Nonlinear dynamics induced by repetitive optical pulse injection
of a semiconductor laser
Fan-Yi Lin1, Shiou-Yuan Tu1, Chien-Chih Huang2, and Shu-Ming Chang2 1Institute of Photonics Technologies, Department of Electrical Engineering,
National Tsing Hua University, Hsinchu 300, Taiwan 2Department of Mathematics,
National Tsing Hua University, Hsinchu 300, Taiwan
The nonlinear dynamics of a semiconductor laser under repetitive optical pulse injec-tion are studied numerically. Different dynamical states, including pulsainjec-tion and oscillainjec-tion states, are found by varying the intensity and the repetition rate of the injection pulses. Through individual period-doubling routes, the laser enters into chaotic pulsation (CP) states and chaotic oscillation (CO) states, respectively. A dynamical mapping of these dynamical states and their corresponding Lyaponov exponents are plotted and examined in the param-eter space. Moreover, the bandwidths of the chaos states are investigated, where bandwidth enhancement of more than four-fold in terms of the relaxation oscillation frequency is found.
Nonlinear dynamical characteristics of semiconductor lasers have been studied intensively in recent years. Diverse dynamical states found have been proposed to be utilized in various applications such as radar [1], lidar [2, 3], and radio-over-fiber communications [4, 5]. For an optically injected laser with a master-slave configuration, bandwidth enhancement and noise suppression phenomena have been observed [6, 7]. By controlling the injection strength and the frequency detuning between the master and the slave lasers, induced periodic oscillations and chaotic oscillations have been obtained [8]. Both period-doubling and break-up of two tori routes to chaos have been reported [9]. However, although many efforts have been made to understand the characteristics of an optically injected semiconductor laser, researches are limited to the condition where the laser is injected with an optical signal of constant intensity. Very few studies have been done on the nonlinear dynamics of a semiconductor laser subjects to a non-constant optical injection.
Non-constant optical injection is important when a transmitter-receiver or a cascaded laser system is considered, where the dynamical output of a transmitter laser can optically inject into a receiver laser inevitably or even intentionally. With a chaotic optical injection, high-frequency broad-band signal generation has been demonstrated [10]. By injecting op-tical pulses at a subharmonic of the cavity round-trip frequency, a long cavity multisection semiconductor laser oscillating at its resonant frequency has been observed [11]. In this letter, we report the complex dynamics of a semiconductor laser induced by optical pulses. By injecting a laser with a train of repetitive pulses, various dynamical states are shown and routes to chaos are identified. The dynamical mapping of the states is plotted and the
corresponding Lyaponov exponents are calculated. Moreover, the bandwidths of those chaos states are also investigated.
The schematic setup of the optical pulse injected semiconductor laser is shown in Figure 1. The laser is injected by a train of optical pulses, where the repetition rate and the intensity of the pulse train are varied as the controllable parameters. The dynamics of this system are simulated using the model described in [12] with the following normalized dimensionless rate equations: da dt = 1 2[ γcγn γsJ˜ ˜ n − γp(2a + a2)](1 + a) + ξi(t)γccos(Ωt + φ), (1) dφ dt =− b 2[ γcγn γsJ˜ ˜ n − γp(2a + a2)]− ξi(t)γc 1 + a sin(Ωt + φ), (2) d˜n dt =−γsn − γ˜ n(1 + a) 2n − γ˜ sJ(2a + a˜ 2) + γsγp γc ˜ J(2a + a2)(1 + a)2, (3)
Here, a is the normalized field, φ is the optical phase, ˜n is the normalized carrier density, γc, γs, γn, and γp are the cavity decay rate, spontaneous carrier decay rate, differential
carrier relaxation rate and the nonlinear carrier relaxation rate, respectively, b is the linewidth enhancement factor, and ˜J is the normalized dimensionless injection current parameter. The dimensionless injection parameter ξi(t) = η|Ai(t)|/(γc|A0|) is the normalized strength of the injection field received by the injected laser and Ω is the frequency detuning. In this study, we assume there is no frequency detuning between the master and the slave laser(Ω = 0). For the repetitive injection pulse train, a Gaussian distribution of ξi(t) with a peak injection
dynamical parameters of a high-speed semiconductor laser [13] are used in the simulation:
γc = 2.4 × 1011s−1, γs= 1.458 × 109s−1, γn = 3 ˜J × 109s−1, γp = 3.6 ˜J × 109s−1, and b = 4.
The lasers are biased at a value of ˜J = 1/3. The relaxation oscillation frequency of the laser is fr = (γcγn+ γsγp)1/2/2π, which is about 2.5 GHz with the parameters used in this
simulation. Second-order Runge-Kutta method is used to solve these coupled rate equations. When a free-running laser is injected by just a single optical pulse, induced oscillations in the laser output field are expected and the laser tends to relax back to its free-running state gradually if no successive pulse is further injected. However, if a train of optical pulses are injected into the laser with the time delay between each successive pulses being shorter than the relaxation time of the laser, the relaxed oscillation will be interrupted while the injected pulses perturb the optical field and phase abruptly. Hence, the nonlinear dynamics of the optical pulse injection system is strongly influenced by the intensity and the repetition rate of the injected pulses.
Figure 2 shows the time series, phase portraits, and power spectra of the dynamical states found in the optical pulse injection system. For peak injection strength ξp and repetition
rate frep (in GHz) of (ξp, frep) = (0.01, 3.0), a period-1 oscillation (P1O) state is found.
As shown in Fig. 2(a), the laser oscillates at the same frequency as the repetition rate of the injection pulses frep. When ξp and frep are both increased to (ξp, frep) = (0.02, 3.5), a
period-2 oscillation (P2O) state is obtained and shown in Fig. 2(b). As can be seen, the laser now oscillates at about 2.33 GHz, while an envelope with a subharmonic frequency of the oscillation frequency is found. Further increases in ξp and frep drive the laser into
a period-4 oscillation (P4O) and chaotic oscillation (CO) states as shown in Figs. 2(c) and (d), respectively. Clearly, the laser follows a period-doubling route into chaos when the parameters of the injected pulses vary. Besides these oscillation states, pulsation states are also found in this pulse injected laser.
Figure 3 shows the time series, phase portraits, and power spectra of the pulsation states found. With (ξp, frep) = (0.13, 3.0), Fig. 3(a) shows the regular pulsing (RP) state found
which the laser pulses repetitively at the frequency of frep. When frep decreases, a period-2
pulsation state (P2P) that has a subharmonic envelope is observed. Further reducing frep
drives the laser pulses with the fourth harmonic frequency (P4P), and goes into chaotic pulsing state (CP) eventually through a similar period-doubling route as in the oscillation counterpart. These pulsation states are distinguished from the oscillation states that the intensity of the pulsation states is higher and drops to zero before next subsequent pulse. To show the regions each dynamical state occupied in the parameter space, a dynamical mapping is plotted in Fig. 4(a).
In Figure 4(a), different regions of dynamical states are identified, where period-doubling routes of both the oscillation states and the pulsation states can be clearly seen. In the mapping, the oscillation states are generally found in the weak injection regime (ξp < 0.1)
while the pulsation states are obtained in the stronger injection regime (ξp> 0.1). Note that
a belt of complex dynamical states, especially the CO and the CP states, is found stretching from the regime of strong injection-low repetition rate (< 2.5 GHz) to the regime of weak injection-high repetition rate (> 2.5 GHz). To quantify the complexity, Fig. 4(b) plots
the corresponding largest Lyaponov exponents of these states. As can be seen, while the RP states in the upper right corner have negative Lyaponov exponents, positive Lyaponov exponents are found for those states showing complex dynamics seen in Fig. 4(a). Within the belt, CO states found in the upper left corner have the largest Lyaponov exponents and thus reveal their high complexities.
While some applications utilize chaos states because of their high complexities for secu-rity reasons, other applications solely demand large-amplitude random signals with broader bandwidths. Therefore, to utilize laser chaos as a broadband source, the bandwidths of the chaos states found in the dynamical mapping are investigated. Figure 5 plots the bandwidths of the chaos states with different parameters of the injected pulse, where the bandwidth is defined as the span of the frequencies which 80% of the energy is contained within. As can be seen, the CO states at weak injection regime that have higher complexity have however narrower bandwidths compared to the CP states at strong injection regime. The bandwidths of the chaos states increase as ξp increases. With our parameters, bandwidth enhancement
ranges from two- to four-fold of the relaxation oscillation frequency is presented.
We have numerically studied the nonlinear dynamics of a semiconductor laser under repetitive optical pulse injection. By injecting a train of repetitive optical pulses, a semi-conductor laser exhibits complex dynamics that it follows a period-doubling route to chaos. Both CO states and CP states are found, among which the CO states show larger Lyaponov
exponents and thus are more complex. The bandwidths of the chaos states are also investi-gated, which increases as the peak intensity of the pulse increases. Bandwidth enhancements ranging from two- to four-fold of the relaxation oscillation frequency of the laser are obtained. This work is supported by the National Science Council of Taiwan under contract NSC 95-2112-M-007-011. Special thanks go to Wen-Wei Lin, whose constant advice and interest were indispensable for this project. The research of Chang and Huang is partly supported by National Science Council in Taiwan grant no. NSC 95-2115-M-007-012-MY3.
Captions
Figure 1 Schematic setup of a semiconductor laser under repetitive optical pulse injection.
The variable attenuator is used to adjust the injection strength and the optical isolator is used to prevent the unwanted feedback.
Figure 2 The time series, phase portraits, and power spectra of different oscillation states
with (ξp, frep) = (a) P1O (0.01, 3.0), (b) P2O (0.02, 3.5), (c) P4O (0.03, 3.8), and (d) CO
(0.04, 4.0). The dashed curve is the waveform of the injected pulses for reference, which is scaled for clarity.
Figure 3 The time series, phase portraits, and power spectra of different pulsation states
with (ξp, frep) = (a) RP (0.13, 3.0), (b) P2P (0.15, 2.8), (c) P4P (0.16, 2.7), and (d) CP
(0.17, 2.3). The dashed curve is the waveform of the injected pulses for reference, which is scaled for clarity.
Figure 4 (a) Dynamical mapping and corresponding (b) Lyaponov exponents of the
dy-namical states.
Figure 5 The bandwidths of the chaotic oscillation (CO) states and chaotic pulsation (CP)
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