Stopping and time reversing a light pulse using
dynamic loss tuning of coupled-resonator
Sunil Sandhu,* M. L. Povinelli, and Shanhui Fan
Department of Electrical Engineering, Ginzton Laboratory, Stanford University, Stanford, California 94305, USA
*Corresponding author: email@example.com
Received July 23, 2007; revised October 15, 2007; accepted October 18, 2007;
posted October 19, 2007 (Doc. ID 85496); published November 12, 2007
We introduce a light-stopping process that uses dynamic loss tuning in coupled-resonator delay lines. We
demonstrate via numerical simulations that increasing the loss of selected resonators traps light in a zero
group velocity mode concentrated in the low-loss portions of the delay line. The large dynamic range achiev-
able for loss modulation should increase the light-stopping bandwidth relative to previous approaches based
on refractive index tuning. © 2007 Optical Society of America
OCIS codes: 130.3120, 200.4740, 210.0210, 230.0230.
Systems that dynamically manipulate the group ve-
locity of light are of great interest in classical and
quantum information processing [1,2]. For this pur-
pose, dynamically tuned delay lines based on cas-
caded optical resonators have been well studied. In
these systems, it was shown theoretically that by
compressing the system bandwidth to zero, a light
pulse can be stopped [3–8].
Dynamically tuned delay lines generally rely on re-
fractive index tuning to achieve the required pulse
delay. However, the input pulse bandwidth is limited
by the tuning range of the refractive index. The oper-
ating bandwidth typically scales as ?n/n . For the
available index modulation strength of ?n/n?10−4
achievable bandwidth is on the order of 20–200 GHz
for ?=1.5 ?m light. In contrast, the strength of loss
modulation in semiconductors can be much larger. In
this Letter, we introduce a light-stopping scheme
based on loss tuning of coupled-resonator delay lines.
The process should increase the allowable input
pulse bandwidth as compared to index modulation
The basic requirements of a light-stopping process
are that the system supports a large-bandwidth state
to accommodate the input pulse bandwidth, which is
then tuned to a narrow-bandwidth state to stop the
pulse . In the following discussion, we give a gen-
eral description of our proposed light-stopping pro-
cess using loss modulation in the system shown in
Fig. 1. The low-loss resonator in the unit cell has a
static loss rate of ?0, while the lossy resonator has a
dynamic loss rate of ??t?. The coupling rate between
neighboring resonators is ?. In the initial state, one
sets ??t?=?0; the system therefore consists of a chain
of essentially lossless resonators ??0??? with cou-
pling rate ? large enough to accommodate the input
pulse bandwidth. Once the pulse is within the sys-
tem, the loss rate ??t? in every other resonator is
tuned to ?stop??. As a result, the system is brought
state. Inthis narrow-
bandwidth state, the low-loss resonators are essen-
tially decoupled from the lossy resonators, and the
signal energy is trapped in the low-loss resonators. In
this light stopping process, as long as one can get a
large dynamic range of loss modulation, one can
choose a large ? and, hence, accommodate a large
pulse bandwidth in the initial state.
During the tuning process, the pulse energy loss
peaks near ??t?=?, which defines a critical-damping
point. To minimize the overall loss that the pulse ex-
periences, the system should spend as little time as
possible in the vicinity of the critical-damping point
[i.e., the dynamic tuning of ??t? should be performed
as rapidly as possible]. Hence, dynamic tuning of the
system to the narrow-bandwidth state is performed
nonadiabatically. This is in contrast to the adiabatic
resonator light-stopping system. The unit cell has length L
and consists of a lossy resonator with dynamic loss ??t? and
a low-loss resonator with static loss ?0. The bottom plot
shows the modulation performed on ??t?, with the dotted
line indicating the critical-damping point.
Top diagram shows a section of the coupled-
November 15, 2007 / Vol. 32, No. 22 / OPTICS LETTERS
0146-9592/07/223333-3/$15.00 © 2007 Optical Society of America
tuning performed in light-stopping systems using in-
dex modulation .
We now perform a detailed numerical analysis of
the light-stopping system in Fig. 1. The system can
be described by the following coupled-mode equa-
tions, which have previously been shown to accu-
rately describe lightpropagation
photonic-crystal resonator systems :
= ?jf0− ??t??ai?t? + j?bi?t? + j?bi−1?t?, ?1?
= ?jf0− ?0?bi?t? + j?ai+1?t? + j?ai?t?.
The unit cell consists of two resonators with mode
amplitudes ai?t? and bi?t?. Each resonator has a
static resonance frequency f0. Tuning of ??t? is per-
formed such that discrete translation symmetry of
the system is always maintained. The complex eigen-
frequencies of the system are given by
f = f0+ j
??t? + ?0
??t? − ?0
where k is the Bloch wave vector. Equation (3) de-
fines two bands that we refer to as the lower band
(? sign) and the upper band (? sign).
Figure 2(a) shows the complex frequency f of the
system in its initial state with loss ??t?=?0??. There
is no splitting of Re?f? at k=?/L, and both bands
have the same loss. Since the upper band has very
high loss for ??t??? [Figs. 2(b) and 2(c)], we focus on
the lower band in the following discussion. The qual-
ity factor Q=f0/2 Im?f? of the lower band has a peak
value of f0/2?0at k=?/L. The peak is independent of
??t? because the k=?/L state has all its energy in the
low-loss resonator. At the critical-damping point ??t?
=? [Fig. 2(b)], the peak has a minimum width in k
space and the overall loss is largest. As ??t? is tuned
above ?, the width of the quality factor peak starts to
increase, which translates to a decrease of loss for
states near k=?/L. For ??t???, the entire lower
band and upper band are associated with the low-loss
and lossy resonators, respectively [Fig. 2(c)]. The fre-
quency bandwidth of both bands becomes very nar-
row, and the group velocity of the signal is zero.
To study pulse propagation during the dynamic
process, the coupled equations [Eqs. (1) and (2)] were
solved numerically for a system of N=70 unit cells. A
Gaussian input signal with bandwidth (full width at
half-maximum of the intensity) of 0.4 THz and center
frequency fc=193 THz was used [Fig. 3(a)]. The cou-
pling constant ?=1.2 THz and resonant frequency f0
=193 THz were chosen so that the pulse would oc-
cupy a frequency width ?fin the linear region of the
system bandwidth, and a k-space region with width
?knear k=?/L, which is the low-loss region for large
??t? [Fig. 2(c)]. Since discrete translational invariance
is maintained during the tuning of ??t?, the pulse oc-
cupies the same k-space region close to k=?/L during
the entire process [Figs. 2(b) and 2(c)]. ??t? was dy-
namically tuned from ?0=193 THz/?2?5000?
?stop=193 THz/?2?15? in 1 ps (Fig. 1), stopping the
pulse [Fig. 3(b)]. After a holding time of 1.33 ps, the
same dynamic process was applied in reverse to re-
trieve the signal. Figure 3(c) shows the pulse output,
which consists of a forward pulse and a backward
time-reversed pulse  of the same amplitude. No-
tice that the pulses are still Gaussian.
We now comment in detail on the loss mechanism
during the dynamic process. During the stopping pro-
cess [i.e., from Fig. 3(a) to Fig. 3(b)], approximately
half the signal energy is lost because half the input
pulse occupies the upper band, which acquires a high
loss as ??t? is tuned to ?stop. The remaining signal en-
ergy loss in this period occurs during the nonadia-
batic ?0→?stoptuning through the critical-damping
point ??t?=?. During the storage period [Fig. 3(b)],
Fig. 2. Band structure for the system shown in Fig. 1. The
left column shows ?Re?f?−f0?, while the right column shows
Q=f0/2 Im?f?. ?fand ?kdenote, respectively, the width in
frequency and in wave vector space occupied by a Gaussian
pulse with a bandwidth of 0.4 THz. (a) Initial state ??t?
??, where Q of both lower and upper bands are equal and
constant for all k vectors. (b) Critical-damping point ??t?
=?. (c) Narrow-bandwidth state ??t???, where Re?f? of
both lower and upper bands are equal and constant for all k
(a) input pulse, (b) pulse in narrow-bandwidth stop state,
(c) output pulses.
Spatial signal profile from coupled-mode analysis:
OPTICS LETTERS / Vol. 32, No. 22 / November 15, 2007
loss arises due to attenuation of the signal in the Download full-text
narrow-bandwidth stop state. The attenuation of the
signal during the narrow-bandwidth state as well as
the slight increase in pulse width can be reduced by
increasing the ratio ?stop/?, which increases the
width in the k space associated with the lower band
quality factor [Fig. 3(c)]. During the release process
[i.e., from Fig. 3(b) to Fig. 3(c)], additional loss results
from the nonadiabatic tuning of ??t?, which leads to
signal leakage into the upper band and a backward
time-reversed pulse [Fig. 3(c)].
We note that in the coupled-mode theory, it is as-
sumed that the loss of the lossy resonators can be in-
creased without any effect on the low-loss resonators.
We have performed finite-difference time domain
(FDTD)  simulations of a 2D coupled resonator
photonic crystal system similar to Fig. 1. We have
verified that the shape of the Q?k? curve in the
narrow-bandwidth state [Fig. 2(c)] agrees with the
coupled-mode theory. Thus, the coupled-mode theory
accurately describes the dynamic light-stopping pro-
cess. Due to field penetration effects, there is an ad-
ditional signal loss not captured in the coupled-mode
theory. However, we have verified that for the ex-
ample shown here, this additional signal loss is mini-
We observe in the simulation that the backward
time-reversed pulse has the same amplitude as the
forward pulse. This effect can be explained as follows:
consider a particular wave vector point k0within the
spatial spectrum of the signal. At the start of the
pulse release process, the signal energy at k0is com-
pletely within the lower band. In addition, the value
of Re?f? for both lower and upper bands is constant
and equal [Fig. 2(c)]. As ??t? is decreased during the
pulse release process, the values of Re?f?k0?? for the
lower and upper bands split when the losses Im?f?k0??
become equal [Eq. (3)]. The band splitting is in equal
and opposite directions. As a result, the signal energy
from the initially flat lower band at k0is distributed
equally among the lower and upper bands. Since dis-
crete translational invariance is maintained through-
out the tuning process, this argument can be gener-
alized to all wave vector points within the spatial
spectrum of the signal.
In the above numerical simulation, the modulation
performed on ??t? corresponds to modulating the ma-
terial loss from ?=14 cm−1to ?=4500 cm−1in a time
of 1 ps (???/v, where v is the group velocity in the
bulk material). In InGaAs/InP quantum wells oper-
ating at room temperature, tuning the material loss
from ?=1500 cm−1to ??0 cm−1has been demon-
strated . Moreover, the intrinsic response time of
this system is in the femtosecond range . It is
likely that a higher contrast ratio of the material loss
can be accomplished at a lower operating tempera-
ture. In silicon devices tuned via free carrier injec-
tion, achieving a material loss of ?=4500 cm−1re-
quires an injected carrier density of ?4?1020cm−3,
which may be achievable in heavily doped structures
. In such a system one needs to remove most of
the carriers to achieve the low material loss value of
?=14 cm−1. This can potentially be achieved with a
reverse bias structure . The modulation time is
fundamentally limited by the carrier lifetime. In sili-
con photonic crystals, the carrier lifetime can be re-
duced by manipulating the degree of surface passiva-
tion to obtain strong surface recombination or by
using ion implantation . For example, a carrier
lifetime of less than 1 ps has been demonstrated in
ion-implanted Si .
This work is supported in part by the United
States AirForce Office
(AFOSR) grant FA9550-05-0414 and the Packard
Foundation. The authors thank D. A. B. Miller and
Onur Fidaner for discussions.
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