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Stopping and time reversing a light pulse using

dynamic loss tuning of coupled-resonator

delay lines

Sunil Sandhu,* M. L. Povinelli, and Shanhui Fan

Department of Electrical Engineering, Ginzton Laboratory, Stanford University, Stanford, California 94305, USA

*Corresponding author: centaur@stanford.edu

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 [9]. For the

available index modulation strength of ?n/n?10−4

−10−3

intransparent semiconductors

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

systems.

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 [3]. 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

toanarrow-bandwidth

[10],the

state.Inthisnarrow-

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

Fig. 1.

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

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tuning performed in light-stopping systems using in-

dex modulation [3].

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-

ratelydescribelightpropagation

photonic-crystal resonator systems [3]:

indynamic

1

2?

dai?t?

dt

= ?jf0− ??t??ai?t? + j?bi?t? + j?bi−1?t?, ?1?

1

2?

dbi?t?

dt

= ?jf0− ?0?bi?t? + j?ai+1?t? + j?ai?t?.

?2?

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

±?2?2cos2?

f = f0+ j

??t? + ?0

2

kL

2?−?

??t? − ?0

2?

2

,

?3?

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 [11] 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)],

to

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

vectors.

Fig. 3.

(a) input pulse, (b) pulse in narrow-bandwidth stop state,

(c) output pulses.

Spatial signal profile from coupled-mode analysis:

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loss arises due to attenuation of the signal in the

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) [12] 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-

mal.

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 [13]. Moreover, the intrinsic response time of

this system is in the femtosecond range [14]. 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

[15]. 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 [16]. 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 [10]. For example, a carrier

lifetime of less than 1 ps has been demonstrated in

ion-implanted Si [17].

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.

of ScientificResearch

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