Dynamical properties of lasers coupled face to face
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Dynamical properties of lasers coupled face to face
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Dynamical properties of lasers coupled face to face
J. Javaloyes
Institut Non Line´ aire de Nice, UMR 6618 CNRS-UNSA, 1361 Route des Lucioles, F-06560 Valbonne, France
Paul Mandel and D. Pieroux
Optique Nonline´ aire The´ orique, Universite´ Libre de Bruxelles, Campus Plaine CP 231, B-1050 Bruxelles, Belgium
Received 14 June 2002; revised manuscript received 19 November 2002; published 10 March 2003
We derive a reduced model to describe two identical lasers coupled face to face. Two limits are introducedin the Maxwell-Bloch equations: adiabatic elimination of the material polarization and large distance betweenthe two lasers. The resulting model describes coupled homogeneously broadened lasers, including semicon-ductor lasers. It consists of two coupled delay differential equations with delayed linear cross-coupling and aninstantaneous self-coupling nonlinearity. The study is analytical and numerical. We focus on the properties of steady and periodic amplitudes of the electric fields. In steady state, there are symmetric, antisymmetric, andasymmetric solutions with respect to a permutation of the two fields. A similar classification holds for theperiodic states. The stability of these solutions is determined partly analytically and partly numerically. Ahomoclinic point is associated with the asymmetric periodic solutions.DOI: 10.1103/PhysRevE.67.036201 PACS number
s
: 05.45.
a, 42.65.Sf, 42.60.Mi, 42.55.Px
I. INTRODUCTION
This paper deals with the nonlinear dynamics of twosemiconductor lasers coupled face to face
F2F
: the outputof each laser is injected, after a suitable attenuation, in theother laser. The emphasis will be on the influence of theunavoidable delay due to the transit time between the twolasers. The subject of F2F coupling is not popular in experi-mental laser physics because the injection of a signal in anamplifier is a fine source of instabilities which are difficult tocontrol. Few studies have been made for gas or solid-statelasers in that configuration. However, with the emergence of chaos synchronization studies in semiconductor lasers moti-vated by the prospect of transmitting coded information
1
,two important factors that hampered the study of the F2Fconfiguration have turned out to be useful sources of newphysics:
i
the injected field induces chaos since the isolatedlasers are operating in a stable regime;
ii
the delay which isthe propagation time between the two lasers has considerableinfluence on the nonlinear dynamics by introducing a newand easily controllable time scale in the system. Its relevanceis also related to the fact that, for semiconductor lasers, thedistance between the two lasers is usually much larger thanthe laser dimensions. Although the derivation of the modelequations will be quite general and not tied to a particulartype of laser, the choice of values for the parameters in thefigures will be for semiconductor lasers.Stable localized synchronization of two different semi-conductor lasers coupled F2F was demonstrated in the peri-odic and quasiperiodic regimes
2
. More recently, synchro-nization of two identical semiconductor lasers coupled F2Fand operating in a chaotic regime was reported
3
. Someformal aspects of synchronization in the F2F configurationhave been considered
4
.Independently of these studies, which specifically focuson semiconductor lasers, there has been a large amount of research on the influence of the delay in the theory of coupled nonlinear oscillators with an emphasis on the Kura-moto model
5,6
. Coupled nonlinear oscillators with a timedelay may display multistability
7
, delay-induced ‘‘death’’in which one oscillator prevents the periodic regime in theother oscillator
8
, and stochastic resonance
9
. Analyticalexpressions for the boundaries of synchronized regimes wereobtained in Ref.
10
. The relevance of these studies to thephysics of semiconductor lasers follows from the proof thatthe Lang-Kobayashi model for a single mode semiconductorlaser with external feedback
11
can asymptotically be re-duced to a phase equation of which the Kuramoto equation isa particular limit
12
. This result was extended to an array of semiconductor lasers, with a systematic study of their syn-chronization properties
13,14
.In comparison with what is known in the theory of ordi-nary and partial differential equations, the mathematics of delay differential equations are underdeveloped due to theinherent difficulty associated with their structure. Few resultsare simple and little is known about generic properties of thisclass of equations. In a recent paper
15
, it has been shownthat in the long delay time limit, the two Lang-Kobayashirate equations, which are the canonical description of a semi-conductor laser with external feedback, may be reduced to asimpler problem which retains the essential features of thefull model. That reduced model consists of a single delaydifferential equation with delayed linearity and instantaneousnonlinearity. The purpose of this paper is to extend that ap-proach to the case of two coupled lasers in a F2F configura-tion.This paper is organized as follows. In Sec. II, we showthat, starting with the Maxwell-Bloch equations for two F2Fcoupled homogeneously broadened lasers, the referencemodel is asymptotically obtained by
i
adiabatically elimi-nating the material polarization,
ii
by introducing the longdelay time limit. The inherent symmetry of the problem sug-gests several combinations of the electric fields. They arereviewed in Sec. IV. The steady state solutions, which are theequivalent of the external cavity modes of the Lang-
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Kobayashi equations, are analyzed in Sec. V. Hopf bifurca-tions arise on these steady solutions and lead to periodicregimes which are analyzed in Sec. VI. These analytic resultsare complemented by a numerical study presented in Sec.VII and conclusions are drawn in Sec. VIII.
II. DERIVATION OF THE REFERENCE MODEL
We consider two single mode lasers, each of which isdescribed by the usual Maxwell-Bloch equations for a homo-geneously broadened laser. The output of each laser is in- jected in the other laser after a suitable attenuation. TheMaxwell-Bloch equations couple the intracavity field
E
˜
j
withthe space average of the material polarization
S
˜
j
and inver-sion
N
˜
j
of the active medium
d
E
˜
j
dt
j
1
i
j
E
˜
j
ig
S
˜
j
j
˜
j
E
˜
3
j
t
3
j
,
1
d
S
˜
j
dt
1
i
j
S
˜
j
ig
E
˜
j
N
˜
j
,
2
d
N
˜
j
dt
N
˜
j
N
˜
eq
2
ig
E
˜
j
*
S
˜
j
S
˜
j
*
E
˜
j
,
3
with
j
1 or 2. The
j
are the damping rates of the twolasing cavities and the
j
are their normalized frequencydetuning. The parameters
˜
j
account for the attenuation of each field before being injected in the other laser. The finitedistance between the two lasers and the finite velocity of light imply that the injected fields are delayed with respect tothe emitted fields by
j
L
j
/
v
, where
L
j
is the optical lengthbetween laser
j
and laser 3
j
, and
v
is the light velocitybetween the two lasers. The active medium is characterizedby the damping rates
and
, the atom-field interactionstrength
g
which is chosen to be real and
j
is the field-matter frequency mismatch for a solid-state laser or the lineenhancement factor for a semiconductor laser. Note thatthere is no generally accepted convention for the sign of
.This leads sometimes to apparent contradictions. It would beeasy to generalize these equations to include different lightvelocities in each directions, different light-matter couplingparameters, and different material damping rates. Given thetwo asymptotic limits we shall introduce, this would notchange the structure of the resulting reduced equations.A first asymptotic limit introduced in this problem isbased on the assumption that the polarization is a variablethat relaxes much faster than the inversion or the field in anyof the two lasers:
,
j
. We also assume that the dif-ference between the two laser frequencies is much smallerthan
so that the beat frequency is a slow variable. As aresult, the material polarization can be adiabatically elimi-nated,
S
˜
j
ig
1
i
j
1
j
2
E
˜
j
N
˜
j
.The remaining dynamical equations
1
–
3
become
d
E
˜
j
dt
j
E
˜
j
1
i
j
g
2
1
i
j
j
1
j
2
N
˜
j
˜
j
E
˜
3
j
t
3
j
,
4
d
N
˜
j
dt
N
˜
eq
N
˜
j
4
g
2
1
j
2
E
˜
j
2
N
˜
j
.
5
We assume that the photon lifetimes inside the cavities,1/
j
, are identical. Therefore time can be usefully rescaledas
t
t
and
T
/
is the dimensionless lifetime of thecarrier within the lasers. Consequently Eqs.
4
and
5
be-come
d
E
˜
j
dt
E
˜
j
1
i
j
g
2
1
i
j
j
1
j
2
N
˜
j
˜
j
E
˜
3
j
t
3
j
,
d
N
˜
j
dt
1
T
N
˜
eq
N
˜
j
4
g
2
1
j
2
E
˜
j
2
N
˜
j
.As a last step, we rescale the dynamical variables and thefeedback rates,
N
˜
j
N
˜
j
(
s
)
1
2
N
j
1
2
N
j
j
1
j
2
/
g
2
,
E
˜
j
12
g
2
1
j
2
E
j
,
˜
j
j
1
j
2
/
1
3
j
2
.We also define the new parameters
N
˜
eq
N
˜
j
(
s
)
1
2
P
j
,
j
j
j
,where
P
j
is the excess pumping rate above threshold and
j
the free running frequency of the laser
j
. All this leads to theset of equations
d
E
j
dt
1
i
j
N
j
E
j
i
j
E
j
j
E
3
j
t
3
j
,
d
N
j
dt
1
T
P
j
N
j
1
2
N
j
E
j
2
.These equations have the form of two coupled Lang-Kobayashi equations if the
j
can be interpreted as fixedconstants. Nevertheless, for a pair of solid-state lasers, the
j
are still functions of the unknown lasing frequency.A further simplification is to assume that the two delaysare identical, i.e.,
j
3
j
. Following the analysis inRef.
15
, we introduce a scaling which is useful in the largedelay limit,
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E
j
E
j
,
N
j
N
j
,
p
j
P
j
/
,
j
j
,
s
t
/
,
j
j
.
6
It yields
E ˙
j
1
i
j
N
j
E
j
i
j
E
j
j
E
3
j
s
1
,
7
T
N ˙
j
p
j
N
j
1
2
N
j
E
j
2
,
8
where the dot denotes the derivative with respect to the di-mensionless time
s
.A second asymptotic limit is the long delay time limit1/
→
0, which leads to a substantial simplification of Eq.
8
since it reduces to
N
j
E
j
2
p
j
. Inserting this relation intothe field equations
7
leads to
E ˙
1
1
i
1
p
1
E
1
2
E
1
i
1
E
1
1
E
¯
2
,
9
E ˙
2
1
i
2
p
2
E
2
2
E
2
i
2
E
2
2
E
¯
1
,
10
with the notation
E
¯
j
(
s
)
E
j
(
s
1).To estimate the validity of the above asymptotic limit, weconstructed numerically the bifurcation diagram of Eqs.
7
and
8
for two values of the ratio
T
/
and the bifucrationdiagram of Eqs.
9
and
10
. The result is shown in Fig. 1.Starting with
10
3
and a random initial condition, weintegrate over 500 delay times to let the system reach its finalstate. We then record the extrema, that is, the intensities atwhich their time derivative cross the zero value. Then,
isslightly increased and the procedure is repeated until
10. Then, we repeat the procedure beginning with
10and decreasing
. The numerical time integration used amodified fourth order Runge-Kutta method with linear inter-polation of the delayed term, which gives a second orderoverall accuracy.With
10
12
and
10
9
, the parameter
T
is equal to10
3
. Accordingly, we chose
equal to 10
4
and 10
5
,corresponding to a propagation length
in vacuum
of 3 and30 meters, respectively. The agreement between the bifurca-tion diagrams obtained with
T
/
10
2
and
T
/
0 is verygood.To make contact with a more standard form of the modelequations, the fields will be described in a reference framerotating at the mean velocity
(
1
2
)/2. It is useful tointroduce the mean and the mismatch functions
x
12
x
1
x
2
,
x
12
x
1
x
2
,where
x
is any of the parameters
j
,
j
,
p
j
, or
j
. Thisleads from Eqs.
9
and
10
to the equations
E ˙
1
1
i
p
E
1
2
E
1
e
i
E
¯
2
F
1
E
1
,
E
¯
2
,
11
E ˙
2
1
i
p
E
2
2
E
2
e
i
E
¯
1
F
2
E
2
,
E
¯
1
,
12
where the functions
F
1
and
F
2
have been defined as
F
1
E
1
,
E
¯
2
i
pE
1
e
i
E
¯
2
i
E
1
p
1
i
E
1
i
p
E
1
2
E
1
,
13
F
2
E
2
,
E
¯
1
i
pE
2
e
i
E
¯
1
i
E
2
p
1
i
E
2
i
p
E
2
2
E
2
.
14
Equations
11
and
12
are two cross-coupled delay dif-ferential equations plus a perturbation
F
j
which vanishes if the two lasers are identical. In that case, there is an obviousanalogy with the reduced equation
E ˙
1
i
p
E
2
E
e
i
E
¯
15
derived in the same long delay time limit from the Lang-Kobayashi equations
15
.
III. PHYSICAL DISCUSSION
An experimental setup has to match the two main hypoth-eses of the present study, as detailed in Sec. II: the long delaylimit with respect to the characteristic times constants of thelasers, and negligible retroreflection at the laser mirrors,since we are interested here only in mutual injection and notin feedback effects.The former can be achieved by coupling the two lasers viaa long single mode telecom fiber whose length should be of the order of 10 m if semiconductor lasers are used. More-over, unavoidable losses due to propagation effects are not a
FIG. 1. Comparing the bifurcation diagrams obtained for
T
/
small but finite and
T
/
0. From top to bottom:
T
/
0.1,0.01,and 0. In each diagram, the exterma of the solutions are plotted. For
T
/
finite, Eqs.
7
and
8
are integrated. For
T
/
0, Eqs.
9
and
10
are integrated.DYNAMICAL PROPERTIES OF LASERS COUPLED FACE . . . PHYSICAL REVIEW E
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problem here, since we only need an injection ratio of theorder of a few percent.As a consequence, the use of a neutraldensity filter
NDF
is also needed in order to control thecoupling strength between the two lasers.The latter hypothesis can be fulfilled by placing a anti-feedback
AF
device between each laser and the fiber. AnAF system will be composed, starting from the emitting faceof each laser, of a quarter wave plate with its axis oriented at45° with respect to a polarizer following it. Doing this, lightentering parallel to the polarizer will end up perpendicular toit after a first passage through the quarter wave plate, a re-flection, and a second passage through the quarter waveplate. Doing this, feedback effects can be suppressed with avery good efficiency of at least 40 dBm.Another method, proposed in Ref.
3
, is to remove thetwo quarter wave plates and to keep just one polarizer and aNDF in order to force the lasers to operate on only onepolarization. However, retroreflection is no more controlledin this case.
IV. EXPLOITING THE SYMMETRIESA. Symmetric and antisymmetric functions
We explore the dynamic of the system assuming the sameset of parameter for both lasers, i.e.,
x
0, so that
F
j
0. This restriction suggests to decompose the fields intosymmetric and antisymmetric combinations,
S
12
E
1
E
2
,
16
D
12
E
1
E
2
,
17
to take advantage of the inherent symmetries of the problem.When Eqs.
11
and
12
are expressed in terms of
S
and
D
,they lead to a pair of coupled equations where the crosscoupling appears only in the nonlinearity,
S ˙
1
i
p
S
2
S
e
i
S
¯
1
i
2
D
2
S
D
2
S
*
,
18
D˙
1
i
p
D
2
D
e
i
D
¯
1
i
2
S
2
D
S
2
D
*
.
19
The system
16
and
17
exhibits the usual gauge invarianceunder the transformation (
S
,
D
)
→
(
Se
i
,
De
i
), common tooptical devices without phase conjugation. Another symme-try, which is trivial here since the two subsystems are iden-tical, is the invariance under the permutation of the fields(
S
,
D
)
→
(
S
,
D
) or (
E
1
,
E
2
)
→
(
E
2
,
E
1
).
B. Parametric representation
To study the bifurcation diagram of Eqs.
16
and
17
, aparametric representation of the fields is useful, as shown inRef.
16
. To simplify the algebra, we concentrate on the case
p
0. We introduce two new complex functions,
X
and
Y
,defined as
S
s
X
s
e
i
s
,
D
s
Y
s
e
i
s
,where
is a free parameter. Inserting these definitions inEqs.
18
and
19
gives
X ˙
1
i
X
2
2
Y
2
X
Y
2
X
*
e
i
(
)
X
¯
i
X
,
20
Y ˙
1
i
Y
2
2
X
2
Y
X
2
Y
*
e
i
(
)
Y
¯
i
Y
.
21
This transformation allows the mapping of periodic and qua-siperiodic solutions of Eqs.
18
and
19
onto steady andperiodic solutions of Eqs.
20
and
21
. As stressed in Ref.
16
, this parametric representation is not unique. If
„
X
(
s
),
Y
(
s
),
…
is a solution of Eqs.
20
and
21
, the one-parameter family
„
X
*
(
s
),
Y
*
(
s
),
*
…
defined by
X
*
s
e
i
(
*
)
s
X
s
,
22
Y
*
s
e
i
(
*
)
s
Y
s
,
23
where
*
is an arbitrary real number, may also verify Eqs.
20
and
21
. This is because all members of the family
„
X
*
(
s
),
Y
*
(
s
),
*
…
correspond to the same physical solution
„
E
1
(
s
),
E
2
(
s
)
…
. For the steady state solutions of Eqs.
20
and
21
, i.e., when
X
(
s
) and
Y
(
s
) are time independent, itfollows from Eqs.
22
and
23
that the parametric represen-tation is unique, i.e.,
*
, while if
X
(
s
) and
Y
(
s
) are
T
p
-periodic solutions,
and
*
are only constrained by therelation
*
2
k
T
p
,where
k
is an integer.
V. EXTERNAL CAVITY MODES
In this section, we analyze the plane wave solutions of Eqs.
16
and
17
, which are the simplest nontrivial solu-tions of these equations. These periodic states are the steadystates of Eqs.
20
and
21
in the parametric representation.From the structure of these equations, it follows directly thatsolutions with the same constant intensity for both fieldsmust be either in phase or dephased by
. In addition, asym-metric modes exist that break the symmetry of the equations.
Strictly speaking
, there are no external cavity modes be-cause there is no external cavity and therefore no feedback atall. Still, we use this term to designate the steady state solu-tions of Eqs.
20
and
21
because they are algebraicallyvery close to the external cavity modes of a semiconductorlaser with external mirror. The formal analogy stems from
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the mutual injection which leads to a generalization of thereduced Lang-Kobayashi equations
15
which are valid inthe case of weak feedback, that is if there is only one reflec-tion against the external mirror. The other justification forusing this terminology is that the bifurcation diagram of these steady states has much in common with the bifurcationdiagram of Eq.
15
.
A. Symmetric and antisymmetric modes
The symmetric external cavity modes
S modes
are
„
X
(
s
),
Y
(
s
),
…
(
x
,0,
). Inserting this definition in Eqs.
20
and
21
leads to the relations
x
2
cos
,
24
cos
sin
.
25
The antisymmetric external cavity modes
AS modes
, de-fined by
„
X
(
s
),
Y
(
s
),
…
(0,
y
,
), lead to the steady staterelations
y
2
cos
,
26
cos
sin
.
27
We first consider the case
0. A single branch of ex-ternal cavity mode emerges from a Hopf bifurcation at
0. This solution is a S mode which is stable in the vicinityof the bifurcation point where it is supercritical. It can beapproximated for
1 by (
x
2
,0,
)
(
,0,
). Increas-ing the feedback strength
, pairs of S modes emerge fromHopf bifurcations on the trivial solution
0 located at
k
3
/2
2
k
where
k
is a positive integer. In the vicinity of the bifurcation point at
k
, the first branch of the pair can beexpressed as
x
2
,0,
k
/(1
k
),0,
k
provided that
k
1 is positive. The other branch is given by
x
2
,0,
k
/(1
k
),0,
k
. It should be noticedthat the first solution is always supercritical while the secondbranch can be either super- or subcritical, depending of thesign of 1
k
. In a similar way, pairs of AS modes emergefrom the trivial solution from another set of bifurcationpoints
k
/2
2
k
. The expression of
y
2
forAS modes isthe same as the expression of
x
2
for the S modes. The prop-erties of the bifurcations leading to the S and to the ASmodes are the same. These results are summarized in Fig.2
a
. For all the figures,
3 and
p
0.For
/2, a mixed pair of
S, AS
modes emerges atthe srcin (
,
)
(0,0) as shown in Fig. 2
b
. For
1, theS mode can be approximated by
x
,0,
2
(1
),0,
and the AS mode by
0,
y
,
0,
2
(1
),
.These branches are both stable and supercritical and theymerge at the srcin. This is shown in Fig. 3
b
. As
in-creases, other mixed pairs emerge from the trivial solution atthe bifurcation points
k
k
,
k
being a positive integer.For those pairs, the S mode can be expressed as
x
2
,0,
k
/
1
k
(
1)
k
,0,
k
(
1)
k
and theAS mode by
0,
y
2
,
0,
k
/
1
k
(
1)
k
,
k
(
1)
k
where
k
1. For the parameters
3 and
p
0, we have
1/
1
. Thus, close to the bifurcation from which theyemerge, the S modes are supercritical
subcritical
and theAS modes subcritical
supercritical
if
k
is odd
even
.Finally, it follows from the definition of the S and ASmodes, Eqs.
24
and
25
and Eqs.
26
and
27
, respec-tively, that adding
to
is equivalent to the mode permu-tation S
AS. Thus the considerations for
0 and
/2
FIG. 2. Bifurcation diagram of the external cavity mode solu-tions. Symmetric modes are in fulllines, antisymmetric modes are indotted line, asymmetric modes arethe loops in dash-dotted lines.Thick lines indicate stable solu-tions, thin lines indicate unstablesolutions.
a
0;
b
/2;
c
;
d
3
/2. Com-mon parameters for all figures:
3 and
p
0. The insets in
a
and
c
are focused on the firstasymmetric solution.DYNAMICAL PROPERTIES OF LASERS COUPLED FACE . . . PHYSICAL REVIEW E
67
, 036201
2003
036201-5
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