two-mode dynamics in qd lasers with optical feedback 

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Semiconductor lasers as damped nonlinear oscillators

In Lorenz equations, all variables have comparable relaxation time-scales, i.e. all time constant parameters have roughly the same order of magnitude. In laser systems, how-ever, this is not the case and some variables can evolve much faster than others. Because chaos requires at least three degrees of freedom [20], the fast relaxation of one variable, which can then be eliminated adiabatically, will prevent the emergence of chaotic dy-namics. Three time constants can be identified in the Lorenz-Haken equations for lasers: the photon lifetime Tp, the polarization relaxation time Tpol and the carrier lifetime or population inversion relaxation time Te. Thus, in 1984, a simple classification of laser systems dependent on these time constants was suggested [9, 16]:
— Class C: when all the time-constants are of the same order of magnitude Tp ∼ Tpol ∼ Te, the lasers are defined as class C. Single-mode class C lasers will, à priori, be described by the full set of Lorenz-Haken equations and are therefore candidates for chaos generation if all conditions are met. Class C lasers are typically NH3 or Ne − Xe lasers.
— Class B: when the polarization relaxes much faster than the other variables, i.e. Tpol << (Tp, Te), the corresponding equation can be adiabatically eliminated. The lasers are then defined as class B and can be modeled by two equations. A large range of devices are classified in this group, including semiconductor lasers, for which we typically find Tp ≈ 3 ps and Te ≈ 1 ns, fiber laser or CO2 lasers.
— Class A: when the photon lifetime largely exceeds the one of the polarization and of the population inversion, i.e. Tp >> (Tpol, Te), only the field equation remains and we obtain class A lasers whose evolution can be described by a single equation. Visible He − Ne lasers and dye lasers are classified in this category.
Although such classification is an obvious over-simplification, it gives useful insight on the dynamical behavior of the lasers. While Class C lasers are supposed to be the most complex ones, potentially chaotic, lasers of classes A & B are expected to be intrinsically stable as one or two variables relax much faster than the others. Thus, lasers of classes A & B will need additional degrees of freedom to generate complex dynamics and/or chaos.

From edge-emitting lasers to VCSELs

The VCSEL structure has been first introduced and demonstrated in 1979 by Soda et al. [28]. As pictured in Fig. 1.7, EELs emit from the edge of the structure while VCSELs emit in the direction perpendicular to the active layer. The smaller active region poten-tially allows for lower threshold – in smaller active medium, the same current densities can be achieved with a lower total injection current – but the net gain is also severely reduced and therefore lower losses are required to achieve laser operation. Because of insufficient mirror reflectivities, the first VCSEL was only lasing at low temperature (77 K) and under large current densities requiring pulsed operation to avoid damages. An essential improvement then came from the introduction of Distributed Bragg Reflectors (DBRs) which allow for reflectivities of the order of 99.9 % – to be compared with the 98 % reflectivity of gold mirrors as used in [28]. Thanks to these new reflectors combined with quantum well technology, VCSELs finally reached continuous wave operation at room temperature in 1989 with a current threshold of only few milliamps [29].
Today, VCSELs are extremely common lasers and are used intensively, in particular for optical telecommunications. In addition to their low threshold and therefore low con-sumption, VCSELs also generally exhibit larger modulation bandwidth which makes them even more suited for telecommunication purposes. Their surface emission layout also greatly simplifies their production as it allows for on-chip testing and creation of 2D arrays. Finally, the typical circular geometry of VCSEL produces a circular beam which can be better coupled into optical fibers. In summary, VCSELs exhibit almost all features required for optical communication, but they also exhibit a major drawback: polarization instabilities.

Experimental observations of polarization instabilities in VCSELs

In EELs, the vertical linear polarization of the output beam is constrained by the geom-etry of the device. In contrast, as shown in Fig. 1.7, VCSELs typically exhibit a circular geometry that does not yield any rule for polarization selection. Nonetheless, a VCSEL will generally emit linearly polarized light at threshold due to the small anisotropies of the laser cavity, but they cannot fully compensate for the lack of geometrical selection. As a result, variations of the injection current, device temperature or strain can give rise to polarization instabilities.
As we will see here, such instabilities represent a severe drawback of the VCSEL struc-ture as they dramatically alter the properties of the devices and can be especially harmful for polarization sensitive applications. For a recent and detailed review about these in-stabilities, see e.g. [30]. A solitary VCSEL usually exhibits two preferred orthogonal linearly polarized (LP) modes slightly separated in frequency – typically Δf = 1 − 15 GHz – due to the bire-frigence of the cavity [32]. At threshold, the laser generally emits linearly polarized light with a direction related to the crystallographic axes of the structure [33–35]. Increasing the current or changing the temperature can however lead to various polarization insta-bilities among which the most striking phenomenon is polarization switching (PS): the first LP mode, otherwise stable since threshold, is destabilized and, after some eventual (a) T = 10◦C gives a PS of type I, (b) T = 15◦C gives a double PS: type I then type II, (c) T = 55◦C gives a PS of type II. HF (LF) is plotted in dashed (solid) line. Taken from [31].
complex dynamical transitions, the laser settles on the orthogonal LP mode. During the last twenty years, this astonishing dynamics has been largely studied and several differ-ent kinds of events, characteristics and features have been reported.

Dynamical properties of quantum dot lasers

From a dynamical viewpoint, the main difference distinguishing QD lasers from their QW counterpart is the description of the carrier dynamics. Whereas in QW devices a single equation for the carriers was sufficient, a proper description of QD devices re-quires to take into account the different dynamics of electrons and holes but also the contribution of the excited state(s) and the eventual wetting layer as additional energy levels [90–92].
Experimentally, it appeared that QD lasers tend to exhibit highly damped relaxation os-cillations compared to regular semiconductor lasers [93], see e.g. Fig. 1.14(a). This high-damping has been linked to the presence of multiple energy levels for carriers and its impact on the carrier dynamics [90, 94]: simulated time-series using the detailed model described in [90] are plotted in Fig. 1.14(b) and clearly show a good agreement with experimental results in (a). At the same time, the large damping suggests that QD laser devices might actually be somewhere between class A and class B lasers. As mentioned earlier, class B lasers are expected to behave like under-damped oscillators while class A lasers would be over-damped oscillators: typically class-B lasers will oscillate toward a stable steady-state whereas class-A lasers will approach it exponentially. Moreover class-A like behaviors of QD devices have been reported both theoretically and experimentally in optically injected devices [95, 96], hence suggesting that despite the increased internal complexity of QD lasers a single equation might be sufficient to describe their evolution.
In terms of stability, the use of QDs instead of QWs has also been expected to make devices with an almost zero α-factor possible. As already mentioned, a smaller α-factor would yield more stable and more robust devices, especially against external perturba-tions such as optical feedback. In practice, a large range of α-factor values have been reported – from close to 0 [97, 98], to almost 60 [99] – and it appeared that different op-erating conditions would lead to largely different values, hence explaining the observed discrepancy [100]. Interestingly, a very recent report also suggests that the same conclu-sion hold for regular semiconductor lasers but with much smaller variations [101]. On the other hand, theoretical investigations showed that a fixed value for the linewidth enhancement factor is, in fact, not accurate enough to describe the carrier induce refrac-tive index changes in QD devices [102]. Even though differences arise only for the most complex dynamics and/or chaotic behaviors, a more detailed and comprehensive model seems to be required in order to obtain a precise modeling of these dynamics.

bifurcation stopolarization switchin g

In this section, we use direct numerical integration combined with continuation tech-niques to identify the bifurcations leading to PSs and to analyze how they are modified when varying the laser parameters.
First, we unveil several PS bifurcation scenarios predicted by the SFM for different val-ues of the birefringence and amplitude anisotropy, including type I and type II PSs with various features. In particular, we highlight that the frequency of the dynamics accom-panying PSs does not necessarily relate to Fsplit, i.e. the birefringence frequency; in fact the frequency is defined by a bifurcation problem and can therefore take a large range of different values that does not necessarily relate to the characteristic frequencies of the system. Secondly, we make a detailed investigation of two specific cases: 1/ a type II PS scenario showing a good agreement with the experimental report of Sondermann et al. [51] – including a transition through EP states and anticorrelated self-pulsing dynam-ics at the birefringence frequency – and 2/ an incomplete switching scenario matching the experimental observations of Olejniczak et al. [53, 54] – including self-pulsations at the relaxation oscillation frequency, two-mode emission for large injection currents and mode-hopping dynamics between non-orthogonal EP states. Thus, we confirm the rele-vance of the SFM to model the dynamics of VCSELs and motivate new experiments.

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Analysis of possible bifurcation scenarios

First, we focus on the evolution of the bifurcations limiting the stability of the three steady-states described previously. Of particular interest is the Hopf bifurcations desta-bilizing the two EP states which lead to self-pulsating solutions at a frequency close to the Hopf frequency – related to the imaginary part of the complex eigenvalues at the bifurcation point. In this section, unless stated otherwise, we will use the following pa-rameter values: γa = {−0.7, 0.7} ns−1, γs = 100 ns−1, γ = 1 ns−1, γp ∈ [0, 100] ns−1, κ = 600 ns−1, α = 3 and µ ∈ [1, 10].
The stability map for the steady-states is depicted in Fig. 2.3 for γa = −0.7 ns−1 and Fig. 2.4 for γa = +0.7 ns−1. Because the sign of γa has a major influence on the system evolution, we need to display the two cases separately. On both figures, the (a)-panel summarizes the parameter regions where the different polarization states are stable in the plane of injection current versus phase anisotropy. The evolution of the Hopf bifur-cation frequencies – i.e. the frequency of the self-pulsing dynamics created close to the bifurcation point – are given in panel (b), along with the splitting Fsplit frequency or birefringence frequency. And in panel (c), we show the evolution of the same frequen-cies but normalized by the relaxation oscillation frequency FRO which varies as a square root of the injection current as described by eq. (2.22). Finally, this overview is completed by a couple of bifurcation diagrams for increasing and decreasing injection currents in order to clarify the dynamical evolution of the system.
The thresholds of the two LPs is not displayed in Figs. 2.3 and 2.4 for simplicity, but the difference between a positive and negative amplitude anisotropy can be simply described as follows: 1/ for γa < 0, a good approximation gives µx,th = 1 + γa/κ and µy,th = 1 − γa/κ. The threshold of X-LP is lower than the one of Y-LP and therefore the laser starts emitting on this state. 2/ for γa > 0 the two approximations remain valid but the situation is reversed: the threshold of Y-LP is lower and X-LP is unstable, hence the laser starts emitting on Y-LP. From Figs. 2.3 and 2.4, we can identify as much as nine different bifurcation scenarios – delimited by different colors and identified by numbers. For scenarios 1 to 4, i.e. for γa < 0, the laser starts emitting on X-LP and we observe the following evolution:
• c a s e 1 – blue area of Fig. 2.3 and bifurcation diagram of panel (d): the laser starts on the X-LP mode and as the injection current is increased a pitchfork bifurcation occurs (black dashed line) which creates the two symmetric EP states described ear-lier. Then the Hopf bifurcations destabilizing the two EP states appear (blue line) leading the system to a stable periodic solution. The frequency of this dynamical state is either close to Fsplit for small γp values or to FRO for higher values of γ p (see Fig. 2.3-(b) and (c)). If the injection current is increased further, the sys-tem experiences a transition through complex dynamical states and chaos until it reaches suddenly the Y-LP stable state as shown in Fig. 2.3(d). Therefore this case corresponds to a type II PS with a large dynamical transition.
For decreasing injection currents, the laser remains on the Y-LP state until it is destabilized by a subcritical Hopf bifurcation (dash-dotted red line). Then, the laser eventually experiences a short dynamical transition before switching back to X-LP or EP steady-states depending on the parameter values.
• c a s e 2 – red area of Fig. 2.3 and bifurcation diagram of panel (e): is similar to case 1, except that the system never reaches the Y-LP steady-state for larger injection current as it never becomes stable. Thus, the laser starts emitting on the X-LP steady-state at threshold which is then destabilized by a pitchfork bifurcation (black dashed line) creating two symmetrical EP steady-states. These two states are how-
ever quickly destabilized by symmetrical Hopf bifurcations (blue line) that lead the system toward self-pulsing dynamics at a frequency close to FRO as shown in Fig. 2.3(c). Next, multiple bifurcations occur for increasing injection currents and lead the system towards a much more complex dynamical state, see Fig. 2.3(e).
• c a s e 3 – yellow area of Fig. 2.3 and bifurcation diagram of panel (f): is identical to case 2 except that three Hopf bifurcations (blue lines) occur on the EP state branch, creating a bistability region between periodic solutions and EP steady-states. This particular hysteresis phenomenon, a bistability between steady-states and self-pulsating solutions, has already been described by Prati et al. [75], and is displayed in Fig. 2.3(f). We can note that the first bifurcation is supercritical and leads the system to a stable periodic solution at a frequency close to the relaxation
oscillation frequency; the second bifurcation is subcritical and the third one is also supercritical but with a frequency much closer to Fsplit. For higher currents a se-quence of bifurcations leads the system to a much more complex dynamical state.

Table of contents :

1 general introduction 
1.1 Semiconductor lasers
1.1.1 From stimulated emission to laser effect
1.1.2 Laser effect in semiconductor materials
1.2 Nonlinear dynamics of laser diodes
1.2.1 Brief history of chaos and lasers
1.2.2 Semiconductor lasers as damped nonlinear oscillators
1.2.3 Impact of time-delayed optical feedback
1.3 Polarization instabilities in vertical-cavity surface-emitting lasers
1.3.1 From edge-emitting lasers to VCSELs
1.3.2 Experimental observations of polarization instabilities in VCSELs .
1.3.3 Theoretical interpretation of polarization instabilities
1.4 Quantum dot laser diodes
1.4.1 Quantum dots as a gain medium
1.4.2 Dynamical properties of quantum dot lasers
1.4.3 Simultaneous emission from the ground and the excited states .
1.5 Outlines of the thesis
2 deterministic polarization chaos 
2.1 Description of the spin-flip model
2.1.1 SFM equations
2.1.2 Steady-states of the SFM
2.1.3 Key measurements
2.2 Bifurcations to polarization switching
2.2.1 Analysis of possible bifurcation scenarios
2.2.2 Scenario of type-II polarization switching
2.2.3 Self-pulsing dynamics without polarization switching
2.3 Deterministic polarization chaos in free-running VCSELs
2.3.1 Experimental characterization of quantum dot VCSELs
2.3.2 Route to polarization chaos
2.3.3 Statistics of the deterministic mode hopping
2.4 Experimental chaos identification
2.4.1 Estimation of the largest Lyapunov exponent
2.4.2 K2-entropy and correlation dimension
2.4.3 Statistical approach and discrimination of colored noise
2.5 Bistability of limit cycles
2.5.1 Experimental observations
2.5.2 Description of the asymmetric SFM
2.5.3 Impact of the asymmetry on the laser dynamics
2.6 Summary and perspectives
3 application of polarization chaos to random bit generation 
3.1 State of the art – optical chaos-based random bit generators
3.2 Experimental setup
3.3 System performances and influence of the post-processing
3.4 Entropy evolution and its link with polarization chaos
3.5 Summary and perspectives
4 two-mode dynamics in qd lasers with optical feedback 
4.1 Background: two-color QD lasers subject to optical feedback
4.2 Mathematical modeling of the QD lasers
4.2.1 Rate equation with separated electron-hole dynamics
4.2.2 Analytical description of the steady-states
4.2.3 Stability analysis
4.3 Solitary QD laser behavior
4.4 Ground-excited state switching with optical feedback
4.4.1 Evolution at low injection currents
4.4.2 Evolution at medium injection currents
4.4.3 Evolution at large injection currents
4.5 Influence of the laser and feedback parameters
4.5.1 Impact of external cavity length variations
4.5.2 Impact of the electron escape rate
4.6 Experimental observation of switching for varying time-delays
4.6.1 Experimental setup
4.6.2 Behavior of the solitary laser diode
4.6.3 Impact of the optical feedback: experimental observations
4.7 Summary and perspectives
5 conclusion and perspectives 
5.1 Main achievements
5.2 Perspectives for future work
a résumé en français 
a.1 Introduction
a.2 Chaos en polarisation dans un laser VCSEL à boite quantique
a.3 Génération de nombres aléatoires à partir du chaos en polarisation
a.4 rétroaction optique sur un laser à boite quantique à deux couleurs
a.5 Conclusion
b linearization of the sfm 
c grassberger-procaccia algorithm implementation 
d dde-biftool details for lasers with feedback 
references 

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