Dynamics of optically injected semiconductor lasers with zero linewidth enhancement factor

The study theoretically investigates the dynamic of an injection-locked semiconductor laser with zero linewidth enhancement factor. The stability map of the injection locking process is drawn, and bifurcation diagrams are generated. Carrier density dynamics are also investigated. The study has reported different periodical and chaotic behaviour and revealed the dynamics of carriers with zero linewidth enhancement factor.


Introduction
Semiconductor lasers are considered class B lasers.These lasers have a distinguished feature regarding their linewidth.The dependence of the refractive index of the active materials on the carrier density causes coupling between phase change and intensity, broadening the laser signal's linewidth.This coupling was first studied by C. Henry in 1982 [1], where he introduced a crucial parameter called LED or Henry factor.This factor plays a significant role in determining the line width of the laser, mode stability, and the chirp under current modulation.Hence, several studies have been devoted to revealing the effect of this factor [2][3][4].
The linewidth enhancement factor (LEF), also known as the Henry factor, helps to differentiate the behaviour of semiconductor lasers considering various types of lasers [1,5].Extensive investigations have been conducted for LEF [6][7][8].A previous study by Guiliani et al. [9] used the self-mixing method and revealed that the α-factor varies for some lasers with the emitted power.However, this variation is associated with the differences in laser linewidth.The α-factor significantly impacts fibre dispersion used in optical fibre communication systems [10].This factor can be measured using optical feedback techniques [11] and self-mixing [9,12].The device characteristics can be enhanced by observing a zero LEF in quantum dot lasers [11].Moreover, optical injection strongly changes the behaviour of the LEF in quantum dot lasers [13].
This factor has also shown a significant effect in modern applications, including fibre dispersion transmission systems [14], intensity noise in semiconductor laser sensors [15], and mode-locked ring lasers [16].Generally, this factor takes higher values in bulk active material lasers, and its value becomes smaller as the quantum confinement increases (quantum wells QW to quantum dots QD) [17].Zero [18,19] and near zero [20,21] LEF have been reported for QD, QW, and Q-dash lasers.LEF also strongly depended on the injection current in tunnel-injection quantum dot lasers [22].The manipulation of the LEF in injection-locked Quantum-Dash Fabry-Perot Laser was demonstrated by the threshold gain shift resulting from high optical injection [23].In QD lasers, controllable [24] and temperature-insensitive [25] LEFs have been demonstrated.This factor was also found to increase with increasing carrier density and decreasing temperature and energy level separation [26].However, the most interesting results are the reporting of negative LEF [24,27].Increasing the p-doping at a low injected current in a QD laser was found to tune the LEF to a negative value [24].The negative value of the LEF was theoretically attributed to the nonparabolicity and many-body effects [27].
One of the previous theoretically demonstrated a significant impact of LEF on the stability map [28].The chaotic behaviour is enhanced when the LEF grows and can be used in cryptographic communications.The LEF also plays an important part in the time-delay signature (TDS) in chaotic semiconductor ring lasers (SRL) to benefit optical chaotic secure communications [29].Another recent study [30] found that altering the LEF causes discontinuities to form due to disaggregation from socalled shrimp-like structures.The chaotic synchronization has been focused on recently because of its potential applications in broadband data transfer and communication security [31,32].The dynamical system in semiconductor lasers, or any other system, experiences transitions to chaos from stability via a process known as routes to chaos [33].A recent study by Locquet [34] provided a thorough assessment of experimental exploration of the paths to chaos in a semiconductor laser subjected to optical feedback from a distant reflector, including period-doubling, quasi-periodic, and subharmonic approaches.These routes, chaos, and other dynamical behaviours are likely to be examined using a bifurcation diagram to show the alteration and reaction of the system [35,36].This bifurcation is used to analyze the dynamics of semiconductor lasers both empirically and theoretically [37][38][39][40].
This study raises a simple question; what is the dynamic of an optically injection-locked semiconductor laser with zero LEF.This question was answered by theoretically investigating the optical injection dynamics of a semiconductor laser in terms of stability map, bifurcations, and carrier density.This could be of important uses in generating chaos in lasers with low LEF, such as quantum dots lasers.

Model
Linewidth enhancement factor (LEF) or α-factor can be expressed by the ratio of the changes in the refractive index ( n) and mode gain ( g m ) caused by changes in carrier density ( N) as follows [1]: To study the dynamics of injected lasers, a simple model was used based on Lang's approach, as described in [19].Our model consists of two lasers: Slave laser (SL) or the injected laser and the Master laser (ML) or the injecting laser with an isolator placed between them as shown in Figure 1.The rate equations were solved using the Runge-Kutta method to reveal the system's dynamics.The stability map is drawn in two dimensions as a function of the injection strength (K) and frequency detuning ( f).At the same time, the bifurcation diagram is recorded by taking the extrema of the electric field of the injected slave laser.The power spectra are taken by applying a fast Fourier transform (FFT) to a certain time window of the slave laser electric field.Finally, the carrier density is normalized to its value at transparency.The parameters used in our simulation are all experimentally characterized as shown in Table 1 [41].

Results and discussion
The investigation started by drawing the stability map (injection level K vs. frequency detuning f) of the slave laser (SL) under a single optical injection, called master laser (ML).Figure 2 shows a map where colours denote the system's dynamics.The white colour indicates the stable locking region.Stability in injection locking can be defined as the SL being nicely locked to the ML, with a side peak or relaxation oscillation frequency (ROF) less than -20 dB relative to the main locked peak [42].An example of this behaviour is shown in Figure 4e.The yellow colour represents the period-one (P1) dynamics, where the SL is not locked to the ML with the strong presence of the latter with other signals at the same spacing of the frequency detuning.Such a dynamic is shown in Figure 4a and Figure 6a.The orange region indicates the quasiperiodic dynamics (QP), where the system exhibits semiperiodic behaviour, as shown in Figure 6c.Finally, the dark red colour indicates the chaotic behaviour, which is confined in very small dots as shown in the figure .The two arrows show the operating points where the bifurcations in Figures 3 and 5 are taken.The first observation is the symmetrical nature of the locking bandwidth (the white region).The asymmetrical locking bandwidth was previously attributed to the non-zero value of the LEF [43].The fluctuation in carrier density modifies the refractive index, leading to a shift in the resonance frequency.As this effect is eliminated by setting the LEF equal to zero, the locking bandwidth appears symmetrical.This symmetry is not only for the locking bandwidth but also shown for the rest of the map, where the behaviour is exactly mirrored between the negative and positive detuning side, which is not reported before, to the best of our knowledge.The disappearance of chaotic islands reported with non-zero LEF can also be noticed [44,45].Although these islands are not shown, there is still some evidence of semichaotic behaviour in the dark red dots shown in the figure.This behaviour is shown in Figure 4c, as will be discussed shortly.
Instead of the chaotic dynamics, the unlocking region seems to be covered by the quasi-periodic behaviour (the orange).These spots are centred around the ROF of the free-running laser (around 8 GHz), as the injected signal enhances the ROF and creates such behaviour.
To further reveal the dynamics of the SL, the bifurcation diagram was generated on the positive detuning side (since both sides of the map are identical) at f = 7.5 and 12.5 GHz, as indicated by the arrows in Figure 2.This bifurcation is taken by recording the extrema of the SL electric field at each specific injection level.Such a bifurcation diagram at 7.5 GHz is shown in Figure 3.It can be seen that the system undergoes P1 dynamics [44], where the ML is strongly evident without locking the SL.This behaviour is shown in Figure 4a (at K = 0.1 as indicated by the label (a) in Figure 3).The SL's power spectrum shows the SL's location and ML.In contrast, the electric field time series and the trajectory of the electric field-population inversion indicate P1 behaviour.For a qualitative analysis the largest Lyapunov Exponents (LLE) [46] were calculated as shown in the lower box in Figure 3.As can be seen in the figure, the LLE takes positive values when the system undergoes chaotic dynamic, and negative values when the system is in a stable state, including periodic behaviours.This is in very good consistency with the bifurcation diagram.
In Figure 5, the bifurcation diagram is drawn at 12.5 GHz.The system, in this case, maintains P1 behaviour up to nearly 0.4 injection level, as shown in Figure 6a and  b, with the increase of the electric field amplitude as the injection level is enhanced.At K = 0.4 (c in Figure 5), the system exhibits period-doubling dynamics, as shown in Figure 6c, in a semi-chaotic manner, with damped peaks.Further increase in injection level drives the system back to the P1 dynamic, as shown in Figure 6d and  e.In some cases, this P1 was chaotic (around K = 0.55, not shown in Figure 6).
As the injection power increased (at K = 0.2, point (b) in Figure 3), period-doubling behaviour (sometimes referred to as period-two P2) was noted [44].This dynamic is shown in Figure 4b.Lower secondary peaks are shown in the middle of P1 spacing as shown in the power spectrum.The time series of the electric  field and its trajectory with population inversion show the period-doubling behaviour.At K = 0.35 (point c in Figure 3), the system exhibits some chaotic behaviour while maintaining in general P1 dynamic, as shown in Figure 4c.Further increase in injection level (K = 0.5, point d in Figure 3) drives the system back to a nice and smooth P1 dynamics as shown in Figure 4d.Finally, at K = 0.6 (e in Figure 3), the system enters the stable locking region as shown in Figure 4e.The LLE is again can determine the stability and chaos in the system as shown in the lower box in Figure 5.As the relation between the LEF and carrier density is very crucial, the investigation revealed the dynamics of carrier density in optically injected semiconductor lasers with zero-LEF.To do so, the stability map was scanned vertically (at constant K) and horizontally (at constant f). Figure 7 shows the normalized carrier density as a function of the frequency detuning at different injection levels.As can be seen from the figure, the dynamics of the carriers seem to be very identical, with the enhancement of carrier variation at zero-detuning (when the ML coincides with SL free-running signal) and at the ROF ( ≈ 8 GHz).This symmetrical behaviour is attributed to the zero-LEF as it was found that nonzero LEF gives asymmetrical characteristics for carrier density, as shown in the inset of the figure where carrier dynamics is shown with LEF = 1.These results are under publication at the time of writing this submission and are consistent with previous studies [41,47].It is also clear that the more power injected, the more carriers vary.
The other observation was that the carrier variation maxima shifted towards the resonance frequency with increasing injection levels.The previous study [48] has shown that ROF increases with LEF = 3.This can be understood as a result of the gain shift resulting from the change in carrier density and hence the change in refractive index as previously mentioned.However, this effect is almost eliminated with zero-LEF, and this sort of shift is not reported before to the best of our knowledge.
In Figure 8, the normalized carrier density was drawn as a function of the injection level for various frequency detuning.As can be seen in the figure, away from the ROF (8 GHz), the variation of carriers is almost negligible (1,5,10 and 12.5 GHz), indicating that carriers show no noticeable dynamics except around the ROF.
The highest variation is noticed at the ROF (green curve) and then around it (purple and blues), as shown in the figure.That means that carrier density variation in the case of zero-LEF is mostly affected by the frequency detuning rather than the injection level.The same investigation was performed for negative frequency detuning values (−2, −5, −7.5 GHz . . .etc.) and found the same dependence, which indicates that the behaviour of the carrier density is identical on both sides of the frequency detuning.
It is observed that paths to chaos nearly always retain themselves despite changes in the stability map.This raises further issues concerning the dynamics and carrier density inside the laser cavity.However, these issues need to be researched further in the future, along with an attempt to validate these findings experimentally.The proposed future model should also include the effect of spontaneous emission noise.

Conclusion
The study has theoretically investigated the dynamics of an injection-locked semiconductor laser with zero-LEF.A very symmetrical map, in terms of stability, chaos, and other behaviours, is reported.In contrast to nonzero LEF, the disappearance of large chaotic islands was noticed.Instead of the chaotic behaviour, most of the unlocking area seems to be covered by quasi-periodic dynamics.The system exhibits different routes to chaos, including the PD and QP routes.
Regarding carrier density dynamics, carriers show maximum variation around the ROF and seem to be enhanced with increasing injection levels.Finally, a slight shift in the maximum variation of carriers was reported towards the resonance frequency with increasing injection level.These results need experimental validation along with theory development.Future studies need to experimentally verify the relation between LEF and routes to chaos.

Figure 1 .
Figure 1.Block diagram of our model.

Figure 2 .
Figure 2. Stability map of the optically injected laser when LEF = 0.

Figure 3 .
Figure 3. Bifurcation diagram (upper box) of the system at 7.5 GHz, the labels (a-e) indicate the points at which the spectra and trajectories in Figure 4 are taken.The lower box shows LLE of the system.

Figure 4 .
Figure 4. Power spectra (first column), SL electric field time series (second column), and trajectories of the electric fieldpopulation inversion (third column) at the points indicated in Figure 3 for K and f = 7.5 GHz.

Figure 5 .
Figure 5. Bifurcation diagram of the system (upper box) at 12.5 GHz, the labels (a-e) indicate the points at which the spectra and trajectories in Figure 4 are taken.The lower box shows LLE of the system.

Figure 6 .
Figure 6.Power spectra (first column), SL electric field time series (second column), and trajectories of the electric fieldpopulation inversion (third column) at the points indicated in Figure 5 for K and f = 12.5 GHz.

Figure 7 .
Figure 7. Normalized carrier density as a function of frequency detuning at different injection levels.The inset shows the carrier dynamics at LEF = 1.

Figure 8 .
Figure 8. Normalized carrier density as a function of injection level at different frequency detunings.

Table 1 .
Parameters of the standard model.