Stabilization of optically coupled lasers with periodic pumping
We study a periodically forced system modeling the synchronization of two optically coupled lasers pumped by an alternating current. A necessary and sufficient condition for existence of a periodic solution is given, as well as a sufficient condition for uniqueness and asymptotic stability.
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irk-123456789-1755122021-02-02T01:29:03Z Stabilization of optically coupled lasers with periodic pumping Torres, P.J. We study a periodically forced system modeling the synchronization of two optically coupled lasers pumped by an alternating current. A necessary and sufficient condition for existence of a periodic solution is given, as well as a sufficient condition for uniqueness and asymptotic stability. Вивчаються перiодично збурюванi системи, що моделюють синхронiзацiю двох оптично зв’язаних лазерiв, що накачуються за допомогою змiнного струму. Наведено необхiднi i достатнi умови iснування перiодичного розв’язку, а також достатню умову для його єдиностi та асимптотичної стiйкостi. 2011 Article Stabilization of optically coupled lasers with periodic pumping / P.J. Torres // Нелінійні коливання. — 2011. — Т. 14, № 3. — С. 392-399. — Бібліогр.: 11 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/175512 517.9 en Нелінійні коливання Інститут математики НАН України |
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We study a periodically forced system modeling the synchronization of two optically coupled lasers pumped by an alternating current. A necessary and sufficient condition for existence of a periodic solution is given, as well as a sufficient condition for uniqueness and asymptotic stability. |
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Article |
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Torres, P.J. |
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Torres, P.J. Stabilization of optically coupled lasers with periodic pumping Нелінійні коливання |
| author_facet |
Torres, P.J. |
| author_sort |
Torres, P.J. |
| title |
Stabilization of optically coupled lasers with periodic pumping |
| title_short |
Stabilization of optically coupled lasers with periodic pumping |
| title_full |
Stabilization of optically coupled lasers with periodic pumping |
| title_fullStr |
Stabilization of optically coupled lasers with periodic pumping |
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Stabilization of optically coupled lasers with periodic pumping |
| title_sort |
stabilization of optically coupled lasers with periodic pumping |
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Інститут математики НАН України |
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2011 |
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http://dspace.nbuv.gov.ua/handle/123456789/175512 |
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Stabilization of optically coupled lasers with periodic pumping / P.J. Torres // Нелінійні коливання. — 2011. — Т. 14, № 3. — С. 392-399. — Бібліогр.: 11 назв. — англ. |
| series |
Нелінійні коливання |
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AT torrespj stabilizationofopticallycoupledlaserswithperiodicpumping |
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2025-07-15T12:49:56Z |
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2025-07-15T12:49:56Z |
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1837717308383952896 |
| fulltext |
UDC 517.9
STABILIZATION OF OPTICALLY COUPLED LASERS
WITH PERIODIC PUMPING*
СТАБIЛIЗАЦIЯ ОПТИЧНО ЗВ’ЯЗАНИХ ЛАЗЕРIВ
З ПЕРIОДИЧНОЮ НАКАЧКОЮ
P. J. Torres
Univ. de Granada
Departamento de Matemática Aplicada,
Facultad de Ciencias, 18071, Granada, Spain
e-mail: ptorres@ugr.es
We study a periodically forced system modeling the synchronization of two optically coupled lasers pumped
by an alternating current. A necessary and sufficient condition for existence of a periodic solution is given,
as well as a sufficient condition for uniqueness and asymptotic stability.
Вивчаються перiодично збурюванi системи, що моделюють синхронiзацiю двох оптично зв’я-
заних лазерiв, що накачуються за допомогою змiнного струму. Наведено необхiднi i достатнi
умови iснування перiодичного розв’язку, а також достатню умову для його єдиностi та асимп-
тотичної стiйкостi.
1. Introduction. In Nonlinear Optics, the synchronization of two optically coupled lasers pumped
by an alternating current is a known fact that has deserved the attention of the specialists from a
theoretical and experimental perspective [1 – 4]. In [3], it is shown that in its simplest formulati-
on, two coupled lasers with periodic pumping behave like an ”equivalent” single laser whose
dynamical behavior is described by the system
τ ġ = g0(t)− g
(
1 + E2
)
,
(1)
Ė =
1
2
(g − g̃th)E.
Here, g is the amplification factor and E is the amplitude of the locked field. The parameter
τ > 0 is the effective relaxation time of the active medium, g0(t) = A(1 + sinωt) is the
2π
ω
-periodic pumping, and g̃th > 0 is the renormalised threshold gain defined by
g̃th = gth + 2M
1−
√
1−
(
∆
M
)2
,
where gth > 0 is the original threshold gain, M > 0 is the coupling coefficient, and ∆ > 0 is
proportional to the off-tuning of the natural frequencies of the cavities.
On the basis of the mentioned references, the theoretical objective is to find an asymptoti-
cally stable periodic solution of system (1). To this purpose, the very recent paper [4] presents
∗ Partially supported by Ministerio de Educación y Ciencia, Spain, project MTM2008-02502.
c© P. J. Torres, 2011
392 ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 3
STABILIZATION OF OPTICALLY COUPLED LASERS . . . 393
a numerical-analytic scheme of investigation of periodic solutions of system (1) by means of
a technique of construction of matrix-valued Lyapunov functions [5]. The aim of this paper is
to contribute to the literature from a different point of view, in such a way that we are able to
identify explicit regions of parameters where there exists a unique periodic solution which is
asymptotically stable.
From now on, the minimal period of the periodic pumping is denoted by T =
2π
ω
.Our main
results are stated below.
Theorem 1. The condition
g̃th < A (2)
is necessary and sufficient for the existence of a T -periodic solution of system (1).
Theorem 2. Assume that (2) holds. Let us fix the constants
m1 :=
1
2
ln
(
A
g̃th
− 1
)
−
√
3
2
√
2
AT, m2 :=
1
2
ln
(
A
g̃th
− 1
)
+
√
3
2
√
2
AT.
Then, under the assumptions
(i) τ ≥ T 2
4
(
AT
τ
+ g̃th
)
e2m2 +
T
2
(e2m2 + 1),
(ii) τ g̃the2m1 ≥ 1
4
(e2m2 + 1)2 +ATe2m2 ,
the T -periodic solution given by Theorem 1 is unique and asymptotically stable.
From this latter result, the following consequence is direct.
Corollary 1. There exists an explicitly computable τ0 (depending on the rest of parameters A,
ω, gth, M, ∆) such that for any τ > τ0, system (1) has a unique T -periodic solution which is
asymptotically stable.
The paper is structured as follows: after this introduction, the existence result is proved in
Section 2, by using a transformation to a Liénard equation and a classical topological degree
argument. Section 3 is devoted to the uniqueness and asymptotic stability. We use a classical
stability criterium by Erbe [6]. Finally, in Section 4 some final remarks are given.
2. A priori bounds and existence. The first step is to write system (1) as an equivalent
Liénard equation. SinceE is the amplitude of the locked field, it is always positive. By introduci-
ng the change of variable E = ex, the second equation of system (1) is written as
ẋ =
1
2
(g(t)− g̃th).
Deriving this equation and substituting into the first equation of system (1), one gets the Liénard
equation
ẍ+ f(x)ẋ+ h(x) =
1
2τ
g0(t), (3)
ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 3
394 P. J. TORRES
where f(x) =
1
τ
(1 + e2x), h(x) =
1
2τ
g̃th(1 + e2x). As it was indicated in the Introduction,
g0(t) = A (1 + sinωt) . Needless to say, Eq. (3) is equivalent to system (1), and from a given
solution x of (3) one can recover the original solution
E = ex, g = 2ẋ+ g̃th. (4)
We begin the study of Eq. (3) with a result on a priori bounds.
Lemma 1. Any eventual T -periodic solution of (3) verifies the bounds
m1 < x(t) < m2 (5)
and
‖ẋ(t)‖ < m3 :=
AT
2τ
(6)
for every t, with m1, m2 as defined in Theorem 2.
Proof. Let us assume that x(t) is a given T -periodic solution. By integrating the equation
over [0, T ] one gets
g̃th
T∫
0
(1 + e2x(t)) dt = AT. (7)
By the integral Mean Value Theorem, there exists t0 ∈]0, T [ such that
x(t0) =
1
2
ln
(
A
g̃th
− 1
)
. (8)
On the other hand, multiplying (3) by ẋ and integrating over a period,
1
τ
‖ẋ‖22 <
1
τ
T∫
0
(1 + e2x)ẋ2dt =
1
2τ
T∫
0
g0(t)ẋdt ≤
1
2τ
‖g0(t)‖2 ‖ẋ‖2, (9)
after a basic application of Cauchy – Bunyakowskii – Schwarz inequality. Hence,
‖ẋ‖2 <
1
2
‖g0(t)‖2 =
A
2
√
3T
2
. (10)
Now, for every t ∈ [0, T ] we have
|x(t)− x(t0)| =
∣∣∣∣∣∣
t∫
t0
ẋ(s)ds
∣∣∣∣∣∣ ≤ ‖ẋ‖1 ≤ ‖ẋ‖2√T <
√
3
2
√
2
AT,
ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 3
STABILIZATION OF OPTICALLY COUPLED LASERS . . . 395
as a result of Cauchy – Bunyakowskii – Schwarz inequality and (10). From this inequality and
(8), (5) is easily obtained.
The next aim is to prove (6). Let us take t∗ ∈ [0, T ] such that x(t∗) = mint x(t), then for
any t ∈]t∗, t∗ + T [ one has
ẋ(t) =
t∫
t∗
ẍ(s) ds = −
t∫
t∗
f(x(s))ẋ(s) ds−
t∫
t∗
h(x(s)) ds+
1
2τ
t∫
t∗
g0(s) ds <
< −
x(t)∫
x(t∗)
f(s) ds+
1
2τ
‖g0‖1 ≤
1
2τ
‖g0‖1 =
AT
2τ
, (11)
where we have used that f, h are positive functions and g0(t) is non-negative. In a similar way,
let us take t∗ ∈ [0, T ] such that x(t∗) = maxt x(t), then for any t ∈]t∗, t∗ + T [ one has
ẋ(t) =
t∫
t∗
ẍ(s) ds = −
t∫
t∗
f(x(s))ẋ(s) ds−
t∫
t∗
h(x(s)) ds+
1
2τ
t∫
t∗
g0(s)ds >
>
x(t∗)∫
x(t)
f(s) ds−
t∗+T∫
t∗
h(x(s)) ds ≥ −AT
2τ
, (12)
where (7) has been used in the last inequality. From (11) and (12), one gets (6).
Lemma 1 is proved.
Obviously, the previous result gives explicit bounds for the eventual T -periodic solutions of
the original system (1), which are specified in the lemma below since they may be of independent
interest for the physical model.
Lemma 2. Any eventual T -periodic solution (g,E) of system (1) verifies the bounds
(
A
g̃th
− 1
) 1
2
e
−
√
3
2
√
2
AT
< E(t) <
(
A
g̃th
− 1
) 1
2
e
√
3
2
√
2
AT
,
(13)
−AT
τ
+ g̃th < g(t) <
AT
τ
+ g̃th
for every t.
Proof. Just use (5), (6) into (4).
Proof of Theorem 1. Let us prove that (2) is necessary and sufficient for the existence of
a T -periodic solution of system (1). The necessity comes from an integration of Eq. (3) over a
whole period, then (7) is obtained, and from there is it evident that g̃th < A.
The sufficient condition follows from classical results on topological degree theory. In fact,
if g̃th < A holds the equation verifies the well-known Landesman – Lazer conditions and for
ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 3
396 P. J. TORRES
instance [7] (Theorem 2), can be directly applied. For completeness, we will give here a sketch
of a different proof. Let us consider the homotopic equation
ẍ+ f(x)ẋ+ h(x) =
1
2τ
[(1− λ)A+ λg0(t)], (14)
with λ ∈ [0, 1]. For λ = 1, it corresponds to Eq. (3). By using the same arguments as in
Lemma 1, one can find uniform bounds (not depending on λ) on the possible T -periodic soluti-
ons of (14) and their derivatives. For λ = 0, we get the autonomous equation
ẍ+ f(x)ẋ+ h(x) =
1
2τ
A,
which is equivalent to the planar vectorial field
F (u, v) =
(
v,−f(u)v − h(u) +
1
2τ
A
)
.
By [8] (Theorem 2), the existence of a T -periodic solution of (3) is proved if the Brouwer
degree of F over a large ball is different from zero. The unique fixed point of F is (u0, v0) =
=
(
1
2
ln
(
A
g̃th
− 1
)
, 0
)
, and after some elementary computations one can see that the Jacobi-
an matrix JF (u0, v0) has positive determinant. Then the Brouwer degree of F over a large ball
is 1 and therefore the proof is finished.
3. Uniqueness and asymptotic stability. In this section, it is assumed that (2) holds. It was
proved in the last section that Eq. (3) has at least one T -periodic solution. The objective is to
prove that, under the assumptions of Theorem 2, such a solution is unique and asymptotically
stable. The following stability result of Erbe [6] (Section 3) will be useful.
Proposition 1. Let us assume that p, q ∈ C(R/TZ) are continuous and T -periodic functions
verifying
(1)
∫ T
0
p(t)dt > 0,
(2)
∫ T
0
q(t)dt+ 2‖p‖∞ ≤
4
T
,
(3) 4q(t) ≥ p(t)2 for every t.
Then, the linear differential equation
ẍ+ p(t)ẋ+ q(t)x = 0 (15)
is asymptotically stable.
For convenience, let us remember that by Lemma 1, any T -periodic solution of (3) verifies
m1 < x(t) < m2, |ẋ(t)| < m3 for every t.
Proof of Theorem 2. As it was noted before, system (1) is equivalent to Eq. (3), hence along
this proof we will work directly with this last equation.
ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 3
STABILIZATION OF OPTICALLY COUPLED LASERS . . . 397
Let us first prove the uniqueness. Assume that x1, x2 are two T -periodic solutions of Eq. (3).
The objective is to prove that the difference d(t) = x1(t)− x2(t) is a solution of a second order
linear equation (15) in the conditions of Proposition 1, then the unique periodic solution of (15)
would be the trivial one, so d(t) ≡ 0.
By subtracting the equations,
d̈+ f(x1)ẋ1 − f(x2)ẋ2 + h(x1)− h(x2) = 0. (16)
By using the Mean Value Theorem
f(x1)ẋ1 − f(x2)ẋ2 = [f(x1)− f(x2)]ẋ1 + f(x2)[ẋ1 − ẋ2] = ḟ(ξ(t))d(t)ẋ1(t) + f(x2(t))ḋ(t)
with ξ(t) a value between x1(t) and x2(t), hence verifying m1 < ξ(t) < m2 for all t. Similarly,
h(x1)−h(x2) = ḣ(ν(t))d(t) with m1 < ν(t) < m2 for all t. Inserting this information into (16),
one finds that d(t) is a solution of a second order linear differential equation like (15) with
p(t) = f(x2(t)) =
1
τ
(
1 + e2x2
)
, q(t) = ḟ(ξ(t))ẋ1(t) + ḣ(ν(t)) =
2
τ
e2ξẋ1 +
g̃th
τ
e2ν .
Let us prove that such coefficients verify the conditions of Proposition 1. First, note that condi-
tion (1) is trivially satisfied because p(t) is positive. On the other hand, by using Lemma 1 and
the monotonicity of the exponential,
−1
τ
(
−2m3e
2m2 + g̃the
2m1
)
< q(t)
1
τ
(2m3 + g̃th) e2m2 , p(t) <
1
τ
(
1 + e2m2
)
.
In consequence,
T∫
0
q(t)dt+ 2‖p‖∞ <
T
τ
(2m3 + g̃th) e2m2 +
2
τ
(
1 + e2m2
)
.
Therefore, (2) holds if
T
τ
(2m3 + g̃th) e2m2 +
2
τ
(
1 + e2m2
)
≤ 4
T
.
After simple computations, one realizes that this is just condition (i) of Theorem 2. Similarly,
q(t) >
1
τ
(
−2m3e
2m2 + g̃the
2m1
)
, p(t)2 <
1
τ2
(
1 + e2m2
)2
.
Hence, condition (3) holds if −8m3e
2m2 + 4g̃the
2m1 ≥ 1
τ
(
1 + e2m2
)2
, and this is equivalent to
condition (ii) of Theorem 2. Therefore, the proof of uniqueness is concluded.
The proof of asymptotic stability is similar. Let x(t) be the unique T -periodic solution of
(3). The linearized equation along x(t) is ÿ + p(t)ẏ + q(t)y = 0, where
p(t) =
1
τ
(
1 + e2x(t)
)
, q(t) =
2
τ
e2x(t)ẋ(t) +
g̃th
τ
e2x(t).
ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 3
398 P. J. TORRES
By performing exactly the same bounds as in the proof of uniqueness, p, q are in the conditions
of Proposition 1. In consequence, the linearized equation along x(t) is asymptotically stable and
the proof is finished.
Proof of Corollary 1. It is clear that conditions (i), (ii) are verified for τ large enough. For
convenience, let us write explicitly a concrete value of τ0. Multiplying (i) by τ, we obtain
τ2 − τ
[
T 2
4
g̃the
2m2 +
T
2
(e2m2 + 1)
]
− 1
4
AT 3e2m2 ≥ 0.
The left-hand side of this inequality is a second-order polynomial, so this is equivalent to assume
that τ is above the positive root of such a polynomial, that is,
τ ≥ R1 :=
T 2
8
g̃the
2m2 +
T
4
(e2m2 + 1) +
1
2
([
T 2
4
g̃the
2m2 +
T
2
(e2m2 + 1)
]2
+AT 3 e2m2
) 1
2
.
On the other hand, (ii) holds if τ ≥ R2 := e−2m1
g̃th
[1
4
(
e2m2 + 1
)2
+ ATe2m2
]
. The proof is
finished by taking
τ0 = max{R1, R2}. (17)
4. Concluding remarks. In this paper, we have proved (Theorem 1) a necessary and sufficient
condition for existence of T -periodic solution of system (1), which model the synchronization of
two optically coupled lasers pumped by an alternating current. Explicit bounds for the solution
are given (Lemma 1).
Besides, a sufficient condition in terms of the involved parameters is given in order to get
uniqueness and asymptotic stability of such a solution (Theorem 2). In Corollary 1, the stability
condition is interpreted in the following way: the effective relaxation time of the active medium
τ should be greater than a given computable quantity τ0. From the physical point of view, this
condition makes sense because τ is much more higher than the unit time (τ >> 1, see [3]).
Surely, the sufficient condition for stability is far from being optimal. To improve it, there
are two possibilities: (1) to apply other stability criteria for the linear second order equation, (2)
to improve the bounds obtained in Lemma 1. The first way opens as many variants as stability
criteria available in the literature (see for instance [9 – 11] and their references). We have chosen
Erbe’s result for the sake of simplicity.
As for the option of improving the bounds in Lemma 1, one can use a recursive procedure
as follows. Once we know that every solution verifies m1 < x(t) < m2, (9) can be improved to
1 + e2m1
τ
‖ẋ‖22 <
1
τ
T∫
0
(1 + e2x)ẋ2 dt =
1
2τ
T∫
0
g0(t)ẋdt ≤
1
2τ
‖g0(t)‖2 ‖ẋ‖2.
Hence, (10) is improved to
‖ẋ‖2 <
1
2(1 + e2m1)
‖g0(t)‖2 =
A
2(1 + e2m1)
√
3T
2
,
ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 3
STABILIZATION OF OPTICALLY COUPLED LASERS . . . 399
and repeating the arguments, one gets that m1
1 < x(t) < m1
2, with
m1
1 :=
1
2
ln
(
A
g̃th
− 1
)
−
√
3AT
2
√
2(1 + e2m1)
, m2
2 :=
1
2
ln
(
A
g̃th
− 1
)
+
√
3AT
2
√
2(1 + e2m1)
.
This trick can be repeated recursively giving rise to monotone and convergent sequences mn
1 ,
mn
2 such that mn
1 < x(t) < mn
2 for every n ∈ N.
Finally, we observe that different bounds for x(t) can be derived by applying (6) and (8) into
the expression x(t) = x(t0) +
∫ t
t0
ẋ(s) ds, thus obtaining
1
2
ln
(
A
g̃th
− 1
)
− AT 2
2τ
< x(t) <
1
2
ln
(
A
g̃th
− 1
)
+
AT 2
2τ
.
Such bound are sharper than m1,m2 in the case when τ is a high value.
In a private communication, professors D. M. Lila and A. A. Martynyuk pointed out that
on this model the physical range for the dimensionless field amplitude E is [0, 1]. In this sense,
the problem would amount to find a non-positive periodic solution x(t) ≤ 0 of Eq. (3). In
view of (8), the necessary condition can be fixed more accurately as g̃th < A ≤ 2g̃th. The
question if this condition is also sufficient for existence of a non-positive periodic solution of
Eq. (3) is an interesting open problem. By using Lemma 2, the more conservative estimate
g̃th < A ≤ g̃th(1 + e−AT
√
3/2) can be given as a sufficient condition. The author warmly thanks
professors D. M. Lila and A. A. Martynyuk for this remark.
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72, Issue 13. — P. 2009 – 2012.
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Received 12.11.10,
after revision — 08.03.11
ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 3
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