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Bistabilities between a steady (or pulsating, chaotic) and different pulsating regimes are investigated for an optically injected semiconductor laser. Both numerical and experimental studies are reported for continuous-wave single-mode semiconductor distributed-feedback lasers emitting at 1.55 μm. Hysteresis are driven by either changing the optically injected power or the frequency difference between both lasers. The effect of the injected laser pumping rate is also examined. Systematic mappings of the possible laser outputs (injection locking, bimodal, wave mixing, chaos or relaxation oscillations) are carried out. At small pumping rates (1.2 times threshold), only locking and bimodal regimes are observed. The extent of the bistable area is either 11 dB or 35 GHz, depending on the varying parameters. At high pumping rates (4 times threshold), numerous injection regimes are observed. Injection locking and its bistabilities are also reported for secondary longitudinal modes.
©2009 Optical Society of America
1. Introduction
A few years after the first experimental demonstration of a laser by Maiman, Stover [1] presented an optical injection experiment with gas lasers. The signal of a laser, called the master laser (ML), was seeded into a second laser called the slave laser (SL). Several studies on optical injection then showed that when the frequencies of both lasers are close together and for an appropriate injected power, the slave laser gets the spectral properties of the master one in terms of frequency and linewidth [2–5]. Afterwards, optical injection has been widely used by scientists to obtain tunable, powerful and narrow linewidth lasers for various applications [6] such as telecommunication, spectroscopy or metrology.
In optics, optical bistability usually refers to optical devices where two resonant transmission states are possible and stable, dependent on the input. Such possibility may be provided by resonant optical amplifiers [7] or by optically injected semiconductor lasers if the pumping rate of the injected laser is very close to threshold [8, 9]. The first optical bistabilities observed by the way of optical injection were obtained using semiconductor Fabry-Perot lasers by Kobayashi [10] in 1981, and then using distributed feedback (DFB) semiconductor lasers by Kawaguchi [11] in 1985. Such bistable behaviors are based on a saturation-induced refractive-index change due to the injected light, as firstly observed in a semiconductor étalon in 1979 [12] and later predicted [13] and demonstrated [14, 15] in a resonant-type semiconductor laser amplifier. Beyond the physics, these bistable behaviors offer potential optical memories applications [16] for ultra-fast processing.
It has been pointed out that the bistability could be associated to a continuous wave output and a periodic orbit (called wave-mixing) [17–19]. This last bistability leads to a hysteresis loop that has been largely analyzed when the injected power or the optical frequency difference between the two lasers (detuning) is swept back and forth.
Wieczorek [20] pointed out the theoretical existence of multistability and of generalized form of bistability, which means the coexistence of two attractors as for example a chaotic attractor and a periodic orbit in a same region of parameters. But from the experimental point of view, it has been only noted by an uncommented mention of an abrupt change in the RIN spectra [20].
In this paper, we concentrate on this generalized form of bistability. To this end, we carry out systematic mapping of the possible long time regimes at both low and high pumping rates of the injected laser. We propose a general view of the bistable properties of an injected laser by carrying out systematic mapping of the injection regimes at different bias currents of the SL when the detuning is increased or decreased. We show a clear hysteresis in these diagrams: the regime at a given operating point (injected power, detuning) is path dependent on the control parameter (for example detuning). Moreover maps are established for larger power (respectively -detuning) extent than previously reported values (~ 5 dB above the reported measurements; respectively 200 GHz-range and more instead of the usual 20 GHz-range for the detuning) [17,18] (resp. [19, 20]). While in [17, 18], only the wave-mixing–locking bistability is described, our study concerns the coexistence of different regimes (or of different type of attractors). Therefore, we present a broader review of the bistable properties of optically injected lasers than previously reported.
The paper is organized as followed: Section 2 details experimental results obtained with a 1.55-μm single-mode semiconductor DFB laser injected by an external continuous-wave signal, section 3 describes a numerical study, which confirms the experimental observations, and section 4 draws a conclusion.
2. Experimental observation of optical bistabilities
2.1. Experimental setup
The experiment consists of a unidirectional coupling from the master laser to the slave laser, imposed by the presence of a double-stage optical isolator (70 dB isolation). Figure 1 displays the sketch of the proposed experiment. The ML is a free-mode-hopping external-cavity single-mode tunable semiconductor laser with a good repeatability and a good stability (frequency jitter of 1 MHz, power fluctuations of 0.01 dB).
The SL is a massive InP/InGaAsP buried-double-heterostructure DFB chip. All components are either pigtailed or fiber-based components. Lasers are properly isolated from external perturbations inside an acoustic box, and all optical components located between the ML and the SL are polarization-maintaining (PM) ones, thus allowing perfect reproducibility of the experiments. The master laser signal is amplified to 18 dBm using an erbium-doped fiber optical amplifier (EDFA). The injected power is then tuned using a variable optical attenuator, thus offering a constant signal to noise ratio of the injected signal.
The control parameters of interest in an optical injection experiment are the injected power Pinj and the detuning Δv=vm -vs, which is the difference between the master frequency (vm) and the slave one (vs). The pumping rate r of the slave is fixed during experiments. This metric is defined as r = I/Ith where I is the bias current and Ith the bias current at threshold. These control parameters are monitored using a power-meter and a lambda-meter as shown in Fig. 1. The optical and electrical spectra of the injected slave are observed using respectively a Fabry-Perot (FP) analyzer and a fast detector coupled to a radio-frequency (RF) spectrum analyzer. The FP is a pigtailed but free-space interferometer with a 135-GHz free spectral range and a finesse of 100, providing a 1.35-GHz resolution. Thus, it is not possible to describe the narrow line of the master laser (whose full width at half maximum is 125 kHz), linewidth transfer and phase locking phenomena have been documented in other publications [2–5, 21, 22]. The bandwidth of the fast detector allows to observe RF spectra of the injected laser from 0 to 15 GHz.
In this article, experimental results are obtained either by fixing the detuning and varying the injected power, or by fixing the injected power and varying the detuning. These experiments are carried out at two different pumping rates of the slave laser, close to threshold (r = 1.2) and far from threshold (r = 4).
2.2. Experimental results close to threshold
The slave is operated at r = 1.2. Fig. 2 gives different FP spectra of the injected slave for a fixed detuning of -20 GHz and for increasing (a) or decreasing (b) values of injected powers between -20 and 0 dBm. The injected power is mainly limited by coupling losses between the fiber pigtail and the laser chip. In this figure, the frequency scale is relative to the free slave frequency. Three regimes can be observed as described in the following.
- Free slave regime: For weak injected powers, the seeded master-light has no effect and the injected slave laser acts as an injection-free laser, as shown in Figs. 2(a)–2(b) for Pinj = −20 dBm.
- Bimodal regime (limit cycle): For intermediate injected powers, the injected laser is bimodal, i.e., both natural and injected lines coexist, as shown in Fig. 2(a) for Pinj = −15 dBm.
- Injection locking: For stronger injected powers, injection locking occurs and the injected slave signal gets the spectral properties of the master signal, as shown in Figs. 2(a)–2(b) for Pinj = 0 dBm.
Fig. 2. Experimental observation of optical bistability for an injected slave laser pumped at 1.2 times the threshold. Experimental spectra of the injected slave laser are observed for an increasing (a) or a decreasing (b) injected power. An experimental mapping of the injection regimes is represented in (c). In (a–c), the black color corresponds to the locking regime, the white corresponds to the bimodal regime or the free slave laser. The grey color corresponds to the bistable area. If the detuning or injected power is decreasing, this color corresponds to the locking regime; otherwise it corresponds to a bimodal regime.
Figures 2(a)–2(b) clearly shows the bistability between locking and bimodal regimes. Specifically, we observe that the locking area appears for an increasing injected power of -7.5 dBm, but disappears for the decreasing injected powers of -17.5 dBm, thus describing a 10-dB-extended bistable area. Figure 2(c) displays a map of the locking area (in black) when the injected signal is varying between -30 and +2 dBm and when the detuning is varying between -300 and +100 GHz. The grey-filled area corresponds to the injection locking regime only for decreasing detuning or injected power. The log scale permits to stress the progressive adi-abatic decrease of the grey area along with the decrease of the injected power. The irregularity and lack of symmetries of this map are striking features. A bistable area only exists at negative detuning. This asymmetry has already been mentioned in the literature and it has been reported [23, 24] that this behavior is related to the non-zero linewidth-enhancement factor αH of the slave laser. The larger αH is, the more asymmetric the map is. Note that the slave laser used in these experiments has an aH-parameter value of nearly 6.
The locking region of the bistable area can be obtained either by a decrease of the detuning or a decrease of the injected power. Once the laser is locked, it maintains locked for a larger interval of detuning. The larger extent of the bistable area described here is 11 dB, for detuning varying between -44 and -7 GHz; or 35 GHz for an injected power of 0 dBm. In Fig. 2(a), we can observe that for injected powers varying from -20 to -7.5 dBm, the master peak is progressively amplified at the expense of the slave peak, until the slave peak suddenly disappears while entering the locking regime. For decreasing detuning as shown in Fig. 2(b), the abrupt transition occurs for an injected power of -17.5 dBm, i.e., when the injected laser leaves the locking regime for a bimodal regime.
Figure 3 shows another map with the same experimental conditions but for a larger detuning, varying from -1,000 to +1,000 GHz. We observe that the master can excite the longitudinal side modes of the slave, as shown by the correlation between the locking area and the free SL spectrum. This spectrum is clearly asymmetric, which is typical of a distributed feedback laser without phase-shift [25]. The side-mode suppression ratio of the free slave laser is greater than 30 dB; therefore, a higher injected power is necessary to obtain injection locking of these secondary modes. In Fig. 3, lines with percentages inside the bimodal regime indicates about the ratio of the total optical power inside the amplified master line. The bistability area decreases as the side mode is further away from the lasing mode. Note that for high injected powers (≥ 3 dBm), a permanent locking regime is reached whatever is the excited mode. This broader view shows clearly that the bistable (or grey) area stops at finite injection power and that it exists for other longitudinal modes.
It is worth mentioning that we have carried out similar experiments in [26] using the same master laser but a distributed feedback fiber laser as a slave laser. DFB fiber lasers are characterized by a linewidth, which is typically an order of magnitude lower than DFB semiconductor lasers. As a consequence, we observed a reduction of the locking area by the same factor [26]. However, a notable feature of the Dfb fiber laser that differs from the semiconductor laser is the fact that the bistable area appears for positive detuning.
2.3. Experimental results far from threshold
The existence of non-locking injection regimes, such as chaos or wave mixing, led Simpson [27] to draw a map of the regimes of an optically-injected laser in 1997, for a laser biased at 1.6 times its threshold. In our experiment, injected slave laser maps are quite similar at 1.6 and 4 times the threshold (4Ith), so we only present the 4Ith-map in this article.
Figure 4 displays a map of the different phenomena which can be observed when the injected signal is varying between -50 and +8 dBm, the detuning is varying between -70 and +30 GHz. Compared to the laser close to threshold, many other injection regimes due to nonlinear effects appear when the slave is far from threshold. The injection regimes are: Locking area (L), wave mixing (1, 2, 4), Chaos (C), and undamped Relaxation oscillation regime (R).
Fig. 3. (a) Experimental mapping and bistabilities of the intermodal injection for a slave laser pumped at 1.2 times its threshold. The black area represents the locking region. The white area represents the bimodal regime. The grey areas represent the bistable locking areas: The locking only occurs for decreasing detuning. Black lines represent the percentage of the power of the master peak when the injected slave is bimodal (100% corresponds to the locking regime). (b) Free slave laser spectrum.
Fig. 4. Experimental mapping and bistabilities of the intramodal injection for a slave laser pumped at 4 times its threshold. “L” represents the locking area, “1” is for single wave mixing, “2” for period-doubling wave mixing, “4” for period-quadrupling wave mixing, “C” for chaos, “R” for undamped relaxation oscillation, and the white area represents the free slave injection regime. Thick black curves represent abrupt transitions of the injected slave spectra.
The bimodal regime is not observed anymore, the white area corresponds from now on to a free slave injection regime, i.e., the slave laser acts as if it was not injected. We observe that the map is less extended than the one close to threshold, except for the bistable area. Figure 4(a) is obtained for a decreasing detuning while Fig. 4(b) corresponds to an increasing one.
The FP spectrum of wave mixing (1) consists in a three-peak spectrum. The main peak is at the free slave frequency vs and has two satellites. One satellite is at the master frequency vm, the second one is the symmetric of vm with respect to vs and is usually less powerful. Note that the frequency pushing effect [4,26,28] appears, meaning a shift of the slave peak from the free slave frequency, far from the master frequency. A characteristic beating peak at the effective detuning (including frequency pushing effect) is observed at the Electrical Spectrum Analyzer (ESA). The region (2) consists in a period-doubling phenomenon: Peaks appear between lines of the single wave mixing (1), giving birth to sub-harmonic oscillations observed at the ESA. The region (2) arises from the region (1). Similarly, region (4) arises from region (2). This behavior is characteristic of a “period doubling bifurcation” phenomenon [29,30]. Note that chaos appears for quite high injected power (> −15 dBm), either for positive or negative detuning. Finally, the characteristics of undamped relaxation oscillation regime (R) are very similar to wave-mixing but in this case, the peak spacing is equal to the relaxation oscillation frequency (ROF) of the SL. Figure 5 illustrates the spectral properties of the possible injection regimes. A more complete description is given in Ref. [21,31–33].
Fig. 4 shows that, for an injected power lower than -17 dBm, the locking area is symmetric with respect to the null detuning axis. For higher injected powers, there is no locking at null detuning, but it exists at positive and mainly at negative detunings, as already reported in the literature [34]. The undamped relaxation oscillation regime takes place [32, 35] between these two areas. One striking point about this map is the wide bistable area, in comparison to the dimensions of the map. The bistable area can be seen by comparing Figs. 4(a) and 4(b). As the close-to-threshold case, hysteresis can be revealed by either varying the detuning or the injected power. The bistability between a locking regime (for decreasing detuning), and free-slave operation, wave mixing (1, 2, 4) or undamped relaxation oscillation regime (for increasing detuning), is clearly seen, especially at negative detuning, thus showing more bistable states than observed at lower pumping rate in Fig. 3. However bistability between different states (chaos-period doubling, undamped relaxation oscillation regime and wave-mixing) may also be observed as shown by the REC bar of fig. 6 (Media 1), which describes optical SL FP spectrum variation versus detuning, highlighting the hysteresis process.
Fig. 5. Experimental FP spectra of different injected SL regimes. The broken vertical line indicates the ML optical frequency (vm).
3. Theory
3.1. Model
Various theoretical approaches have been proposed to numerically study optically-injected semiconductor lasers. Most studies are based on asymptotic analysis [17, 19] or bifurcation analysis [20, 36]. These methods have been largely described and we will focus on numerical simulations. The behavior of a single-mode injected laser can be described by the following normalized rate equations, which are given for the normalized carrier density Δn(t) and for the complex electric field E(t) [35] by:
where Einj is the complex electric field of the ML, which is injected in the SL cavity, τc is the round-trip time in the laser cavity, aH is the linewidth-enhancement factor. The carrier lifetime τe is determined at threshold. GN is the differential gain and gd the normalized one. jb is the normalized pumping rate of the SL (jb = r − 1) and finally nsp is the spontaneous emission rate.
Fig. 6. Single frame movie (Media 1) of SL spectrum variation vs detuning. The top-right inset represents the previously described mapping at 4-times the threshold. The bottom bar indicates the different regimes for decreasing and increasing detuning at a glance.
Experimental measurements have been achieved in order to numerically determine parameter values [33]. A particular attention has been paid in order to perfectly fit the αH parameter, the relative intensity noise versus the microwave frequency, the Light-Intensity curve as well as the dependency of the relaxation oscillation frequency on the bias current. This match has been done for the laser, which is mapped at 4 times its threshold. However, we have not used the same laser chip for mapping at 1.2 times due to a laser breakdown. Nevertheless, both lasers are of the same kind and have similar behavior when submitted to optical injection.
Table 1 sums up numerical values we use.
3.2. Numerical mapping
In this subsection, the nonlinear dynamics of the simulated injected laser is mapped as it has been done in Section 2. To this end, we numerically solve equations (1) and (2). Then, we compute the FFT of the electrical field E(t) and we analyze it to determine in which injection regime the laser is. The overall result is shown in Figs. 7 and 8, when the SL pumping rate is respectively 1.3 and 4.
At low pumping rate (see Fig. 7), the injected laser mainly acts as an optical amplifier. As previously said (see Section 2), only frequency-locking or amplification are observed. Moreover, this figure shows that the hysteresis are well numerically reproduced. The SL is frequency-locked (grey area) only for a decreasing detuning. Note that the bistable area corresponding to the coexistence of a stationnary state and of a periodic orbit is observed at only one boundary, which corresponds to an abrupt transition between two regimes (locked-periodic), due to saddle-node bifurcation [20]. This property still remains at higher current. The bistable domain is presumably bounded at the right by an homoclinic bifurcation where the period of the bimodal or wave-mixing limit-cycle becomes infinite [20]. However further numerical and analytical work is needed to demonstrate that it is indeed the bifurcation mechanism.
Table 1. Some parameters values for laser diode used.
Fig. 7. Numerical mapping and bistabilities of the intramodal injection at 1.3 times its threshold. The black area represents the locking area. The grey area represents the bistable area: The injected SL is frequency-locked only for decreasing detuning.
By comparing this map with Fig. 2(c), a good qualitative agreement is achieved between experimental and numerical results. However, differences can be observed on the extension and location of the bistable area. As explained before, this mismatch is due to the fact that we do not measure the laser parameters before it broke down. Nevertheless, the general trend is observed.
Fig. 8. Numerical mapping of the intramodal injection at 4 times its threshold. Symbols and colors are the same as in Fig. 4. Figure (a) is obtained for a decreasing detuning while Fig. (b) is for an increasing one.
For the SL biased at a higher current (4 times its threshold), numerical simulations lead to more complex dynamics (wave-mixing, chaos, etc.). Figure 8(a) gives the mapping for a decreasing detuning while Fig. 8(b) is obtained for an increasing one.
At this pumping rate, good qualitative and quantitative agreements are achieved between experimental and numerical maps. Such an adequation between experiments and simulation has already been reported in [37]. The original result of our paper is that the bistabilities are well numerically reproduced and that the effect of the slave laser pumping bias current is studied. As far as we know, it is the first demonstration of such a result. By comparing Figs. 4 and 8, minor differences on the extension and location of the different regimes can be seen. Small errors onto the value of the different parameters may explain these small discrepancies as well as the sensitivity of the maps to the injection bias current, as it can be experimentally checked.
4. Conclusion
In this paper, we investigated the bistability between different states and show that it leads to hysteresis when the detuning is driven back and forth. Major report concerns bistability between a locked steady state and pulsating intensity oscillations. This form of bistability typically appears near the steady state locking boundary at negative detuning. The pulsating intensities correspond to either a bimodal limit-cycle for low pumping rates or different forms of wave-mixing and relaxation oscillations at high pumping rates. Nevertheless, the areas of the bistable domain do not change significantly as the pumping rate is increased. From negative to positive detunings, the bistable domain is bounded at the left by the locking boundary of the steady state (saddle-node bifurcation point). This point is clearly noted as we decrease the detuning from positive values.
Acknowledgements
Authors would like to thank Guy Michel Stéphan and our “Laser Physics Group” colleagues from the lab FOTON-ENSSAT for helpful discussions.
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