978-1-4673-4714-3/13/$31.00 ©2013 IEEE 670

2013 Ninth International Conference on Natural Computation (ICNC)

Control Of Chaos In A Semiconductor Laser Using Photoelectric Nonlinear Feedback

Sen-Lin YAN Dept. of physics and electronic engineering

Nanjing Xiaozhuang University Nanjing, China

Abstract—A novel nonlinear chaos-control method of a semiconductor laser is presented by adding a control path of photoelectric delayed nonlinear feedback. Chaos-control physical models of an injected semiconductor laser are studied by using photoelectric nonlinear and linear delayed controllers. And chaos-control equations are analyzed. A route to a periodic state or a cycle-one from chaos is found. Chaos-control can be realized. When we adjust the delayed time of the nonlinear and linear photocurrents, the chaotic laser can be conducted to show a single-cycle, a dual-cycle, a cycle-3, a cycle-4 or other multi-cycles. When we modulate the photocurrent nonlinear controller, the chaotic laser can be brought into a single-period, a dual-period, a period-3, a period-4 or other multi-periodic states.

Keywords- chaos; nonlinear-control; semiconductor laser; photocurrent

I. INTRODUCTION Nonlinear phenomena exist widely in actual world.

Nonlinearity can result in chaos under some conditions. Chaos is a kind of ubiquitous and complicated nonlinear behavior. Chaos is also a kind of objective phenomenon in nature. It is very sensitive to its starting condition and has the characteristic of random variety. Long chaotic behavior cannot be forecasted. Since “OGY” method of chaos-control was presented [1], chaos and chaos-control theories have fast developed [2-6]. Many chaos-control techniques have been presented to convert a chaotic motion to a periodic regular motion [2,3]. Chaotic oscillation is educed to some stable periodic orbits by the application of a small perturbation or proportional feedback [4-6]. Both experimentally and numerically, It has been recently shown that chaotic lasers can be stabilized or suppressed by using the current modulation and the optical feedback, and so on [7-9]. Y.Liu, N. Kikuchi and J. Ohtsubo obtained controlling behavior of a semiconductor laser with external optical feedback. G. Levy and A. A. Hardy studied control and suppression of chaos in flared laser systems in a numerical analysis. L.G. Luo and P. L. Chu realized suppression of self-pulsing in erbium-doped fiber lasers. S. L. Yan controlled chaos in a semiconductor laser via weak optical positive feedback and modulating amplitude. S. L. Yan studied on control of chaos in an Er-doped fiber dual-ring laser via external optical injection and shifting optical feedback light. However, few reports on control-chaos of lasers with nonlinear feedback are presented. In this paper we present a novel chaos-control method of photoelectric nonlinear feedback (PNF).

Result will be very helpful to the study of laser system, chaos-control techniques and photoelectric applications.

II. MODEL A model block of chaos-control of an injected laser under

the condition of PNF is illustrated in Fig.1. The injected laser is illustrated in the left where Pm implies the external injection light. Due to the injection light into the laser, the laser output light becomes of quasi-periodic or chaotic state [7,8]. To realize to control the chaotic laser, a chaos-control path of photoelectric nonlinear delayed negative feedback is devised to show in the right of Fig.1. In the chaos-control path, PD is the photoelectric detector. TA is the photoelectric delayed time controller. RL is the linear controller. RN is the nonlinear controller. When the photocurrents passing through the controllers are negatively feed to the current, one can make use of PNF to control chaotic laser.

Laser PD

I

Pm P RL

RN

TA

Chaos-control path

-

Figure 1. Schematics of PNF chaos-control of laser setup.

Considering the photon number and phase of the optical

field and the carrier number of the laser, we can use the lang-kobayashi equations to describe the chaos-control dynamics and variable behavior of injected semiconductor laser under PNK effect [7]

( ) 2 cos( )p mL

dP kG P PPdt

γ φτ

= − + （1a）

1 ( ) sin( )2

mc p m

L

Pd kGdt Pφ β γ φ ω

τ= − + − −Δ （1b）

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2 2[1 ( ) / ( ) / ]u uI P t P P t pdNdt q

γ τ χ τ− − − −=

2eN GPγ− − （1c）

where P and φ are the photon number and phase of the optical field, N is the carrier number of laser.

( / )( )/ 1 /g th sG v a V N N P P= Γ − + is expressed as the nonlinear mode gain, where Г=V/Vp is the mode confinement factor, V is the active volume, Vp is the volume of the optical mode, vg is the group velocity of photon in the cavity, a is the gain constant, Nth=nthV is the carrier number at transparency with density nth, Ps is the photon number in the saturation. γp=vg(αm+αint) is the photon loss rate where αm is the cavity loss and αint is the internal loss rate. τL=2ngL/c is the round-trip time in the cavity length L, c is the vacuum light velocity and ng=c/vg is the group refractive index. k is the optical injection factor. βc is the linewidth enhancement factor. I is the drive current. q is the unit charge. γe=Anr+ B(N/V)+C(N/V)2 is the total carrier loss rate where Anr is the nonradiative recombination rate, B is the radiative recombination coefficient taking into account the carrier loss induced by spontaneous emission and C is the Auger recombination coefficient. Δωm can be regarded as the frequency detuning between the external injection light and the output light. τ is the delayed time of photocurrents. γ is the photoelectric linear delayed feedback factor. χ is the photoelectric nonlinear delayed feedback factor. When PNF is used to perform on the system, we can adjust the delayed time or modulate photocurrents to control the chaotic laser.

We discuss simply physics mechanism of PNK chaos-control. From Eqs.(1), the detuning is obtained by

[ cos( ) sin( )]m m cω ρ β φ φΔ = − + （2a） where ρm=k×(Pm/P0)1/2/τL is the normalized injection factor with the stable output P0. And we obtain

2 220 0

0 0 0 0[1 / / ] 0u u

eI P P P p N G P

qγ χ γ− − − − = （2b）

where 0 0 0( / )( )/ 1 /g th sG v a V N N P P= Γ − + and γe0=Anr+B(N0/V)+

C(N0/V)2 where N0 is the unmoving point. From Eq.(2b), we find effect of the nonlinear feedback on the stable output while the nonlinear feedback affects the normalized injection factor. This results in PNK effect on both the nonlinear mode gain and the linewidth enhancement factor. And the frequency property is affected. Due to chaos being sensitive to the system parameter, the chaos-control method will be valid. So the above analysis may be considered as physics mechanism of PNK chaos-control. The parameters are used in paper [7].

III. CHAOS-CONTROL VIA ADJUSTING THE DELAYED TIME OF NONLINEAR AND LINEAR PHOTOCURRENTS

Dynamics of an injected semiconductor laser was widely

studied. A route to chaos from a stable point in the laser was found. We give a dynamical result of semiconductor laser due to an external injection. Figure 2 illustrates chaotic attractor and output of the injected laser when I=20mA, Δωm=0.1GHz and Pm=0.16Ps. Both nonlinear and linear photocurrents are

used to perform chaos-control of the laser. We can adjust the delayed time of two photocurrents to control the chaotic laser. When we adjust the delayed time as 0.1ns and take γ=χ=0.33, laser dynamical behavior is conduce to becomes of a limit orbit as shown in fig.3, namely, chaotic laser can be controlled to a single-period. When τ=0.3ns, chaotic laser can be conduced to a dual-period shown by fig.4. Taking τ=0.6ns, laser behavior can be educed to a period-3 shown by fig.5. And taking τ=1ns, laser state can be found to be educed to another period-3. Figure 6 shows that chaotic laser is controlled to a period-4 for τ=1.2 ns. And we find that chaotic laser is controlled to other multi-period for τ=1.5 ns. So chaos-control can be realized by adjusting the delayed time of the nonlinear and linear photocurrents by aid of PNF.

(a) Chaotic attractor

(b) The laser output

Figure 2. Chaotic attractor and output of the injected laser.

Figure 3. Limit-orbit.

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Figure 4. Dual-cycle

Figure 5. Cycle-3

Figure 6. Cycle-4

We give other chaos-control results following as figures 8, 9, 10 and 11. Figure 7 illustrates another chaotic behavior of the laser when Δωm=0.5GHz. We find a route to a periodic cycle from chaos when we perform to control the chaotic laser via PNF. Taking γ=2χ=0.33 and the delayed time as 1.2, 1, 0.5 and 0.2ns, chaotic laser can be conduce to a cycle-3, another cycle-3, a dual-cycle and a cycle-one shown in figs. 8, 9, 10 and 11, respectively. It implies that chaos-control can be obtained via the method of PNF.

(a) Another chaotic attractor

(b) The laser output

Fig.7 Another chaotic attractor of the injected laser and its output.

Figure 8. A cycle-3

Figure 9. Another cycle-3

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Figure 10. A dual-cycle

Figure 11. A cycle-one

IV. CHAOS-CONTROL VIA ADJUSTING ONLY THE NONLINEAR PHOTOCURRENT

When we us only the nonlinear photocurrent to perform chaos-control, Eq.(1c) is rewritten as

2 22[1 ( ) / ]u

eI P t pdN N GP

dt qχ τ γ− −= − − （3c）

So another chaos-control physical model of the nonlinear photocurrent delayed feedback is obtained from Eqs.(3c), (1a) and (1b). We adjust only the nonlinear photocurrent to control chaotic laser. When we take χ=0.6 and alter the delayed time of the photocurrent as 0.1, 0.3 0.6, 1 and 1.2ns, chaotic laser can be controlled into a single-period, a dual-period, a period-3, another period-3 and a period-4 state, respectively, in figs 12.

(a) A single periodic state

(b) A dual-periodic state

(c) A period-3 state

(d) A period-4 state

Figure 12. The laser is conduced to show many behaviors

.

(a) A period-5

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(b) A period-4

(c) A period-3

(d) Another period-3

(e) A period-2

(f) A period-one

Figure 13. Another chaos-control result

We give another chaos-control result shown in figures 13 where the chaotic laser with Δωm=0.5GHz. We find a route to a periodic state from chaos when we perform to control the chaotic laser via PNF. Taking γ=0, χ=0.5 and the delayed time as 2, 1.6, 1.2 0.9, 0.6 and 0.3ns, figs.12 show that the laser oscillates in a period-5, a c period-4, a period-3, another period-3, a dual-period and a period-one, respectively. It implies that chaos-control can be realized.

V. CONCLUSION This paper demonstrated successfully that PNF method can

be used in control of chaotic laser. The laser can be stabilized in a variety of multi-periodic cycles. The method enriches control technology of chaotic laser. It provides a new way in applications of feedback laser system. And it is broadens control method of other chaotic systems. The result is helpful for study of chaos-control and photoelectric feedback lasers.

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[4] Y.Liu, N. Kikuchi and J. Ohtsubo, “Controlling behavior of a semiconductor laser with external optical feedback,” Phys.Rev.E, vol.51, pp.2697-2700,1995.

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