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VOLUME 88, NUMBER 6 P HYS ICAL REVIEW LETTER S11 FEBRUARY 2002

Multipulse Excitability in a Semiconductor Laser with Optical Injection

Sebastian Wieczorek,1Bernd Krauskopf,2,* and Daan Lenstra1

1Department of Physics and Astronomy, Vrije Universiteit Amsterdam,

De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands

2Department of Engineering Mathematics, University of Bristol,

Bristol BS8 1TR, United Kingdom

(Received 3 October 2001; published 25 January 2002)

An optically injected semiconductor laser can produce excitable multipulses. Homoclinic bifurcation

curves confine experimentally accessible regions in parameter space where the laser emits a certain

number of pulses after being triggered from its steady state by a single perturbation. This phenomenon

is organized by a generic codimension-two homoclinic bifurcation and should also be observable in other

systems.

DOI: 10.1103/PhysRevLett.88.063901 PACS numbers: 42.65.Sf, 05.45.–a, 42.55.Px

A system is called excitable when it produces a large

nonlinear response to a small but sufficiently large pertur-

bation from its stableequilibrium. When perturbed above a

certain threshold, an excitable system makes a large excur-

sion in phase space leading to a large amplitude pulse. It

then settles back to the stable equilibrium in what is called

the refractory period. Here we use the example of an opti-

cally driven semiconductor laser to show that an excitable

response to a single stimulus may have a form of not just a

single pulse but of a certain fixed number of pulses. How

many pulses are produced depends on the laser’s operating

conditions.

The notion of excitability was first introduced in biol-

ogy to describe the spiking behavior of nerve cells [1] and

later found in reaction-diffusion systems [2]. More re-

cently different types of excitability were found in optical

systems, ranging from nonlinear cavities with temperature-

dependent absorption [3] to lasers with saturable absorber

[4], optical feedback [5], and optical injection [6]. Inlasers

essentially twotypes of excitability are found [4]: the clas-

sic case of excitability on an invariant circle, and excitabil-

ity in the vicinity of a homoclinic bifurcation, found first in

lasers with saturable absorber [4] and then in multisection

distributed feedback lasers [7]. All these examples of ex-

citability were associated with a single-pulse response to

a single perturbation. A double-pulse excitable response

observed in a laser with optical feedback was explained as

the result of noise on a two-dimensional model [5].

Our main result is that a multipulse response to a single

perturbation is a natural and deterministic phenomenon.

It is an example of excitability near a homoclinic bifur-

cation, but in this case near an n-homoclinic bifurcation,

where the homoclinic orbit (to a saddle focus) closes up

not after the first but after the nth global loop; see Figs. 2

and 4 below. The key element is a generic bifurcation

structure of tongues formed by n-homoclinic bifurcation

curves in parameter space. These tongues are organized

by special points called Belyakov bifurcation points [8,9],

and inside each such tongue a different number of pulses

will be triggered. We show multipulse excitability here

for the example of an optically injected semiconductor

laser, but, in fact, this phenomenon may occur in any at

least three-dimensional system with Belyakov bifurcation

points. Other examples of systems with Belyakov points

are an atmospheric circulation model [10], atritrophic food

chain model [11], and a reduced model of a multisection

semiconductor laser [12].

We work with an optically injected single-mode semi-

conductor laser because it is a technologically important

example of a forced nonlinear oscillator [13] for which as-

tonishingly accurate experimental verification of various

types of dynamics was demonstrated, both at local and

global scale [14]. A single-mode class-B laser with opti-

cal injection is described well by the rate equations

?E ? K 1 ?1

? n ? 22Gn 2 ?1 1 2Bn??jEj22 1?,

which are a three-dimensional dynamical system for the

complex electric field amplitude E ? Ex1 iEyand the

population inversion n; see Ref. [13] for more details.

The two experimental control parameters are the injected

field rate K and its detuning v (measured in units of the

characteristic relaxationoscillation frequency vr)from the

free-running laser frequency. We focus here on the case of

a semiconductor laser, so that the self-modulation parame-

ter a is larger than one, and we fix the laser parameters to

the realistic values a ? 2, B ? 0.015, and G ? 0.035.

When K and v are changed, the solutions of Eqs. (1)

generally change as well. Qualitative changes of the sys-

tem’s dynamics, the so-called bifurcations, can be detected

and continued in the two-dimensional ?K,v?-plane, for ex-

ample, with the package AUTO [15]. The resulting bifur-

cation curves form a bifurcation diagram by dividing the

?K,v?-plane into regions with different dynamical behav-

ior of the laser; see Ref. [13] for details of the bifurcation

diagram of Eqs. (1).

Here we focus on the locking region inside which the

laser operates at constant power and the injected light fre-

quency. Locking is represented by a stable equilibrium in

2?1 1 ia?n 2 iv?E ,

(1)

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VOLUME 88, NUMBER 6P HYS ICAL REVIEW LETTERS11 FEBRUARY 2002

the three-dimensional phase space ?Ex,Ey,n?. The lock-

ing region in the ?K,v?-plane plotted in Fig. 1 is confined

between the Hopf bifurcation curve H and the saddle-

node bifurcation curve SN. When the locking region is

approached from above, the laser typically shows intensity

oscillations at the relaxation oscillation frequency ?vR.

At H the corresponding stable periodic orbit disappears in

a Hopf bifurcation, leading to a stable equilibrium, which

is the locked state. When the locking region is approached

from below, the laser produces oscillations at a frequency

close to the detuning v. Close to SN the flow on the cor-

responding periodic orbit slows down near the point where

the saddle node will appear and then makes a quick ex-

cursion along the other part of the periodic orbit. At SN

two points, an attractor and a saddle focus, appear and the

laser locks.

It wasgenerally believed thatin Eqs. (1)the saddle-node

bifurcation always takes place on a periodic orbit. (This

is also known as Adler’s locking mechanism.)

show thatthis does nothappen along the wholeof the curve

SN. As is also shown in Fig. 1 there are special points

A1and A2where a homoclinic bifurcation curve labeled

h1touches SN (codimension-two saddle-node homoclinic

points), forming what we call a homoclinic tooth. As a

consequence, between A1and A2the saddle-node bifurca-

tion does not take place on the periodic orbit. The same

is true between A3and A4and at many more homoclinic

teeth (not shown Fig. 1) that become smaller and smaller

for larger K. In Eqs. (1) the size of the teeth depends cru-

cially on the parameter a: they appear only for values of

a . 1 and grow in size as a increases.

Here we

FIG. 1.

two homoclinic teeth. In this and all figures of the ?K,v?-plane,

v is in units of vRwhile K is dimensionless. The gray curves

correspond to subcritical bifurcations, along which no attractors

bifurcate.

The locking range of Eqs. (1) in the ?K,v?-plane with

Near h1but outside a homoclinic tooth, the dynamics is

as sketched in Fig. 2(a). On h1the one-dimensional un-

stable manifold Wuof the saddle-focus equilibrium s co-

alesces with the two-dimensional stable manifold Wsof s,

forming a one-homoclinic orbit; see Fig. 2(b). What the

dynamics looks like inside the homoclinic tooth depends

crucially on whether the stable or the unstable direction of

the saddle focus is stronger [16]. The dashed curve ns in

Fig. 1 marks where the saddle focus is as attracting as re-

pelling (neutral saddle). Along the parts of h1below ns the

attracting direction of the saddle focus is stronger and the

homoclinic orbit bifurcates into an attracting periodic orbit

asin Fig. 2(c1). This is oftencalled asimple Shil’nikov bi-

furcation. The two-dimensional stable manifold Wsforms

the boundary between the basin of the locked state a and

that of the attracting periodic orbit. On the other hand,

when h1is crossed above ns, then the unstable manifold

makes two loops above s and converges to the stable equi-

librium a as in Fig. 2(c2). This is often called a chaotic

Shil’nikov bifurcation, because it implies the existence of

an infinite number of saddle periodic orbits of different pe-

riods close to h1. Furthermore, there are homoclinic bifur-

cation curves hnfor any n ? 1,2,3,... of n-homoclinic

orbits to the saddle focus s. They lie inside and near h1,

creating a cascade of bifurcations that is not entirely un-

derstood yet [9]. The codimension-two Belyakov points

B1and B2are special as they form the boundary on h1

between these types of Shil’nikov bifurcations.

We now reveal the homoclinic bifurcation structure

associated with B1and B2in Eqs. (1) and show how this

introduces regions in the ?K,v?-plane with multipulse

excitability.In Fig. 3(a) is plotted an enlargement of

the biggest homoclinic tooth with some of the additional

curves hn(computed with the HOMCONT part of AUTO

[15]). There are two types of such curves: those that ac-

cumulate in the form of tongues at B1above ns, and those

that cross the neutral-saddle line ns [8]. We find that many

hntongues cross ns and extend to the region of simple

Shil’nikov bifurcation. Every intersection point of such a

homoclinic bifurcation curve with ns is a new Belyakov

point. Tongues that approach SN grow larger and are, in

fact, experimentally accessible. (The experimental reso-

lution is ?100 MHz, while the detuning v is in units of

vR? 30 rad?s [14].) Some of the curves hneven con-

nect to the curve SN and create new segments where the

FIG. 2.

just after (c1),(c2) the Shil’nikov bifurcation along h1for the

simple case (c1) and the chaotic case (c2).

Sketches of the phase portraits before (a), at (b), and

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VOLUME 88, NUMBER 6P HYS ICAL REVIEW LETTERS11 FEBRUARY 2002

FIG. 3.

tooth in Fig. 1 showing tongues of multipulse excitability.

Successive enlargements near the bigger homoclinic

saddle-node bifurcation takes place on a limit cycle with

n loops. Figure 3(b) shows a further enlargement near

SN with more n-homoclinic bifurcation curves. For pa-

rameters between tongues and below ns the laser is either

locked to the input signal or shows self-pulsations.

For values of K and v inside each tongue bounded by

hnand below ns we find multipulse excitability. Figure 4

shows this for ?K,v? inside the tongues bounded by the

curves h1, h2, h3, and h4from Fig. 3(b). Plotted are the

unstable manifolds Wuof the saddle (left column) and

the intensity response to a single small perturbation (right

column), calculated with DSTOOL [17]. The mechanism of

multipulse excitability is the following. The upper branch

of the unstable manifold Wuof the saddle makes n loops

above the stable manifold Wsand then dives under Wsto

end up at the stable equilibrium; see Fig. 4 (left column).

A single very small perturbation of the stable locked state

FIG. 4.

and as intensity response to a single perturbation (right column)

for ?K,v? from inside h1(a), h2(b), h3(c), and h4(d). From

(a) to (d) ?K,v? takes the values ?0.45,20.93?, ?0.472,20.98?,

?0.48,20.97?, and ?0.455,20.95725?.

Multipulse excitability in ?E,n?-space (left column)

results in the immediate return to the locked state. How-

ever, if the perturbation is strong enough to kick the system

to the other side of Ws, then it follows the upper branch of

Wsand, hence, produces an n-pulse response; see Fig. 4

(right column). The threshold for excitability is given by

Ws. Since, near the saddle, this is roughly given by the dis-

tance between the saddle and the attractor, this threshold

becomes smaller closer to the curve SN. We finally point

out that for certain parameter values near h1above ns the

excitable response can be chaotic and unpredictable, be-

cause the trajectory may wander between a huge number

of coexisting unstable orbits before it settles back to the

stable equilibrium.

In conclusion, we presented a deterministic mechanism

for multipulse excitability that appears to be experimen-

tally accessible in a real laser. We identified and described

codimension-twohomoclinic Belyakov bifurcations and an

ensuing cascade of n-homoclinic bifurcation tongues as

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VOLUME 88, NUMBER 6P HYS ICAL REVIEW LETTERS 11 FEBRUARY 2002

responsible for this phenomenon. Therefore, one should

also find this type of multipulse excitability in other sys-

temsfrom applications in parameter regions with Belyakov

points. In a real system there is always some noise so that

some pulses may be triggered while the system follows

the unstable manifold before the locked state is reached

again [5]. As a result, a single perturbation inside an hn

tongue may occasionally result in more than n pulses as

a response. Furthermore, close to SN (spontaneous emis-

sion) noise itself may be enough to trigger pulses. Finally,

we mention that the phenomenon of coherence resonance

must be expected: a minimum of the jitter of a noise-

triggered pulse train for a particular noise level [4]. How

this general phenomenon manifests itself in the presence

of multipulse excitability remains an interesting question

for future research.

The research of S.W. was supported by the Founda-

tion for Fundamental Research on Matter (FOM), which

is financially supported by the Netherlands Organization

for Scientific Research (NWO). B.K. is supported by the

EPSRC.

*Corresponding author.

Email address: B.Krauskopf@bristol.ac.uk

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