15 May 1997

OPTICS C~~M~NICATI~NS

ELSEVTER Optics Communications 138 (1997) 91-94

Remote synchronization of high-frequency chaotic signals in semiconductor lasers for secure communications

Lutfur Rahman, Guifang Li * , Feng Tian Rociaesrer imtitute of Technology, Departnzent of Electricd Engineering, 79 Lomb Memorial Driue, Rochester, NY 14623, USA

Received 27 November 1996; accepted 6 January 1997

Abstract

We show that a solit~ly-stably remote se~conductor laser can be driven into a chaotic state synchronized iyith the chaotic dynamics OF a local master laser. The master laser, which is also solitarily stable, acquires its own chaotic dynamics when laterally coupled with other semiconductor lasers. Synchronization between the chaotic states of two remotely located lasers is robust for a wide range of parameter values. Transients towards synchronization take place within a time of the order of X0 ns.

Keywords: Chaos; Laser diodes; Nonlinear dynamics

Synchronization of chaotic signals has been an interest- ing topic of research in the last few years [I-S]. Both electrical [l-4] and optical [5-81 systems have been pro- posed to achieve synchronized chaos. Previous investiga- tions in optical systems produced only locally synchro- nized chaos with center frequency in the kKz range [5,6]. Recently, Mirasso and coworkers f&l proposed the use of the external cavity semiconductor laser (ECLD) for the generation of remote high-frequency (GHz range) synchro- nized chaos. In their scheme, two ECLDs with slightly different parameters were coupled to each other in the drive-response configuration with one ECLD serving as the master and the other as the slave. Parameter values were chosen such that both ECLDs were chaotic individu- ally. Synchronization is achieved by injecting the chaotic signal from the master ECLD into the slave ECLD. Exact synchronization is, however, only possible provided that the slave ECLD has a different photon lifetime but other- wise the same parameters as the master ECLD. This is

* Corresponding author. E-mail: gxleee@rit.edu.

necessary to account for the change in dynamics of the slave ECLD due to external injection. Since a change in the photon lifetime is inevitably accompanied by changes in other laser diode parameters, it is expected to be diffi- cult to fabricate such master-slave pairs even on the same chip. We propose here a scheme for generating remote and ~gh-frequency synchronized chaos that involve laser diodes with identical parameters in theory, and is robust enough to allow for slight parameter variations in practice.

The configuration of the proposed scheme is shown in Fig. 1. It consists of a local array of N (Z 2) single-mode semiconductor lasers and a remote laser joined by an injection fiber and an optical isolator. The local array of lasers are solitarily stable but exhibit rich dynamic behav- ior inclu~ng chaos when coupled by means of their evanescent fields. The remote laser #r is designe’d to be nearly identical to laser #l in the local array. The optical isolator is used to ensure the unidirectional injection from laser #2 in the local array into the remote laser #r. We show subs~uendy that, with proper injection from laser #2, the remote laser #r can synchronize with the chaotic output of laser #l.

The dynamic evolution of each laser in the local array

~30-4018/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved. PII 50030-4018~97)00042-4

92 L. Rahman et al. /Optics &ommur~icntions 138 (1997191-94

Remote Synchronized Chaotic Signal

0 / Laser#r /A

Remote Laser

Fig. 1. Scheme for remote synchronization using semiconductor lasers.

is described by the rate equations for the normalized field amplitudes Ej and normalized carrier densities N, [7]

dE. $=(l +ici!)$cj-iTl(<j_, +Ej+l), (1)

’ dNi dr --r,

[P-N,-(1+21\1~):Ejl”],

where j = 1,2,. . . ,N, E, = EN+, = 0, r is the normalized time (by the photon lifetime fP>, ‘;_, is the normalized carrier lifetime t,, a! is the linewidth enhance factor, p is the normalized pump rate relative to threshold, and q1 is the normalized lateral coupling strength. Explicit forms of normalizations are as follows:

Ej = (+gt,) 1’2 Aj, q = $gnthTp( nj/nth - l),

p = &v,T,( P/PC, - I), Pth = QJt,, vl= UnTpK.

Above the complex field and canier density of laser #j are Aj and nj, respectively, g is the differential gain, nth is the threshold carrier density, P is the injection current, ug is the group velocity and K is the lateral coupling constant between adjacent lasers in the local array. With weak lateral coupling, the lasers are stable and evolve via relaxation oscillations into constant steady states ( Ej = fi, Nj = 0). With sufficiently strong coupling, however, the relaxation oscillations evolve into sustained oscillations and then chaos. The rate equations for the remote laser diode are:

2 = (1 +ity)N,.E,.-iqrEz, (3)

The injection rate from laser #2 into the remote laser #r, v?, can be tuned to be nearly identical to the lateral coupling, vl, between the lasers in the local array.

We will first present the results for a 3-element local array. First, exact sync~onization is possible provided that the parameters of these 3 elements and the remote laser as well as 7,. and vl are chosen to be the same. With the following choice of parameters, fP = 1 ps, p = 0.05, a = 5, TV = 2000, remote synchronization can be obtained for a wide range of coupling strengths whereby E, and E,, even though chaotic, follow each other. Fig. 2 shows the magni-

tudes of the fields E, , Er and the difference between them as El and E, evolve from different initial conditions towards synchronization for 7, = vr = 3.2 X 10s4. Syn- chronization is achieved within about 20 ns under a wide range of initial conditions. Second, synchronization is ro- bust with respect to slight variations in parameters of the local array and remote laser. For example, with a 10% difference in coupling strength, v1 = 1 .I vr and 71 = 3.2 X

IO-” the synchronization error [IE,(t)1’ -

lE,~t~ill,,,/~lE,ff~/‘l,, is about 10%. Another measnre of the robustness is the synchronization signal-to-noise ratio (SNR) defined as

where I(t) = l~(t>/*, and B (l/B) is the bandwidth (time) of observation. In Fig. 3, SNR,,, is plotted as a function of time for the above case with 1 GHz observation band- width. The synchronization SNR oscillates between 40 to 120 dB. There is a 4 dB decrease (2 dB increase) if the

I . .

II 1 I

0

I -

-__-+_ 1

0.00 20.00 40.00

Time (ns)

Fig. 2. Evolution of the magnitude of (a) local field E,, (b) remote field E,. and (c) the synchronization error /E, /- I&i in time. Parameter values are q = q. = 3.2 X 10m4, p = 0.05, T? = 2000,

and a=5.

L. Rahman et al. /Optics Communications 138 (1997) 91-94 93

20 ’ 50 60 70 80 90

Time (ns)

Fig. 3. The synchronization SNR as a function of time for 10%

difference in coupling strength and an observation bandwidth of 1

GHz.

signal bandwidth is changed to B = 100 MHz (B = 5

GHz). That is, on a longer time scale, the quality of synchronization is better.

To determine the stability of remote synchronization, we calculated the conditional Lyapunov exponents (CLEs) for the remote subsystem [4]. Fig. 4 shows these CLEs as functions of vr for the same parameters as used for Fig. 2. Remote synchronization is stable and robust when all CLEs are negative. In region (I), 3.7 X low5 < vr < 3.2 X

lop4 (not completely shown), laser #l and #r are syn- chronized but non-chaotic (periodic orbits and on-route to chaos). In region (II), 3.2 X 10m4 < ql < 5.0 X 10e4, re- mote chaotic synchronization is obtained for most values of coupling strength except a periodic-orbit window. The dips in the largest exponent in region (II) of Fig. 4 correspond to periodic solutions. For region (III), ql > 5.0 X 10m4, except a window of synchronized periodic orbits,

.--& 0.4 E

3 0.2

: 0.0

c” ; -0.2

w & c -0.4 2

3 -0.6

4 * -0.8

0.0002 0.0004 0.0006 0.0008

Normalized Coupling Strength

Fig. 4. The conditional Lyapunov exponents for the remote sub- system as a function of the coupling strength. When the largest

CLE becomes positive, remote synchronization of chaotic signals fails.

chaotic dynamics of laser #l and #r are not synchroniza- tion and mutually uncorrelated. The CLEs also determine the rate of approach to synchronization. For q1 = 3.2 X lop4 the CLEs are (-0.21, - 023, - 0.25) bits/ns. The largest CLE -0.21 bits/ns corresponds to an average exponential decay rate of 0.6 dB/ns for the synchroniza- tion error j E, I- 1 E,l, which is in agreement with direct numerical simulations.

A careful comparison of the rate equations can lead to the conclusion that whenever lasers #l and #2 are in different chaotic states, the remote laser #r can be syn- chronized with laser #l. As a result, many local-array configurations are possible. The simplest local-array con- figuration consists of two elements only. For otherwise the same parameter as above, the two-element local array undergoes self-sustained pulsations starting at qr = 5.5 X

10-5. When ql is increased, the local array undergoes a sequence of period-doubling bifurcations and becomes chaotic when qI = 4.57 X 10e4. The output from laser #2 can now be injected into a remote laser which should be nearly identical to laser #I. For the range of coupling strengths 4.57 X lo-” < v1 < 5.01 X 10-4, laser #l and #r are chaotic and synchronized. The conditional Lya- punov exponents for the remote-laser subsystem are nega- tive signifying stable remote synchronization. The basin of attraction for remote synchronization is, however, small. The remote laser #r and laser #l need to be started with their respective fields and carrier densities within about 10% of each other to realize synchronized chaos. This is in contrast to the 3-element configuration which has a much larger parameter range and basin of attraction for remote chaotic synchronization.

In summary, we have demonstrated synchronization of the chaotic state of the remote laser diode to that of one of the laser diodes in the local array. With identical lasers, synchronization is exact. For typical parameter values, the rate of approach to synchronization is 0.6 dB/ns. If the parameter differences are within a few percent, the syn- chronization SNR is greater than 40 dB. High-frequency remote synchronized chaos found here may have potential application in high-speed secure communications.

Acknowledgements

This research is supported in part by the NSF CAREER Program under contract #ECS-9622298, the ONR Young Investigator Program under contract #N00014-95-1-0966, and the DOD DURIP Program under contract #N0014-96- l-1106.

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94 L. Rahman et al. /Optics Communications 138 (1997) 91-94

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