diver-
gent reflected beam by the finite apertures of the ob-
We plot the amplitude ?R?t?/R0in Fig. 2(a) for two
single-sphere sites, where R0=0.6 is the probe reflec-
tivity, demonstrating a reproducible response. Ul-
trafast electron diffusion produces a sharp spike at
t=0.Alongitudinal acoustic strain pulse of center fre-
quency fp=35 GHz propagates in the metal film and
reflects off the film/sphere interface. This pulse pro-
duces a first series of echoes [see the inset of Fig.
2(a)]. The amplitude of successive echoes, of interval
?t1=76 ps, decreases in time owing to attenuation in
the film and strain transmission to the sphere with a
coefficient of 2Z2/??1vL1+Z2?=0.28. We determine the
film thickness from vL1?t1/2=160 nm (ignoring small
corrections from Gouy phase shifts). The tri-polar
shape of the echoes is characteristic of the strain de-
tection through the photoelastic effect in the metal
film . The transmitted strain pulse reflects off the
SiO2/PDMS interface and propagates back to the
metal layer, where it produces a second series of ech-
oes [downward arrow in Fig. 2(a)]. The interval be-
tween the two series of echoes, ?t2=1.88 ns, yields a
reasonable value for the longitudinal sound velocity
in fused silica, vL2=2D/?t2=5.3±0.3 km/s .
?R?t? shows oscillations from the sphere vibration,
superimposed on a slow thermal relaxation. These
oscillations are long-lived as is evident from their
presence at t?0 owing to previous pump pulses. We
plot in Fig. 2(b) the amplitudes and phases from tem-
poral Fourier analysis, with the 11 ns window giving
a 94 MHz frequency resolution. Six resonances [see
arrows in Fig. 2(b)] are detected at frequencies of f1
=0.57, f2=0.73, f3=0.93, f4=1.12, f5=1.30, and f6
=1.58 GHz. Reproducible phase data are obtained as
shown in the inset of Fig. 2(b). The phase leads of the
resonances f1and f4are −?100±25?° and −?155±15?°
with respect to a cosine, respectively.
To interpret these results, consider the vibration of
a free homogeneous isotropic sphere . Owing to
the axial symmetry of the excitation, only spheroidal
modes of azimuthal mode number m=0 are gener-
ated. Modes associated with bodily motion of the
whole sphere such as bouncing are too low in fre-
quency to be detected with the present set-up .
Using the classical elastic dispersion equation for
spheroidal modes , we determine the angular fre-
lar mode numbers, respectively. Knowledge of the
longitudinal and shear sound velocities vL2and vT2in
silica is not required for comparison with the data. It
is sufficient to use D/vL2=?t2/2 and D/vT2=??1
−2??/2?1−???1/2D/vL2, where ?=0.17 is the Poisson’s
ratio in silica . We plot ?l
1(b) for both theory and experiment. As m=0, l de-
fines the number of nodes on the surface of the
sphere along a fixed-azimuth-? perimeter ?D. So one
can define the acoustic wavelength to be ?=?D/l in
the high frequency limit and consider ?l
sus l as the dispersion curve for surface waves. Fre-
quencies associated with Rayleigh or whispering gal-
lery modes (WGs) are connected with dotted lines
Assuming that ?R arises solely from the photoelas-
tic effect, the results of Fig. 2(a) imply that the am-
plitudes ?1and ?2of the longitudinal strain in the in-
depth direction due to the acoustic pulse of center
frequency fpand to the sphere vibration, respectively,
are of similar order: ?2/?1?1. The dispersion curves
indicate that the measured vibrational frequencies
can only correspond to modes n?2 for which the sur-
face displacement is u2??2D. (The radial displace-
ment varies on a length scale ?D for sufficiently
small l mode numbers; see .) So the ratio of the
surface displacement u2to that due to the acoustic
n, with n and l being the radial and angu-
nD/2vT2versus l in Fig.
spheres. Downward arrow, second set of acoustic pulses. In-
set, zoom-in on the first set of acoustic pulses (upward ar-
rows). (b) Amplitude and phase of Fourier spectra related
to (a). Arrows, observed resonances. (c) ?R/R0for two dif-
ferent paired spheres (displaced). (b) Amplitude of Fourier
spectra related to (c).
(Color online) (a) ?R/R0for two different single
December 1, 2009 / Vol. 34, No. 23 / OPTICS LETTERS
pulse u1??1vL1/2fp is u2/u1?2Dfp/vL1?102. This Download full-text
high ratio suggests that a nonnegligible surface dis-
placement should contribute to ?R from probe beam
defocusing . Even though we did not introduce an
iris in our setup to maximize this effect, the finite ap-
ertures in the 100? objective act as one. If we as-
sume that the lowest-order modes n=0 yield the
highest surface displacement, they are likely to pro-
duce the highest amplitudes in ?R. So we identify the
measured frequencies f1–f6as the following ?n,l?
modes: (0,2), (0,1), (0,3), (0,4), (0,5), and (0,6). We
compare the measured normalized frequencies with
the theory in Fig. 1(b). The good agreement suggests
that we are indeed detecting n=0 modes. The acous-
tic wavelength ??2w1is limited by the pump spot di-
ameter w1, so only modes with l??D/2w1?8 are de-
As a further experiment we pumped and probed on
the top of one of the spheres of a contacting pair.
?R?t?/R0and the corresponding Fourier spectra for
two different such sites are shown in Figs. 2(c) and
2(d), respectively. The amplitudes of modes (0,2) and
(0,4) (of frequencies f1and f4, respectively) are par-
ticularly enhanced compared to the single-sphere
case, and their phase leads become −?75±25?° and
To elucidate this behavior, we solve the equation of
motion for a free isotropic sphere and calculate the
radial displacement ul
cal coordinates ?r,?,??,
nof each mode ?n,l? in spheri-
r?Pl?cos ??, ?1?
n?r,?? = Al
+ l?l + 1?
spherical Bessel function of the first kind jldescribes
the r dependence of ul
fixed-azimuth-? perimeter defined by the Legendre
polynomial Pl?cos ??. We illustrate in Fig. 3 the cal-
culated deformation of the surface of the sphere,
D/2+?P4?cos ??, subjected to a radial displacement of
arbitrary amplitude ? for mode (0,4). From the defi-
nition of Plwe see that for modes of even l—notably
modes (0,2) and (0,4)—the displacement along the
equatorial perimeter ??=?/2? is nonzero, and the
contact between the two spheres is subject to an os-
cillating stress that should couple their resonances. A
n/vT2are the longitudinal
n. Each mode ?n,l? of amplitude
nproduces a radial surface displacement along a
resulting shift in the frequencies  of modes (0,2)
and (0,4) closer to a harmonic of the laser repetition
rate could explain the enhancements observed . A
detailed analysis should allow the phase and relative
amplitudes of the modes to be extracted.
In conclusion, we have demonstrated for the first
time, to our knowledge, that—in spite of the high sur-
face curvature—the acoustic modes of individual
micrometer-sized spheres can be efficiently generated
and detected using focused ultrashort laser pulses.
The measurement of the mode frequencies should
prove to be a useful tool for the quality control of the
sphere elastic properties. The measured phase of the
vibrational modes, when combined with a solution of
the elastic wave equation that accounts for the peri-
odic pulsed laser excitation and the spatiotemporal
thermoelastic source, should allow the investigation
of the vibration generation mechanism and elucidate
the dynamics of the mechanical coupling between
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(Color online) Left, radial deformation of the
OPTICS LETTERS / Vol. 34, No. 23 / December 1, 2009