Direct observation of stopped light in a whispering-gallery-mode microresonator
A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Strekalov, and L. Maleki
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California 91109-8099, USA
Received 7 August 2006; published 17 August 2007
We introduce the definition of group velocity for a system with a discrete spectrum and apply it to a linear
resonator. We show that a positive, negative, or zero group velocity can be obtained for light propagating in the
whispering-gallery modes of a microspherical resonator. The associated group delay is practically independent
of the ring-down time of the resonator. We demonstrate “stopped light” in an experiment with a fused silica
microsphere.
DOI: 10.1103/PhysRevA.76.023816 PACS numbers: 42.60.Da, 42.25.Bs, 42.25.Gy, 42.25.Hz
The group velocity is usually introduced in a class of
problems where responsivity of a material can be considered
as a continuous function of frequency. For instance, prob-
lems of propagation of narrowband light in systems like
atomic media belong to this class 1,2. However, for the
broad class of systems with discrete spectra, the common
definition of the group velocity sometimes is not valid. An
optical resonator is an example of such a system. We show
here that the discreteness of the spectrum brings different
features to the notion of the group velocity defined as the
velocity of a train of optical pulses; namely, such a train can
be delayed by a linear resonator for much longer than the
ring-down time of the resonator. Such a delay is impossible
for a single pulse interacting with a linear lossless resonator,
even though linear resonators as well as their chains can
introduce a significant group delay 3–6.
This peculiarity arises when a conceptual transition is
made from the framework of a distributed resonator to that
of a lumped resonator. While distributed resonators possess
an infinite number of modes, only a finite number of modes
are usually considered for their spectral studies, and often
only a single mode is retained for the sake of simplicity.
Such an approximation silently transforms the distributed ob-
ject to a lumped one, discarding multiple phenomena, one of
which is the subject of our study.
Propagation of slow light in a dispersive medium can be
characterized by the propagation of a beat note envelope of
two plain monochromatic electromagnetic waves E
1
= E
˜
exp-i
1
t + ik
1
z + c.c. and E
2
= E
˜
exp-i
2
t + ik
2
z + c.c.
in the medium 2. The beat note of the waves is described
by |E
1
+ E
2
|
2
=2|E
˜
|
2
1+cos
1
-
2
t - k
1
- k
2
z. The veloc-
ity of its propagation, V
g
=
1
-
2
/ k
1
- k
2
, corresponds to
the conventional definition of the group velocity / k if
1
→
2
and the wave vector k is a continuous function of
frequency .
Let us consider a lumped model of a resonator with a
transfer function
H =
+ i -
0
- i -
0
, 1
where is the full width at half maximum and
0
is the
frequency of the resonance. A monochromatic signal with
frequency acquires a phase shift argH when passing
through the resonator. The resonator delays the beat note of
waves E
1
and E
2
by an amount of time
g
1
,
2
2/ . The
maximum delay is achieved for
1
→
2
, and, as a general
rule,
g
1
,
2
1
-
2
2. As a result of the above con-
ditions, there is a general belief that the group delay intro-
duced by a linear resonator cannot exceed the ring-down
time of the resonator.
In what follows we show that this conclusion is not valid
for a distributed resonator such as a microsphere resonator
that supports whispering-gallery modes WGMs. The group
delay in such a resonator can exceed its ring-down time sig-
nificantly. This happens because a WGM resonator belongs
to the class of systems with a discrete optical spectrum. Such
a resonator can be described using a lumped model within
each spectral line. However, the model is not valid if the
light interacts with multiple modes. The usual definition of
the group velocity / k does not hold in this case. For
example, the expression V
g
=
1
-
2
/ k
1
- k
2
is the only
correct definition of the group velocity for a bichromatic
field. A similar method should be applied to describe the
propagation of a generalized optical field that has a discrete
spectrum, e.g., a train of pulses, in a distributed resonator.
The spectrum of the field consists of a series of arbitrarily
narrowband for an arbitrarily long train lines enveloped by
the Fourier transform of an individual pulse. The number of
“significant” spectral lines that are not too strongly sup-
pressed by the envelope is given by essentially the duty cycle
of the pulse train. This number may be just a few for a dense
series of smooth e.g., Gaussian pulses. The group velocity
of the train can be extremely small if the spectrum of the
resonator with which the pulses interact is properly engi-
neered.
Let us now turn to a more formal discussion. We present
the electric field inside the microsphere resonator as
E = e
-it
+ c.c., 2
where the spatial field distribution has the general form
=
¯
P
l
m
cos J
l+1/2
k
l,q
re
im
/
r . 3
, , r are the spherical coordinates; the indices l, m, and q
determine the spatial distribution of the field, m =0,1,2,...
and q = 1 , 2 , . . . are the azimuthal and radial quantization
numbers, respectively, and l =0,1,2,... is the orbital mode
number. , , r is the mode spatial profile, and
PHYSICAL REVIEW A 76, 023816 2007
1050-2947/2007/762/0238164 ©2007 The American Physical Society 023816-1