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Cluster states in a ring of four coupled semiconductor lasers
Communications in Nonlinear Science and Numerical Simulation 12 (2007) 942–952 www.elsevier.com/locate/cnsns
Cluster states in a ring of four coupled semiconductor lasers L. Poinkam Meffo, P. Woafo *, S. Domngang Department of Physics, Faculty of Sciences, University of Yaounde I, Box 812, Yaounde, Cameroon Received 31 August 2005; received in revised form 5 October 2005; accepted 5 October 2005 Available online 22 November 2005
Abstract By varying the coupling strength, clusters synchronization is found in a ring of four mutually coupled semiconductor lasers with optical feedback. The numerical simulation confirms the mathematical analysis and shows the degree of synchronization in terms of the parameters differences between the lasers. 2005 Elsevier B.V. All rights reserved. PACS: 05.45.Xt; 05.45.Gg Keywords: Semiconductor lasers; Chaos; Cluster states
1. Introduction Chaotic synchronization is one of the extensively studied areas of research, in the last few years [1] since the work of Pecora and Carroll [2]. The problem of chaos synchronization of two or many devices is important in various branches of physics, engineering and biology. In particular in the field of telecommunication, it has been studied for its applications in the masking of communication [3–7]. Since then, several schemes of chaotic synchronization to secure communication have been proposed [8,9]. Amount these, the synchronization of chaotic semiconductor (SC) lasers has been extensively studied [10–13] since they are well suited for high-speed communication networks based on optical fibres. Recently, Locquet et al. [13] found two different synchronization regimes in two external cavity lasers coupled unidirectionally in the master–slave scheme. Following this work, one can ask which types of synchronization states exist in a network of such lasers coupled mutually. Synchronization states in a network could lead to potential applications such as switching in communication technology as suggested recently by Chembo and Woafo [14]. In this paper, we report on the existence of clusters synchronization (type abab) in a ring of four mutually coupled semiconductor lasers with optical feedback.
*
Corresponding author. Tel.: +237 9980567; fax: +237 222262. E-mail address: [email protected] (P. Woafo).
1007-5704/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2005.10.002
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The paper is organized as follows. In the next section, the model of four mutually coupled semiconductor lasers with optical feedback is described. Section 3 deals with the numerical simulation to show the existence of cluster states. In Section 4, an analytical approach is conducted for the stability of the synchronized states. Conclusion is given in Section 5. 2. The model The system shown in Fig. 1 consists of four identical semiconductor lasers with optical feedback from external mirrors. We assume that the mirrors are positioned such that the external cavity length (defined as the distance between the laser facet and the mirror) is the same for all the four lasers. The output of each laser is injected into the other one via an optical isolator (OI). Each laser is subject to external optical feedback and to optical injection. The injection is described by adding a suitable forcing term to the standard laser equations. Following the Lang Kobayashi approach [15], the rate equations for the lasers are: dEi 1 g½N i N 0 1 ¼ Ei ðtÞ þ ci Ei ðt sÞ cos½ðx0 sÞi þ /i ðtÞ /i ðt sÞ 2 1 þ sE2i ðtÞ sp dt þ cext Eiþ1 ðt sc Þ cos½ðx0;i x0;iþ1 Þt þ /i ðtÞ /iþ1 ðt sc Þ þ x0;iþ1 sc þ cext Ei1 ðt sc Þ cos½ðx0;i x0;i1 Þt þ /i ðtÞ /i1 ðt sc Þ þ x0;i1 sc d/i a g½N i N 0 1 Ei ðt sÞ ¼ sin½ðx0 sÞi þ /i ðtÞ /i ðt sÞ ci 2 1 þ sE2i ðtÞ sp Ei ðtÞ dt Eiþ1 ðt sc Þ cext sin½ðx0;i x0;iþ1 Þt þ /i ðtÞ /iþ1 ðt sc Þ þ x0;iþ1 sc Ei ðtÞ Eiþ1 ðt sc Þ sin½ðx0;i x0;i1 Þt þ /i ðtÞ /i1 ðt sc Þ þ x0;i1 sc Ei ðtÞ dN i I N i ðtÞ ½N i ðtÞ N 0 2 E ðtÞ ¼ g e ss dt 1 þ sE2i ðtÞ i
ð1Þ
ð2Þ
ð3Þ
where i = 1, 2, 3, 4 refers to the lasers, with X0 = X4 and X5 = X1.
Fig. 1. Circuit diagram of a ring of four coupled semiconductor lasers. (1)–(4) refers to the lasers, (OI) is the optical isolator and M is the mirror.
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cext and ci are respectively the external injection rate and the feedback rate of the laser i. Ei(t), /i(t) and Ni(t) are respectively the amplitude of the electric field (expressed in arbitrary units), the slowly varying phase of the electric field and the carrier number for laser i. The internal parameters of the lasers are the gain g = 1.5 · 104 s1, the gain saturation coefficient s = 5 · 107, the carrier number at transparency N0 = 1.5 · 108, the line width enhancement factor a = 5, the photon lifetime sp = 2 ps, and the carrier lifetime ss = 2 ns. The operating parameters are the injection current I = 44 mA (e is the elementary charge), the external cavity round-trip time s = 0.2 ns, the light propagation time between the output facet of one laser and the facet through which light is injected to the second laser sc = 10 ns. Finally, the phase mismatch after one round trip is (w0s) = 2.5 rad (mod 2p). The semiconductor lasers parameters values are taken from Ref. [13]. For theses parameters values, each laser operates in the chaotic regime. 3. Numerical simulation Assuming the state abab exists (lasers 1 and 3 synchronize as do lasers 2 and 4), the set of Eqs. (1)–(3) reduces to the 2 · 3D partially synchronous invariant sub manifold defined by ~ r1 ¼ ~ r3 and ~ r2 ¼ ~ r4 where ~ ri ¼ ðEi ; /i ; N i Þ. Using the analytical formalism of Ref. [16], we make the following transformation ~ R1 ¼ ~ r1 þ~ r3 ¼ ðEt1 ; /t1 ; N t1 Þ and ~ R2 ¼ ~ r2 þ~ r4 ¼ ðEt2 ; /t2 ; N t2 Þ. Therefore, the motion of this state is described by the following equations: " # t t 1 ðN N Þ 1 t 1 0 gt E_ 1 ¼ Et1 þ cEt1 ðt sÞ cosðw0 s þ /tt1 ðtÞ /tt1 ðt sÞÞ t t 2 2 s p 1 þ S ðE1 Þ þ 2cext Et2 ðt sc Þ cosðw0 s þ Dwt þ /tt1 ðtÞ /tt2 ðt sc ÞÞ " # t t t t a ðN N Þ 1 Et1 ðt sÞ t 1 0 /_ 1 ¼ g sinðw0 s þ /tt1 ðtÞ /tt1 ðt sÞÞ 2c t 2 t t s E 2 ðtÞ p 1 þ S ðE1 Þ 1 4cext
Et2 ðt sc Þ sinðw0 s þ Dwt þ /tt1 ðtÞ /tt2 ðt sc ÞÞ Et1 ðtÞ
2I N t gt ðN t1 N t0 Þ t 2 t N_ 1 ¼ 1 ðE Þ e ss 2 1 þ S t ðEt1 Þ2 1 " # t t _Et ¼ 1 gt ðN 2 N 0 Þ 1 Et þ cEt ðt sÞ cosðws þ /tt ðtÞ /tt ðt sÞÞ 2 2 0 2 2 2 2 sp 2 1 þ S t ðEt2 Þ þ 2cext Et1 ðt sc Þ cosðw0 s þ Dwt þ /tt2 ðtÞ /tt2 ðt sc ÞÞ " # t t t t _/t ¼ a gt ðN 2 N 0 Þ 1 2c E2 ðt sÞ sinðw0 s þ /tt ðtÞ /tt ðt sÞÞ 2 2 2 2 sp Et2 ðtÞ 2 1 þ S t ðEt2 Þ 4cext
Et1 ðt sc Þ sinðw0 s þ Dwt þ /tt2 ðtÞ /tt1 ðt sc ÞÞ Et2 ðtÞ
2I N t gt ðN t2 N t0 Þ t 2 t N_ 2 ¼ 2 ðE Þ e ss 2 1 þ S t ðEt2 Þ2 2
ð4Þ
ð5Þ ð6Þ
ð7Þ
ð8Þ ð9Þ
where gt = g/2, at = 2a, St = s/4, /tt = /t/2, N t0 ¼ 2N 0 and c = ci. One can thus consider the dynamics of the infinitesimal perturbation of ~ R1 and ~ R2 to calculate the largest Lyapunov exponent k1 and k2 for the perturbations of the states a and b respectively. As Fig. 2 clearly reveals, the cluster solution corresponds to a chaotic one.
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0.27 0.26 0.25 0.24
λ1 0.23 0.22 0.21 0.2 0.19 0
2
4
6 γ ext (ns-1)
8
10
12
Fig. 2. The largest Lyapunov exponents k1 (for the cluster a) versus the coupling parameter cext in the submanifold (abab).
To confirm the above analytical treatment, the set of equations (1)–(3) is solved numerically, the four lasers starting with different initial conditions. We assume that the parameters of the four lasers are identical as given here before and we set ci = 11 ns1. The control parameter is the coupling rate coefficient cext. The indicator of the clusters synchronized states is the quantity gc ¼ hjE4 ðtÞ E2 ðtÞj þ jE3 ðtÞ E1 ðtÞji
ð10Þ
where the brackets hÆi stand for the time average. If gc = 0, we are in the clusters state where E1 = E3 and E2 = E4: lasers 1 and 3 synchronize their behaviour as lasers 2 and 4, but with E1 5 E2. In Fig. 3, the variations of g are represented as a function of cext varying from 0 to 10 ns1. It appears that gc remains near the value gc = 0 in all the range considered. Fig. 4 confirms the clusters states in the phase diagrams (E1, E3) and (E2, E4) for cext = 5 ns1. The corresponding synchronization errors versus time are shown in Fig. 5. When the synchronization is achieved, each laser behaves chaotically as represented in Fig. 6. This chaotic behaviour is observed over the range of cext considered (see the bifurcation diagram of Fig. 7). Since during the fabrication, it is difficult to have lasers with the same parameters, there is a particular interest to take into account the influence of the parameters mismatch on the existence of clusters states. Indeed, as chaos communication is concerned, the parameters mismatch has a strong influence on the bit error rate [17]. To analyze this influence in this report, the parameters of the lasers have been varied randomly within 3% and the corresponding gc is plotted in Fig. 8(a). Comparing with Fig. 3, one finds that the perfect synchronization obtained for high values of cext is degraded by the parameters mismatch. In the temporal domain, the degradation manifests itself by strong bursts or deviations as it appears in Fig. 8(b). However as shown in Fig. 8(a),
Fig. 3. Variations of gc as a function of cext.
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800 750 700
E1(t)
650 600 550 500 450
(a)
400 400 450 500 550 600 650 700 750 800 E 3(t)
800 750 700
E 2 (t)
650 600 550 500 450 400
(b)
350 350 400 450 500 550 600 650 700 750 800 E 4(t)
Fig. 4. (a) Amplitude of the electric field of laser 1 versus the amplitude of the electric field of laser 3 showing synchronization with cext = 5 ns1. (b) Amplitude of the electric field of laser 2 versus the amplitude of the electric field for laser 4 showing synchronization with cext = 5 ns1.
despite the parameters mismatch, there exists a domain around cext = 1.5 ns1 where good synchronization is achieved. 4. Linear stability of the cluster state To investigate the linear stability of the existed abab state, we assume that in the synchronization manifold, lasers 1 and 3 are governed by the same dynamics (seat a) as lasers 2 and 4 (seat b). Therefore, their dynamics is described by the following equations where we have used the complex notation to reduce the number of variables: dE1 ðtÞ 1 þ ja gðN 1 N 0 Þ 1 ¼ ð11Þ E1 þ cE1 ðt sÞ expðjw0 sÞ þ 2cext E2 ðt sc Þ expðjw0 sc Þ dt 2 sp 1 þ sjE21 j dN 1 ðtÞ I N 1 gðN 1 N 0 Þ 2 ¼ ð12Þ jE1 j dt e ss 1 þ sjE21 j dE2 ðtÞ 1 þ ja gðN 2 N 0 Þ 1 ¼ ð13Þ E2 þ cE2 ðt sÞ expðjw0 sÞ þ 2cext E1 ðt sc Þ expðjw0 sc Þ dt 2 sp 1 þ sjE22 j dN 2 ðtÞ I N 2 gðN 2 N 0 Þ 2 ¼ jE2 j ð14Þ dt e ss 1 þ sjE22 j According to the above set of equations, the abab state can be regarded as a system of two mutually coupled semiconductors lasers with optical feed back. Let us assume that in their synchronization manifold, there is a stationary state
E1(t)- E3(t)
L.P. Meffo et al. / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 942–952 6e-08 5e-08 4e-08 3e-08 2e-08 1e-08 0 -1e-08 -2e-08 -3e-08 -4e-08 0
E 2(t)- E 4(t)
(a)
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10 20 30 40 50 60 70 80 90 t(ns)
1e-06 8e-07 6e-07 4e-07 2e-07 0 -2e-07 -4e-07 -6e-07 0
10
20
30
40
50
60
70
80
90
t(ns)
(b) 1
E1(t)
Fig. 5. (a) Time series of the synchronisation error for cext = 5 ns showing synchronization between the outputs of lasers 1 and 3. (b) Time series of the synchronisation error cext = 5 ns1 showing synchronization between the outputs of lasers 2 and 4.
800 750 700 650 600 550 500 450 400 0
10
20
800 750 700 650 600 550 500 450 400 350 0
10
20
30
E2(t)
(a)
(b)
30
40
50 t(ns)
40 50 t(ns)
60
60
70
70
80
80
90
90
Fig. 6. (a) Time series of the electric field of laser 1 for cext = 5 ns1. (b) Time series of the electric field of laser 2 for cext = 5 ns1.
Ei ðtÞ ¼ Eist ;
N i ðtÞ ¼ N ist
around which the chaotic lasers evolve. Eist and Nist are given by the following equations:
ð15Þ
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Fig. 7. Bifurcation diagram of laser 1 versus cext.
gc
180 160 140 120 100 80 60 40 20 0
0
1
2
3
4
(a)
5 6 7 γ ext (ns-1)
8
9 10
200 150
E 1(t)-E3 (t)
100 50 0 -50 -100 -150
0
10
(b)
20
30 t(ns)
40
50
60
Fig. 8. (a) Variations of gc versus cext in case of parameters mismatch. (b) Time series of the synchronization error showing bursts in the case of parameters mismatch (same as in a) for cext = 5 ns1.
E1st ¼
1þja
2
2cext E1st expðjw0 sc Þ gðN 0 N 2st Þ 1 þ c expðjw0 sÞ sp 2 1þsjE22st j
2
bIss ð1 þ s E1st Þ þ egN 0 ss E21st c
¼ e½1 þ s E21st þ g E21st ss
bIss ð1 þ s E22st Þ þ egN 0 ss E22st c
¼ e½1 þ s E2 þ g E2 ss
E2st ¼
N 1st N 2st
2cext E2st expðjw0 sc Þ gðN 0 N 1st Þ 1 þ 1þs E2 c expðjw0 sÞ sp j 1st j
1þja
2st
2st
ð16Þ
ð17Þ
ð18Þ ð19Þ
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Inserting Eqs. (18) and (19) into Eqs. (17) and (18), one obtains
2
E1st
4c2ext
¼
E2 ð1 þ a2 ÞA2 2c bA þ c2 1 1 2st 1
2
2
E2st
4cext
¼
E2 ð1 þ a2 ÞA2 2c bA þ c2 2 1 1st 2
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ð20Þ ð21Þ
where
" # 1 gðeN 0 Iss Þ 1
þ A1 ¼ 2 e 1 þ ðgss þ sÞ E21st
sp " # 1 gðeN 0 Iss Þ 1
þ A2 ¼ 2 e 1 þ ðgss þ sÞ E22st
sp
ð22Þ ð23Þ
and b ¼ cosðwsÞ þ a sinðwsÞ
ð24Þ
Combining Eqs. (20) and (21), one obtains after some arrangements the following nonlinear algebraic equation a8 A81 þ a7 A71 þ a6 A61 þ a5 A51 þ a4 A41 þ a3 A31 þ a2 A21 þ a1 A11 þ a0 ¼ 0 where the coefficients ai are given in Appendix.
Fig. 9. Variation of the stationary solutions as a function of cext: (a) E1st; (b) E2st; (c) N1st; (d) N2st.
ð25Þ
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Eq. (25) is solved numerically and the stationary states E1st, E2st, N1st and N2st are plotted in Fig. 9 as functions of the external feedback coupling. Considering a small perturbation from the synchronized manifold in the form x(t) = xst + dxexp(kt) with x = (Ei, Ni), we obtain the following linearized equations for the perturbation in matrix form: 2
a1 þ m k d1 6 q1 k þ p1 6 6 4 n 0 0 0
n 0
0 0
a2 þ m k q2
d2 k þ p2
32
3 dE1 76 dN 7 76 1 7 76 7¼0 54 dE2 5 dN 2
where # # " " 1 þ ja gðN 1st N 0 Þ 1 1 þ ja gðN 2st N 0 Þ 1
a1 ¼ ; a2 ¼ ; 2 sp 2 sp 1 þ s E21st
1 þ s E22st
# # " " 1 þ ja gE1st 1 þ ja gE2st
; d2 ¼
; d1 ¼ 2 2 1 þ s E21st
1 þ s E22st
" m ¼ c expðksÞ expðjw0 sÞ; "
# gE22st 1
þ ; p2 ¼ 1 þ s E22st ss
q1 ¼
n ¼ 2cext expðksc Þ expðjw0 sc Þ; 2gðN 1st N 0 ÞE1st
; 1 þ s E21st
q2 ¼
# gE21st 1
þ p1 ¼ ; 1 þ s E21st ss
2gðN 2st N 0 ÞE2st
1 þ s E22st
The determinant D of the coefficient matrix is given by D ¼ k4 ða1 þ a2 þ 2m p1 p2 Þk3 ½p2 ða2 þ mÞ d 2 q2 ða1 þ m p1 Þða2 þ m p2 Þ n2 þ p1 ða1 þ mÞ d 1 q1 k2 þ ½ðp2 ða2 þ mÞ d 2 q2 Þða1 þ m þ p1 Þ þ ða2 þ m p2 Þðp1 ða1 þ mÞ d 1 q1 Þ þ n2 ðp1 þ p2 Þk þ ðp1 ða1 þ mÞ d 1 q1 þ p2 ða2 þ mÞ d 2 q2 þ p1 p2 n2 Þ
ð26Þ
Let us set k = + jx where and x represent the decay rate and oscillation frequency of the linear mode, respectively. Hence, the synchronized state abab is stable if < 0. Such analysis has been used in Ref. [18] for the stability of synchronization of two semiconductor lasers with optical feedback coupled in the master–slave scheme. Fig. 10 shows the variation of the different values of (real parts of the solution of the equation D = 0) as cext varies. The cluster states are thus stable in the range of cext considered.
Fig. 10. Distribution of the real part of the eigenvalue of the determinant of the stability matrix versus cext for the state abab.
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5. Conclusion This paper has considered a network of four mutually coupled semiconductor lasers with optical feedback. The coupling between the four lasers is ensured by injecting a fraction of each laser output into the neighbouring lasers via optical isolator. Numerical simulations have shown that cluster states of the type abab do exist even in the case of parameters mismatch. An analytical investigation has shown that the cluster states are stable against perturbations. As the chaos masking communication engineering is concerned, the type of synchronization observed in this paper can be assimilated to switching states in a chaos secure network. Indeed, two secure and exclusive groups of working stations can be considered corresponding to lasers 1 and 3 for the first group and lasers 2 and 4 for the second group. Extension of the study is possible in a ring of more than four lasers. Appendix. Coefficients of the algebraic equation (25) a8 ¼ pe0 g3al ;
a7 ¼ 2e0 e1 pg3al 6cbpe0 g2al ;
a6 ¼ pe0 g3 þ 12e0 e1 pcbg2al þ pe1 g3al þ g2al g1
a5 ¼ q0 g2al þ 2e0 e1 pg3 g2 pe0 6g6 4cbg1 gal a4 ¼ g0 pe0 þ g3 pe1 2e0 e1 pg2 þ ð2gal b2 þ 4c2 b2 Þg1 4cbgal q3 þ g2al g8 32gal ga4 þ q0 e2 gal a3 ¼ 2e0 e1 pg0 6c2 bpe0 g2 pe1 4c3 b1 g1 þ ð2gal c2 þ 4c2 b2 Þq3 4gal cbg7 þ 2ga4 cb gal ga1 þ 2q0 ½gal ðe2 e3 e2 cbÞ a2 ¼ c6 pe0 12c5 bpe0 e1 þ pe1 g0 þ c4 g1 þ ð2gal c2 þ 4c2 b2 Þðq3 þ g7 Þ þ gal ðq0 e23 ga3 Þ c2 ga4 þ 2cbga1 þ 16c4ext v2 e22 4q0 e2 e3 cb a1 ¼ 12c2 e0 e1 p 6pe1 c5 b þ c4 q3 4c3 bg7 2cbðq0 e23 ga3 Þ c2 ga1 ga5 þ 2q0 c2 e2 e3 a0 ¼ pe1 c6 þ c4 g8 þ c2 ½q0 ðe22 þ e23 Þ ga3 16v2 e23 c4ext with e0 ¼ gsp ðeN 0 Iss Þ þ e; gal ¼ 1 þ a2 ; q¼
e1 ¼ 2esp ;
!
1 þ a2 ck1 þ þ c2 ; 4s2p sp
p¼
ð1 þ a2 Þu 2ucbsp ; sp
g1 ¼ q00 e0 e2 16e20 c4ext ;
q00 ¼ q þ 2pv;
e2 ¼ 2eðgss þ sÞsp ; v¼
4c2ext ; gss þ s
h ¼ gal u2 ;
g2 ¼ 12c3 bgal þ 8c3 b3 ;
u¼
e3 ¼ eðgss þ sÞ 2gc2ext ðeN 0 Iss Þ eðgss þ sÞ
pe0 ¼ pe20 ;
pe1 ¼ pe21
g3 ¼ 3c2 g2al þ 12c2 bgal ;
g4 ¼ 12e0 e1 pcbg2al
g0 ¼ 3c4 gal þ 12c4 b2 g7 ¼ q00 e1 e3 16e21 c4ext ; q3 ¼ q00 ðe0 e3 þ e1 e2 Þ ga3 ¼ 32ve1 e3 c4ext ;
g6 ¼ pcbe21 g2al ; 32e0 e1 c4ext ;
q0 ¼ qv þ pv2 þ h
ga1 ¼ 32vc4ext ðe0 e3 þ e1 e2 Þ;
ga4 ¼ 32ve0 e2 c4ext ;
ga2 ¼ 32ve1 e2 c4ext
ga5 ¼ 32ve3 e2 c4ext
References [1] [2] [3] [4]
Boccaletti S, Kurths J, Osipov G, Valladares DL, Zhou CS. The synchronization of chaotic systems. Phys Rep 2002;366:1–101. Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64:821–4. Bunner MJ, Just W. Synchronization of time delay systems. Phys Rev E 1998;58:R4072–5. Pecora LM, Carroll TL, Johnson G, Mar D. Volume-preserving and volume-expanding synchronized chaotic systems. Phys Rev E 1997;56:5090–100. [5] Colet P, Roy R. Digital communication with synchronized chaotic lasers. Opt Lett 1994;19:2056–8. [6] Perez G, Cerdeira H. Extracting messages masked by chaos. Phys Rev Lett 1994;74:1970–3.
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[7] Woafo P. Transitions to chaos and synchronization in a nonlinear emitter–receiver system. Phys Lett A 2000;267:31–9. [8] Mirasso CR, Colet P, Garcia-Fernandez P. Synchronization of chaotic semiconductor lasers: application to encoded communications. IEEE Photon Technol Lett 1996;8:299–301. [9] Chembo Kouomou Y, Woafo P. Stability and optimization of chaos synchronization through feedback coupling with delay. Phys Lett A 2002;298:18–28. [10] Spencer P, Mirasso CR, Colet P, Shore A. Modeling of optical synchronization of chaotic external-cavity VCSELS. IEEE J Quantum Electron 1998;34:1673–9. [11] Goedgebuer P, Larger L, Porte H. Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode. Phys Rev Lett 1998;80:2249–52. [12] Larger L, Goedgebuer JP, Delorme F. Optical encryption system using hyperchaos generated by an optoelectronic wavelength oscillator. Phys Rev E 1998;57:6618–24. [13] Locquet A, Rogister F, Sciamanna M, Megret P, Blondel M. Two types of synchronization in unidirectionally coupled chaotic external-cavity semiconductor lasers. Phys Rev E 2001;64:045203. [14] Chembo Kouomou Y, Woafo P. Cluster synchronization in coupled chaotic semiconductor lasers and application to switching in chaos-secured communication networks. Opt Commun 2003;223:283–93. [15] Lang R, Kobayashi K. External optical feedback effects on semiconductor injection laser properties. IEEE J Quantum Electron 1980;16:347–55. [16] Zhang Y, Hu G, Cerdeira HA, Chen S, Braun T, Yao Y. Partial synchronization and spontaneous spatial ordering in coupled chaotic systems. Phys Rev E 2001;63:026211. [17] Chembo Kouomou Y, Colet P, Gastaud N, Larger L. Effect of parameter mismatch on the synchronization of chaotic semiconductor lasers with electro-optical feedback. Phys Rev E 2004;69:056226; Chembo Kouomou Y, Colet P, Larger L, Gastaud N. Mismatch-induced bit error rate in optical chaos communications using semiconductor lasers with electro-optical feedback. IEEE J Quantum Electron 2005;41:156–63. [18] Murakami A, Ohtsubo J. Chaos synchronization based on a continuous chaos control method in semiconductor lasers with optical feedback. Phys Rev E 2001;63:066203.
Cluster states in a ring of four coupled semiconductor lasers L. Poinkam Meffo, P. Woafo *, S. Domngang Department of Physics, Faculty of Sciences, University of Yaounde I, Box 812, Yaounde, Cameroon Received 31 August 2005; received in revised form 5 October 2005; accepted 5 October 2005 Available online 22 November 2005
Abstract By varying the coupling strength, clusters synchronization is found in a ring of four mutually coupled semiconductor lasers with optical feedback. The numerical simulation confirms the mathematical analysis and shows the degree of synchronization in terms of the parameters differences between the lasers. 2005 Elsevier B.V. All rights reserved. PACS: 05.45.Xt; 05.45.Gg Keywords: Semiconductor lasers; Chaos; Cluster states
1. Introduction Chaotic synchronization is one of the extensively studied areas of research, in the last few years [1] since the work of Pecora and Carroll [2]. The problem of chaos synchronization of two or many devices is important in various branches of physics, engineering and biology. In particular in the field of telecommunication, it has been studied for its applications in the masking of communication [3–7]. Since then, several schemes of chaotic synchronization to secure communication have been proposed [8,9]. Amount these, the synchronization of chaotic semiconductor (SC) lasers has been extensively studied [10–13] since they are well suited for high-speed communication networks based on optical fibres. Recently, Locquet et al. [13] found two different synchronization regimes in two external cavity lasers coupled unidirectionally in the master–slave scheme. Following this work, one can ask which types of synchronization states exist in a network of such lasers coupled mutually. Synchronization states in a network could lead to potential applications such as switching in communication technology as suggested recently by Chembo and Woafo [14]. In this paper, we report on the existence of clusters synchronization (type abab) in a ring of four mutually coupled semiconductor lasers with optical feedback.
*
Corresponding author. Tel.: +237 9980567; fax: +237 222262. E-mail address: [email protected] (P. Woafo).
1007-5704/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2005.10.002
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The paper is organized as follows. In the next section, the model of four mutually coupled semiconductor lasers with optical feedback is described. Section 3 deals with the numerical simulation to show the existence of cluster states. In Section 4, an analytical approach is conducted for the stability of the synchronized states. Conclusion is given in Section 5. 2. The model The system shown in Fig. 1 consists of four identical semiconductor lasers with optical feedback from external mirrors. We assume that the mirrors are positioned such that the external cavity length (defined as the distance between the laser facet and the mirror) is the same for all the four lasers. The output of each laser is injected into the other one via an optical isolator (OI). Each laser is subject to external optical feedback and to optical injection. The injection is described by adding a suitable forcing term to the standard laser equations. Following the Lang Kobayashi approach [15], the rate equations for the lasers are: dEi 1 g½N i N 0 1 ¼ Ei ðtÞ þ ci Ei ðt sÞ cos½ðx0 sÞi þ /i ðtÞ /i ðt sÞ 2 1 þ sE2i ðtÞ sp dt þ cext Eiþ1 ðt sc Þ cos½ðx0;i x0;iþ1 Þt þ /i ðtÞ /iþ1 ðt sc Þ þ x0;iþ1 sc þ cext Ei1 ðt sc Þ cos½ðx0;i x0;i1 Þt þ /i ðtÞ /i1 ðt sc Þ þ x0;i1 sc d/i a g½N i N 0 1 Ei ðt sÞ ¼ sin½ðx0 sÞi þ /i ðtÞ /i ðt sÞ ci 2 1 þ sE2i ðtÞ sp Ei ðtÞ dt Eiþ1 ðt sc Þ cext sin½ðx0;i x0;iþ1 Þt þ /i ðtÞ /iþ1 ðt sc Þ þ x0;iþ1 sc Ei ðtÞ Eiþ1 ðt sc Þ sin½ðx0;i x0;i1 Þt þ /i ðtÞ /i1 ðt sc Þ þ x0;i1 sc Ei ðtÞ dN i I N i ðtÞ ½N i ðtÞ N 0 2 E ðtÞ ¼ g e ss dt 1 þ sE2i ðtÞ i
ð1Þ
ð2Þ
ð3Þ
where i = 1, 2, 3, 4 refers to the lasers, with X0 = X4 and X5 = X1.
Fig. 1. Circuit diagram of a ring of four coupled semiconductor lasers. (1)–(4) refers to the lasers, (OI) is the optical isolator and M is the mirror.
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cext and ci are respectively the external injection rate and the feedback rate of the laser i. Ei(t), /i(t) and Ni(t) are respectively the amplitude of the electric field (expressed in arbitrary units), the slowly varying phase of the electric field and the carrier number for laser i. The internal parameters of the lasers are the gain g = 1.5 · 104 s1, the gain saturation coefficient s = 5 · 107, the carrier number at transparency N0 = 1.5 · 108, the line width enhancement factor a = 5, the photon lifetime sp = 2 ps, and the carrier lifetime ss = 2 ns. The operating parameters are the injection current I = 44 mA (e is the elementary charge), the external cavity round-trip time s = 0.2 ns, the light propagation time between the output facet of one laser and the facet through which light is injected to the second laser sc = 10 ns. Finally, the phase mismatch after one round trip is (w0s) = 2.5 rad (mod 2p). The semiconductor lasers parameters values are taken from Ref. [13]. For theses parameters values, each laser operates in the chaotic regime. 3. Numerical simulation Assuming the state abab exists (lasers 1 and 3 synchronize as do lasers 2 and 4), the set of Eqs. (1)–(3) reduces to the 2 · 3D partially synchronous invariant sub manifold defined by ~ r1 ¼ ~ r3 and ~ r2 ¼ ~ r4 where ~ ri ¼ ðEi ; /i ; N i Þ. Using the analytical formalism of Ref. [16], we make the following transformation ~ R1 ¼ ~ r1 þ~ r3 ¼ ðEt1 ; /t1 ; N t1 Þ and ~ R2 ¼ ~ r2 þ~ r4 ¼ ðEt2 ; /t2 ; N t2 Þ. Therefore, the motion of this state is described by the following equations: " # t t 1 ðN N Þ 1 t 1 0 gt E_ 1 ¼ Et1 þ cEt1 ðt sÞ cosðw0 s þ /tt1 ðtÞ /tt1 ðt sÞÞ t t 2 2 s p 1 þ S ðE1 Þ þ 2cext Et2 ðt sc Þ cosðw0 s þ Dwt þ /tt1 ðtÞ /tt2 ðt sc ÞÞ " # t t t t a ðN N Þ 1 Et1 ðt sÞ t 1 0 /_ 1 ¼ g sinðw0 s þ /tt1 ðtÞ /tt1 ðt sÞÞ 2c t 2 t t s E 2 ðtÞ p 1 þ S ðE1 Þ 1 4cext
Et2 ðt sc Þ sinðw0 s þ Dwt þ /tt1 ðtÞ /tt2 ðt sc ÞÞ Et1 ðtÞ
2I N t gt ðN t1 N t0 Þ t 2 t N_ 1 ¼ 1 ðE Þ e ss 2 1 þ S t ðEt1 Þ2 1 " # t t _Et ¼ 1 gt ðN 2 N 0 Þ 1 Et þ cEt ðt sÞ cosðws þ /tt ðtÞ /tt ðt sÞÞ 2 2 0 2 2 2 2 sp 2 1 þ S t ðEt2 Þ þ 2cext Et1 ðt sc Þ cosðw0 s þ Dwt þ /tt2 ðtÞ /tt2 ðt sc ÞÞ " # t t t t _/t ¼ a gt ðN 2 N 0 Þ 1 2c E2 ðt sÞ sinðw0 s þ /tt ðtÞ /tt ðt sÞÞ 2 2 2 2 sp Et2 ðtÞ 2 1 þ S t ðEt2 Þ 4cext
Et1 ðt sc Þ sinðw0 s þ Dwt þ /tt2 ðtÞ /tt1 ðt sc ÞÞ Et2 ðtÞ
2I N t gt ðN t2 N t0 Þ t 2 t N_ 2 ¼ 2 ðE Þ e ss 2 1 þ S t ðEt2 Þ2 2
ð4Þ
ð5Þ ð6Þ
ð7Þ
ð8Þ ð9Þ
where gt = g/2, at = 2a, St = s/4, /tt = /t/2, N t0 ¼ 2N 0 and c = ci. One can thus consider the dynamics of the infinitesimal perturbation of ~ R1 and ~ R2 to calculate the largest Lyapunov exponent k1 and k2 for the perturbations of the states a and b respectively. As Fig. 2 clearly reveals, the cluster solution corresponds to a chaotic one.
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0.27 0.26 0.25 0.24
λ1 0.23 0.22 0.21 0.2 0.19 0
2
4
6 γ ext (ns-1)
8
10
12
Fig. 2. The largest Lyapunov exponents k1 (for the cluster a) versus the coupling parameter cext in the submanifold (abab).
To confirm the above analytical treatment, the set of equations (1)–(3) is solved numerically, the four lasers starting with different initial conditions. We assume that the parameters of the four lasers are identical as given here before and we set ci = 11 ns1. The control parameter is the coupling rate coefficient cext. The indicator of the clusters synchronized states is the quantity gc ¼ hjE4 ðtÞ E2 ðtÞj þ jE3 ðtÞ E1 ðtÞji
ð10Þ
where the brackets hÆi stand for the time average. If gc = 0, we are in the clusters state where E1 = E3 and E2 = E4: lasers 1 and 3 synchronize their behaviour as lasers 2 and 4, but with E1 5 E2. In Fig. 3, the variations of g are represented as a function of cext varying from 0 to 10 ns1. It appears that gc remains near the value gc = 0 in all the range considered. Fig. 4 confirms the clusters states in the phase diagrams (E1, E3) and (E2, E4) for cext = 5 ns1. The corresponding synchronization errors versus time are shown in Fig. 5. When the synchronization is achieved, each laser behaves chaotically as represented in Fig. 6. This chaotic behaviour is observed over the range of cext considered (see the bifurcation diagram of Fig. 7). Since during the fabrication, it is difficult to have lasers with the same parameters, there is a particular interest to take into account the influence of the parameters mismatch on the existence of clusters states. Indeed, as chaos communication is concerned, the parameters mismatch has a strong influence on the bit error rate [17]. To analyze this influence in this report, the parameters of the lasers have been varied randomly within 3% and the corresponding gc is plotted in Fig. 8(a). Comparing with Fig. 3, one finds that the perfect synchronization obtained for high values of cext is degraded by the parameters mismatch. In the temporal domain, the degradation manifests itself by strong bursts or deviations as it appears in Fig. 8(b). However as shown in Fig. 8(a),
Fig. 3. Variations of gc as a function of cext.
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800 750 700
E1(t)
650 600 550 500 450
(a)
400 400 450 500 550 600 650 700 750 800 E 3(t)
800 750 700
E 2 (t)
650 600 550 500 450 400
(b)
350 350 400 450 500 550 600 650 700 750 800 E 4(t)
Fig. 4. (a) Amplitude of the electric field of laser 1 versus the amplitude of the electric field of laser 3 showing synchronization with cext = 5 ns1. (b) Amplitude of the electric field of laser 2 versus the amplitude of the electric field for laser 4 showing synchronization with cext = 5 ns1.
despite the parameters mismatch, there exists a domain around cext = 1.5 ns1 where good synchronization is achieved. 4. Linear stability of the cluster state To investigate the linear stability of the existed abab state, we assume that in the synchronization manifold, lasers 1 and 3 are governed by the same dynamics (seat a) as lasers 2 and 4 (seat b). Therefore, their dynamics is described by the following equations where we have used the complex notation to reduce the number of variables: dE1 ðtÞ 1 þ ja gðN 1 N 0 Þ 1 ¼ ð11Þ E1 þ cE1 ðt sÞ expðjw0 sÞ þ 2cext E2 ðt sc Þ expðjw0 sc Þ dt 2 sp 1 þ sjE21 j dN 1 ðtÞ I N 1 gðN 1 N 0 Þ 2 ¼ ð12Þ jE1 j dt e ss 1 þ sjE21 j dE2 ðtÞ 1 þ ja gðN 2 N 0 Þ 1 ¼ ð13Þ E2 þ cE2 ðt sÞ expðjw0 sÞ þ 2cext E1 ðt sc Þ expðjw0 sc Þ dt 2 sp 1 þ sjE22 j dN 2 ðtÞ I N 2 gðN 2 N 0 Þ 2 ¼ jE2 j ð14Þ dt e ss 1 þ sjE22 j According to the above set of equations, the abab state can be regarded as a system of two mutually coupled semiconductors lasers with optical feed back. Let us assume that in their synchronization manifold, there is a stationary state
E1(t)- E3(t)
L.P. Meffo et al. / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 942–952 6e-08 5e-08 4e-08 3e-08 2e-08 1e-08 0 -1e-08 -2e-08 -3e-08 -4e-08 0
E 2(t)- E 4(t)
(a)
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10 20 30 40 50 60 70 80 90 t(ns)
1e-06 8e-07 6e-07 4e-07 2e-07 0 -2e-07 -4e-07 -6e-07 0
10
20
30
40
50
60
70
80
90
t(ns)
(b) 1
E1(t)
Fig. 5. (a) Time series of the synchronisation error for cext = 5 ns showing synchronization between the outputs of lasers 1 and 3. (b) Time series of the synchronisation error cext = 5 ns1 showing synchronization between the outputs of lasers 2 and 4.
800 750 700 650 600 550 500 450 400 0
10
20
800 750 700 650 600 550 500 450 400 350 0
10
20
30
E2(t)
(a)
(b)
30
40
50 t(ns)
40 50 t(ns)
60
60
70
70
80
80
90
90
Fig. 6. (a) Time series of the electric field of laser 1 for cext = 5 ns1. (b) Time series of the electric field of laser 2 for cext = 5 ns1.
Ei ðtÞ ¼ Eist ;
N i ðtÞ ¼ N ist
around which the chaotic lasers evolve. Eist and Nist are given by the following equations:
ð15Þ
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Fig. 7. Bifurcation diagram of laser 1 versus cext.
gc
180 160 140 120 100 80 60 40 20 0
0
1
2
3
4
(a)
5 6 7 γ ext (ns-1)
8
9 10
200 150
E 1(t)-E3 (t)
100 50 0 -50 -100 -150
0
10
(b)
20
30 t(ns)
40
50
60
Fig. 8. (a) Variations of gc versus cext in case of parameters mismatch. (b) Time series of the synchronization error showing bursts in the case of parameters mismatch (same as in a) for cext = 5 ns1.
E1st ¼
1þja
2
2cext E1st expðjw0 sc Þ gðN 0 N 2st Þ 1 þ c expðjw0 sÞ sp 2 1þsjE22st j
2
bIss ð1 þ s E1st Þ þ egN 0 ss E21st c
¼ e½1 þ s E21st þ g E21st ss
bIss ð1 þ s E22st Þ þ egN 0 ss E22st c
¼ e½1 þ s E2 þ g E2 ss
E2st ¼
N 1st N 2st
2cext E2st expðjw0 sc Þ gðN 0 N 1st Þ 1 þ 1þs E2 c expðjw0 sÞ sp j 1st j
1þja
2st
2st
ð16Þ
ð17Þ
ð18Þ ð19Þ
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Inserting Eqs. (18) and (19) into Eqs. (17) and (18), one obtains
2
E1st
4c2ext
¼
E2 ð1 þ a2 ÞA2 2c bA þ c2 1 1 2st 1
2
2
E2st
4cext
¼
E2 ð1 þ a2 ÞA2 2c bA þ c2 2 1 1st 2
949
ð20Þ ð21Þ
where
" # 1 gðeN 0 Iss Þ 1
þ A1 ¼ 2 e 1 þ ðgss þ sÞ E21st
sp " # 1 gðeN 0 Iss Þ 1
þ A2 ¼ 2 e 1 þ ðgss þ sÞ E22st
sp
ð22Þ ð23Þ
and b ¼ cosðwsÞ þ a sinðwsÞ
ð24Þ
Combining Eqs. (20) and (21), one obtains after some arrangements the following nonlinear algebraic equation a8 A81 þ a7 A71 þ a6 A61 þ a5 A51 þ a4 A41 þ a3 A31 þ a2 A21 þ a1 A11 þ a0 ¼ 0 where the coefficients ai are given in Appendix.
Fig. 9. Variation of the stationary solutions as a function of cext: (a) E1st; (b) E2st; (c) N1st; (d) N2st.
ð25Þ
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Eq. (25) is solved numerically and the stationary states E1st, E2st, N1st and N2st are plotted in Fig. 9 as functions of the external feedback coupling. Considering a small perturbation from the synchronized manifold in the form x(t) = xst + dxexp(kt) with x = (Ei, Ni), we obtain the following linearized equations for the perturbation in matrix form: 2
a1 þ m k d1 6 q1 k þ p1 6 6 4 n 0 0 0
n 0
0 0
a2 þ m k q2
d2 k þ p2
32
3 dE1 76 dN 7 76 1 7 76 7¼0 54 dE2 5 dN 2
where # # " " 1 þ ja gðN 1st N 0 Þ 1 1 þ ja gðN 2st N 0 Þ 1
a1 ¼ ; a2 ¼ ; 2 sp 2 sp 1 þ s E21st
1 þ s E22st
# # " " 1 þ ja gE1st 1 þ ja gE2st
; d2 ¼
; d1 ¼ 2 2 1 þ s E21st
1 þ s E22st
" m ¼ c expðksÞ expðjw0 sÞ; "
# gE22st 1
þ ; p2 ¼ 1 þ s E22st ss
q1 ¼
n ¼ 2cext expðksc Þ expðjw0 sc Þ; 2gðN 1st N 0 ÞE1st
; 1 þ s E21st
q2 ¼
# gE21st 1
þ p1 ¼ ; 1 þ s E21st ss
2gðN 2st N 0 ÞE2st
1 þ s E22st
The determinant D of the coefficient matrix is given by D ¼ k4 ða1 þ a2 þ 2m p1 p2 Þk3 ½p2 ða2 þ mÞ d 2 q2 ða1 þ m p1 Þða2 þ m p2 Þ n2 þ p1 ða1 þ mÞ d 1 q1 k2 þ ½ðp2 ða2 þ mÞ d 2 q2 Þða1 þ m þ p1 Þ þ ða2 þ m p2 Þðp1 ða1 þ mÞ d 1 q1 Þ þ n2 ðp1 þ p2 Þk þ ðp1 ða1 þ mÞ d 1 q1 þ p2 ða2 þ mÞ d 2 q2 þ p1 p2 n2 Þ
ð26Þ
Let us set k = + jx where and x represent the decay rate and oscillation frequency of the linear mode, respectively. Hence, the synchronized state abab is stable if < 0. Such analysis has been used in Ref. [18] for the stability of synchronization of two semiconductor lasers with optical feedback coupled in the master–slave scheme. Fig. 10 shows the variation of the different values of (real parts of the solution of the equation D = 0) as cext varies. The cluster states are thus stable in the range of cext considered.
Fig. 10. Distribution of the real part of the eigenvalue of the determinant of the stability matrix versus cext for the state abab.
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5. Conclusion This paper has considered a network of four mutually coupled semiconductor lasers with optical feedback. The coupling between the four lasers is ensured by injecting a fraction of each laser output into the neighbouring lasers via optical isolator. Numerical simulations have shown that cluster states of the type abab do exist even in the case of parameters mismatch. An analytical investigation has shown that the cluster states are stable against perturbations. As the chaos masking communication engineering is concerned, the type of synchronization observed in this paper can be assimilated to switching states in a chaos secure network. Indeed, two secure and exclusive groups of working stations can be considered corresponding to lasers 1 and 3 for the first group and lasers 2 and 4 for the second group. Extension of the study is possible in a ring of more than four lasers. Appendix. Coefficients of the algebraic equation (25) a8 ¼ pe0 g3al ;
a7 ¼ 2e0 e1 pg3al 6cbpe0 g2al ;
a6 ¼ pe0 g3 þ 12e0 e1 pcbg2al þ pe1 g3al þ g2al g1
a5 ¼ q0 g2al þ 2e0 e1 pg3 g2 pe0 6g6 4cbg1 gal a4 ¼ g0 pe0 þ g3 pe1 2e0 e1 pg2 þ ð2gal b2 þ 4c2 b2 Þg1 4cbgal q3 þ g2al g8 32gal ga4 þ q0 e2 gal a3 ¼ 2e0 e1 pg0 6c2 bpe0 g2 pe1 4c3 b1 g1 þ ð2gal c2 þ 4c2 b2 Þq3 4gal cbg7 þ 2ga4 cb gal ga1 þ 2q0 ½gal ðe2 e3 e2 cbÞ a2 ¼ c6 pe0 12c5 bpe0 e1 þ pe1 g0 þ c4 g1 þ ð2gal c2 þ 4c2 b2 Þðq3 þ g7 Þ þ gal ðq0 e23 ga3 Þ c2 ga4 þ 2cbga1 þ 16c4ext v2 e22 4q0 e2 e3 cb a1 ¼ 12c2 e0 e1 p 6pe1 c5 b þ c4 q3 4c3 bg7 2cbðq0 e23 ga3 Þ c2 ga1 ga5 þ 2q0 c2 e2 e3 a0 ¼ pe1 c6 þ c4 g8 þ c2 ½q0 ðe22 þ e23 Þ ga3 16v2 e23 c4ext with e0 ¼ gsp ðeN 0 Iss Þ þ e; gal ¼ 1 þ a2 ; q¼
e1 ¼ 2esp ;
!
1 þ a2 ck1 þ þ c2 ; 4s2p sp
p¼
ð1 þ a2 Þu 2ucbsp ; sp
g1 ¼ q00 e0 e2 16e20 c4ext ;
q00 ¼ q þ 2pv;
e2 ¼ 2eðgss þ sÞsp ; v¼
4c2ext ; gss þ s
h ¼ gal u2 ;
g2 ¼ 12c3 bgal þ 8c3 b3 ;
u¼
e3 ¼ eðgss þ sÞ 2gc2ext ðeN 0 Iss Þ eðgss þ sÞ
pe0 ¼ pe20 ;
pe1 ¼ pe21
g3 ¼ 3c2 g2al þ 12c2 bgal ;
g4 ¼ 12e0 e1 pcbg2al
g0 ¼ 3c4 gal þ 12c4 b2 g7 ¼ q00 e1 e3 16e21 c4ext ; q3 ¼ q00 ðe0 e3 þ e1 e2 Þ ga3 ¼ 32ve1 e3 c4ext ;
g6 ¼ pcbe21 g2al ; 32e0 e1 c4ext ;
q0 ¼ qv þ pv2 þ h
ga1 ¼ 32vc4ext ðe0 e3 þ e1 e2 Þ;
ga4 ¼ 32ve0 e2 c4ext ;
ga2 ¼ 32ve1 e2 c4ext
ga5 ¼ 32ve3 e2 c4ext
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