September 15, 1995 / Vol. 20, No. 18 / OPTICS LETTERS
1835
Splitting of high-Q Mie modes induced by light backscattering
in silica microspheres
D. S. Weiss,* V. Sandoghdar, J. Hare, V. Lef èvre-Seguin, J.-M. Raimond, and S. Haroche
Laboratoire Kastler-Brossel, Ecole Normale Supérieure, 24 rue Lhomond, F-75231 Paris Cedex 05, France
Received May 3, 1995
We have observed that very high-Q Mie resonances in silica microspheres are split into doublets. This splitting
is attributed to internal backscattering that couples the two degenerate whispering-gallery modes propagating
in opposite directions along the sphere equator. We have studied this doublet structure by high-resolution
spectroscopy. Time-decay measurements have also been performed and show a beat note corresponding to
the coupling rate between the clockwise and counterclockwise modes. A simple model of coupled oscillators
describes our data well, and the backscattering eff iciency that we measure is consistent with what is observed
in optical f ibers.  1995 Optical Society of America
High-Q Mie resonances in spherical dielectric resonators, so-called whispering-gallery (WG) modes,1 have
led to a wide range of activity in optics, starting with
experiments on microdroplets by Ashkin and Dziedzic2
in the 1970’s. More recently, the combination of the
strong conf inement of the electromagnetic field in
these modes and their long lifetime has attracted new
interest in view of cavity quantum electrodynamics experiments dealing with a few tightly coupled atoms and
photons.3 Along these lines, we have proposed and
thoroughly analyzed an experiment that leads to the
quantized def lection of an atomic beam grazing a WG
mode.4 Highly confined and long-lived photon modes
are also ideal for producing low-threshold laser effects
in microspheres.5,6 Finally, high-Q WG modes may be
used to realize quantum nondemolition measurements,
as pointed out by Braginsky et al.7 We have observed
that, for Q values above a few 108 , each resonance
is generally split into a doublet. We show here that
this splitting is due to backscattering inside the sphere
that couples the clockwise (CW) and counterclockwise
(CCW) WG modes and lifts their degeneracy. A similar line-splitting effect was observed in a fiber-ring
resonator in which large amounts of backscattering
were artif icially induced.8 In our microresonators a
backscattering eff iciency of 10210 per round trip is
enough to split the resonances. The sensitivity of our
experimental setup has allowed us to investigate fully
this effect, which was only brief ly mentioned in Ref. 9.
Our experimental setup is shown in Fig. 1. It is
an improved version of the apparatus described in
Ref. 10. We now operate in a vacuum-tight chamber where microspheres are produced and studied.
We obtain reproducible silica resonators, 40–300 mm
in diameter, by melting the end of a fiber with two
counterpropagating CO 2 laser beams. At a pressure
of approximately 2 3 1026 mbars, spheres can be
kept for several weeks without degradation. Efficient coupling to high-Q strongly confined WG modes
is achieved with a high-index prism, through frustrated total internal ref lection. The beam of a diode
laser stabilized on a grating and a Fabry-Perot cavity (linewidth 50 kHz) is coupled into the sphere by
tunneling of the evanescent field across the piezo0146-9592/95/181835-03$6.00/0
controlled gap between the prism and the sphere (resolution 2 nm). The power coupled into a mode is in the
100-nW range at critical coupling. We detect resonances at the exit of the prism with a phase-modulation
technique at 40 MHz, using an electro-optic crystal
(EO) and a lock-in amplifier.11 When the laser frequency is scanned, each resonant WG mode gives rise
to three lines 40 MHz apart, providing a convenient
frequency scale. For Q values above a few 108 , each
line generally splits into a doublet. This is shown for
one side component in Fig. 2(a); the splitting is Dn ­
1 MHz and the width (FWHM) is w ­ 270 kHz, corresponding to a quality factor Q ­ 1.4 3 109 . These
measurements have been repeated on many different
modes and spheres. Although the measured Q’s are
always about the same for highly conf ined WG modes,
their splittings range from 300 kHz up to a few megahertz. They are intrinsic properties of the modes, independent of the laser power.
WG modes are quasi-bound states of light with high
angular momentum numbers l and m and low radial number n. In perfect spheres, resonance frequencies do not depend on m. However, our resonators
have a typical 1023 eccentricity, which lifts the degeneracy between n and l modes with different jmj
Fig. 1. Schematic of the experiment: PZT, piezoelectric
transducer; BS, beam splitter; PMT, photomultiplier tube.
 1995 Optical Society of America
1836
OPTICS LETTERS / Vol. 20, No. 18 / September 15, 1995
Fig. 2. Doublet resonances recorded at the same time
(a) in the forward direction, by the phase-modulation
technique, and ( b) in the backward direction. They lie at
the same frequencies and show the same splitting Dn ­
1 MHz and the same width w ­ 270 kHz. The forward
signal is proportional to the field amplitude, whereas the
backward signal is proportional to the CCW field intensity.
Fits calculated with our coupled-oscillators model are
superimposed onto the signals. The small amount of
backward light, ref lected from the prism output face and
coupled into the sphere (approximately 1%), explains the
imbalance of the two peaks.
values12 and leads to separations of approximately
500 MHz between adjacent modes much larger than
the splittings discussed here. Therefore these systematic closely spaced doublets cannot originate from this
shape effect.
It is usually assumed that the symmetry attached
to CW and CCW propagation and represented by the
6m Kramers degeneracy is satisfied in microspheres
or microdroplets. However, interaction between these
modes and subsequent splitting of their eigenfrequencies can be caused by internal backscattering. We assume here that, as in the case of silica optical fibers,13
backscattering is due to the presence of a collection of
independent microscopic dipoles much smaller than l
(defects of the silica network or residual impurities, for
instance). These Rayleigh scatterers naturally give
rise to a coupling between the two cavity modes under consideration, since they radiate equally backward
and forward. Whereas the power radiated in other
directions is lost outside the sphere, the backward
response to an incoming field builds up a counterpropagating field by a multiple interference effect, which is
nothing other than the buildup of a mode in a lowloss cavity. Solving Maxwell’s equations in the slowly
varying envelope approximation, we easily derive the
coupling rate V between the CW and CCW fields. It
is related to the average linear polarizability a of the
Rayleigh scatterers and to their number density rsc
in the mode volume V by sVyvd ­ s1y2dsrsc a 2 yVd1/2 ,
where v is the angular frequency of the modes. This
coupling rate increases as the square root of the total
number of dipoles in the mode, because we have assumed randomly distributed scatterers. In this linear
model, V is independent of the field intensity. Nonlinearities, such as the Kerr effect, could conceivably
couple the two modes in an intensity-dependent way.
However, such effects are clearly ruled out because
we observe no change in the mode splittings with a
2-order-of-magnitude change in intensity.
In this coupled-oscillators model, the new eigenmodes are superpositions of the CW and CCW fields,
and their eigenfrequencies differ by Vyp. In turn,
the CW and CCW fields are the symmetric and
antisymmetric superpositions, respectively, of the
new eigenmodes. Therefore, when it is not directly
excited, the counterpropagating field can reach a
sizable value only if the loss rate vyQ is small compared with the exchange rate V, i.e., if the width of
the resonance is smaller than the doublet splitting,
which is in itself independent of the cavity finesse.
The Rayleigh scatterers assumed in this model also
radiate power outside the modes; this loss can be
described by the Rayleigh extinction coeff icient
as ­ s8p 3 y3dsrsc a 2 yl4 d. We calculate that this contributes to less than 10% of the observed Q’s. To
account for the measured Q values, we incorporate
into our model an imaginary part of the bulk material
refractive index.
To test our coupled-oscillators model, we set up a
beam splitter and a photomultiplier tube in order to
detect backscattered light (see Fig. 1). When tuning
the laser frequency, we observed resonant doublets
on the backward light, at the same frequencies and
with the same widths as in the forward direction (see
Fig. 2). The imbalance of the two peaks can be explained simply by a small amount of parasitic light
ref lected back from the exit face of the prism that
coupled directly to CCW modes. This changes the balance between the two modes and slightly distorts the
shape of one of the resonant peaks. We observed that
the backward resonant signal progressively disappears
when the gap between the prism and the sphere is
made smaller. This occurs because coupling to the
prism broadens the resonance until its width becomes
larger than the splitting. This is well described by our
model, which requires a large splitting-to-width ratio
for a sizable CCW field to be built up.
We have also performed time-decay measurements
on our system. We use the lock-in amplifier dispersive signal to lock the diode laser, so that one sideband
of the excitation field (approximately 5% of the total
intensity) is on resonance. When the rf signal feeding the electro-optic phase modulator is switched off,
the output light in the forward direction is the sum of
the ref lected excitation field (at the carrier frequency)
and the field scattered by the sphere into the prism.
The decay of the field amplitude in the sphere is then
recovered by demodulation of the photodiode signal
at 40 MHz. Figure 3 shows such a signal averaged
over 20,000 runs. The laser sideband was locked at
the center of the doublet to excite the two eigenmodes
symmetrically. As expected, one observes a damped
oscillation. The beat-note period T ­ 0.92 ms is related to the doublet splitting by the predicted relation T ­ 2yDn. To get enough signal, we reduced the
sphere -prism gap so that the time decay of the field
amplitude was only 2t ­ 0.51 ms, where t is the photon
lifetime. When the Fourier transform of the damped
beat-note signal is performed, it nicely matches the
doublet observed by continuous spectroscopy, as shown
in the inset of Fig. 3. Locking the laser sideband on
September 15, 1995 / Vol. 20, No. 18 / OPTICS LETTERS
1837
cally lock a diode laser to a narrow sphere resonance
and reduce its spectral width. We plan to use this
method to obtain a compact and robust device with a
fiber coupler that we recently developed.14
We acknowledge the financial support of a France
Telecom CNET -CNRS contract (93.1B.180). Laboratoire Kastler-Brossel is a Unité de Recherche de l’Ecole
Normale Supérieure et de l’Université Paris 6, associée
au Centre National de la Recherche Scientifique.
*Present address, Department of Physics, University
of California, Berkeley, Berkeley, California 94720.
References
Fig. 3. Time-decay signal with the laser sideband locked
at the center of the doublet shown in the inset. The
two eigenmodes are then equally excited and give rise to
a beat note of period T ­ 0.92 ms according to the fit
shown by the smooth solid curve. To get enough signal,
we decreased the prism– sphere gap, and the decay time
was only 2t ­ 0.51 ms. The measured beat-note period
and the decay time agree well with the data obtained by
continuous spectroscopy. This is clearly seen in the inset,
where the Fourier transform of the temporal signal (dashed
curve) is superimposed onto the experimental doublet.
one peak of a doublet gives a pure exponential decay
with a time constant in good agreement with the resonance width.
All our experimental observations are well explained
by our model for backscattering. Assuming mode volumes of the order of 1000 mm3 , the range of splittings measured in our experiments corresponds to
a backscattered intensity of approximately 10210 per
round trip. This is consistent with the backscattering efficiency and the Rayleigh attenuation of 2 dBykm
measured in silica optical fibers.13 In both cases
this phenomenological approach gives access only to
the average s rsc a 2 d and does not allow us to identify the nature of the scatterers. The mode splitting reported here results from the very features
of microspheres that make them currently of interest, that is, strong confinement of light in long-lived
modes. Among the possible applications of high-Q microspheres, the backscattered field studied here can
provide an elegant feedback source with which to opti-
1. S. C. Hill and R. E. Benner, in Optical Effects Associated With Small Particles, P. W. Barber and R. K.
Chang, eds. (World Scientific, Singapore, 1988).
2. A. Ashkin and J. M. Dziedzic, Phys. Rev. Lett. 38, 1351
(1977).
3. S. Haroche in Fundamental Systems in Quantum
Optics, Les Houches Summer School, Session LII1,
1990, J. Dalibard, J. M. Raimond, and J. Zinn-Justin,
eds. (North-Holland, Amsterdam, 1992), pp. 769 – 940.
4. F. Treussart, J. Hare, L. Collot, V. Lef èvre, D. S. Weiss,
V. Sandoghdar, J. M. Raimond, and S. Haroche, Opt.
Lett. 19, 1651 (1994).
5. G. Chen, M. M. Mazumder, Y. R. Chemla, A. Serpengüzel, R. K. Chang, and S. C. Hill, Opt. Lett. 18,
1993 (1993).
6. B. Lü, Y. Wang, Y. Li, and Y. Liu, Opt. Commun. 108,
13 (1994).
7. V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko,
Phys. Lett. A 137, 393 (1989).
8. R. J. C. Spreeuw, J. P. Woerdman, and D. Lenstra,
Phys. Rev. Lett. 61, 318 (1988).
9. V. S. Ilchenko and M. L. Gorodetsky, Laser Phys. 2,
1004 (1992).
10. L. Collot, V. Lef èvre-Seguin, M. Brune, J. M. Raimond,
and S. Haroche, Europhys. Lett. 23, 327 (1993).
11. G. C. Bjorklund, Opt. Lett. 5, 15 (1980).
12. H. M. Lai, C. C. Lam, P. T. Leung, and K. Young, J.
Opt. Soc. Am. B 8, 1962 (1991).
13. S.-C. Lin and T. G. Giallorenzi, Appl. Opt. 18, 915
(1979).
14. N. Dubreuil, J. C. Knight, D. K. Leventhal, V.
Sandoghdar, J. Hare, and V. Lef èvre, Opt. Lett. 20,
813 (1995).