Sustained oscillations accompanying polarization switching in a laser diode 

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Bifurcation diagram of the Lorenz’s equations

Although chaotic dynamics is easily found from direct simulations of Eqs. (1.3)-(1.5), many different dynamics and complex waveforms can also be obtained when varying the simulation parameters. We show in Fig. 1.13 different solutions obtained when varying the parameter r. For r Æ 0 in Fig. 1.13 (a), the system is in its quiescent state with no fluctuation of its variables. Increasing r in Fig. 1.13 (b), chaotic fluctuation in observed. For r Æ 160 in Fig. 1.13 (c), X oscillates quasi-periodically at two different frequencies and finally for high values of r in Fig. 1.13 (d), a periodic train of pulses is observed.

Unlocking nonlinear dynamics in semiconductor lasers

As we have seen, class B lasers are intrinsically stable. During the last 40 years, many techniques have been employed to unlock nonlinear dynamics as it can be useful for various applications (see Section 1.5). In Fig. 1.16, we summarize the main configurations that allow rich nonlinear dynamics in semiconductor lasers [39]. For all of them, the idea is to increase the number of degrees of freedom of the system from 2 to 3 or even infinite for some cases.
• Direct modulation in Fig. 1.16 (a): Under certain conditions, direct modulation of the laser injection current can induce chaotic pulsing. It requires a modulation frequency of the order of the relaxation oscillation frequency with a relatively high modulation depth [40, 41]. In Ref. [42], the polarization properties of VCSEL have been numerically explored under the influence of direct modulation, and chaos was achieved on a broader range of parameters than found in EELs.
• Optical injection in Fig. 1.16 (b): A laser called master injects its light into the cavity of another laser, the slave. A large variety of dynamic can be unlocked when varying the optical frequency detuning between the master and the slave or by increasing the injected power [43, 44]. Some studies have focused on different polarization states of the injected light : either orthogonal to the naturally emitted light of the slave laser [45] or parallel [46]. In other study, the master laser is settled on an already chaotic dynamic [47]. Optical injection might also induce the so-called injection locking where the laser emission wavelength locks on the master laser wavelength. Injection locking has been shown to enhance the modulation frequency limit of the slave laser [48]. • Feedback in Fig. 1.16 (c-d) : There are two main configurations of feedback : the optical feedback and the opto-electronic feedback. In optical feedback experiment, a fraction of the laser light is re-injected in the cavity. In optoelectronic feedback experiment, the emitted light is detected by a photodiode and re-injected in the laser through its injection current. In both cases, the feedback induces a time-delay ¿ which competes with the internal time-scale of the lasers. However they differs in the sense that opto-electronic feedback is incoherent i.e. it does only influence the carrier population while optical feedback is said to be coherent as it interferes with the electrical field. We will later develop the case of optical feedback.
• It is also worth mentioning that chaos has been also achieved in solitary VCSELs. In some cases, a nonlinear coupling can occur between two-polarization modes, hence unlocking series of bifurcations when changing the injection current or the temperature or the strain inside the cavity. This has been reported in a quantum-dots VCSEL [49] where the active region is made of tiny dots of active material or in quantum-well VCSELs subjected to a mechanical strain [50].

Optical feedback and route to chaos

Among the above mentioned techniques to unlock nonlinear dynamics in semiconductor lasers, optical feedback is probably the easiest way as it only requires a simple mirror as shown in fig. 1.17. In this configuration, the external mirror (placed at a distance L of the laser) and the output facet of the laser form an external cavity. As discussed above, the round-trip time of the light inside the external cavity induces a delay ¿ Æ 2L c that interplays with the internal time scale of the laser i.e. the relaxation oscillation period TRO Æ 1/ fRO.
Another parameter of importance in external cavity laser is the feedback ratio. It is defined as the ratio between the power of the feedback light and the total output power of the laser. It can be easily tuned by introducing a variable attenuator in the light path.

Chaos for secure communication

As described in Section 1.3.1, lasers are ubiquitous in today telecommunication system. Cisco has predicted to reach by 2021 an annual IP traffic of 3.3 ZB (3.3 trillions of GB) compared to the 1.2 ZB in 2016 with more than 58% of the world population connected to internet [64]. This expansion of telecommunication combined to the democratization of digital money transaction requires safer data transmission.
Although the most promising technology is the quantum-key distribution (QKD) with an impressive demonstration in 2017 by the Chinese satellite Micius [65] , chaos-based telecommunication can also improve the data security and can be easily incorporated in actual infrastructure [66].
The chaos-based telecommunication relies on the synchronization of two similar laser diodes through optical injection [67–69], one at the emitter side and the other one at the receiver side. While the emitter’s laser (master) is forced into chaotic oscillation through e.g. optical feedback, the receiver’s laser (slave) is injected with the light provided by the first one. When both lasers are almost identical and operate in similar conditions (current and temperature), the receiver’s laser can exhibit identical chaotic oscillations compared to the emitter’s. The chaotic signal is used here as a carrier and the message is carefully encoded in the chaotic carrier. Since synchronization occurs between the two chaotic signals (at the emitter and at the receiver), a simple substraction of the chaotic output of the receiver with the input of the receiver (chaos+message) yields the message decoding. In addition, as the synchronization can only be achieved by the use of an identical laser, decryption of the message can be hardly performed by a spy. This experiment has already been conducted in the city of Athens, Greece [66].

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Chaotic LIDAR for ranging

LIDAR usage is expected to grow quickly in the next few years in crowded environment such as cities and highways. As a result, the probability for on LIDAR device to detect pulses from other LIDAR (causing failures of detection also called ghost images) also increases. In addition, classic LIDAR are vulnerable to jamming i.e. when someone intentionally shines a LIDAR detector, also causing ghost images. In order to prevent such issue, randomly-modulated LIDARs have been proposed [70]. It consist on the emission of randomly modulated optical power through an external modulator. Then, the position of a target is deduced from the cross-correlation between the emitted and the back-scattered signal. As a result, other sources of light contributed solely to noise. However, this technique still has two drawbacks : first, the electronic of modulation needs to be fast which is then very costly, second, it often relies on a pre-designed sequence of random bits that can be detected and reproduced by a jammer.
By contrast, chaotic lidar (CLIDAR) can overcome such problematic as it doesn’t require fast electronic and chaos is by nature unpredictable [71]. It can be achieved by an optical feedback [71] or from an optical injection [72]. However, improvements of CLIDAR is still required for better energy-efficiency. Indeed, usually, the detection electronic is relatively slow (Ç 1 GHz) while chaotic laser tends to distribute its energy around the relaxation oscillation frequency (¼ 10 GHz) which results in energy losses by the equivalent detection low-pass filter. In 2018, Cheng et al. [73] proposed a solution based of an homodyne interference arm that redistributes the energy toward low-frequency regions improving the signal-to-noise ratio by 20dB.

Table of contents :

1 Introduction 
1.1 From maser to laser : the story of Charles H. Townes
1.2 Physics of lasers
1.2.1 Principle
1.2.2 Semiconductor lasers
1.2.3 From edge to surface-emitting lasers
1.2.4 Relaxation oscillations
1.3 Some Applications of lasers
1.3.1 Optical communications
1.3.2 LIDAR for optical ranging and aerosol detection
1.4 Nonlinear dynamics and chaos in laser diodes
1.4.1 The strange-attractor of Lorenz
1.4.2 Bifurcation diagram of the Lorenz’s equations
1.4.3 Analogy between lasers and the Lorenz’s system
1.4.4 Classification of lasers Class C lasers Class B lasers Class A lasers
1.4.5 Unlocking nonlinear dynamics in semiconductor lasers Different approaches Optical feedback and route to chaos
1.5 Applications of chaos in laser diodes
1.5.1 Chaos for secure communication
1.5.2 Chaotic LIDAR for ranging
1.6 Conclusion, objectives and outlines
2 Polarization instabilities in VCSELs and non-local correlation property in low-frequency fluctuation regime 
2.1 Polarization properties in VCSELs
2.1.1 Polarization instabilities in VCSELs in comparison with EELs
2.1.2 Types of polarization switching Polarization selection from gain competition Polarization selection from gain competition and losses
2.1.3 Prediction of polarization switching from spin relaxation process
2.1.4 Application of polarization instabilities Random bits generation Logical gates High-frequency oscillation generation Reservoir computing based on two polarization modes
2.2 VCSEL under optical feedback and low-frequency fluctuation regime
2.2.1 Steady-States and External Cavity Modes
2.2.2 Low-Frequency Fluctuation mechanism
2.2.3 LFF in polarization modes of VCSELs
2.2.4 Observation of double-peak structures in the literature
2.3 Experimental investigation
2.3.1 Experimental setup
2.3.2 LFF : Double peak and correlation features in the RF spectrum
2.4 Physical origin of the double-peak structure
2.4.1 Phase-space dynamic and mode/antimode interaction
2.4.2 Double-peak structure with a single-mode model
2.5 Conclusion
3 Vectorial Rogue Wave in VCSEL light polarizations 
3.1 Observations of Rogue Wave
3.1.1 Freak wave in oceanography
3.1.2 Identification of rogue waves
3.1.3 Rogue Waves in optics and motivation
3.2 Observations of Extreme Events in VCSELs
3.2.1 Two types of extreme events in VCSELs
3.3 Extreme events statistics
3.3.1 Deviation from a Gaussian distribution
3.3.2 Waiting times between successive events
3.4 Generation rate of extreme events
3.4.1 Vectorial Extreme events
3.4.2 Noise effect on the generation rate of EEs
3.4.3 Differences between Type-I and Type-II LFF
3.5 Modal competition effect on the EEs generation rate
3.6 Conclusion and perspective
4 Sustained oscillations accompanying polarization switching in a laser diode 
4.1 Square-wave modulation in optics
4.2 Bifurcation to high-frequency oscillation in square-wave regime
4.2.1 Description of the experimental setup
4.2.2 An overview on the observed dynamics
4.2.3 Effects of the feedback ratio and the injection current
4.2.4 Influence of the delay
4.3 Numerical Investigation
4.3.1 Rate equations for EEL subjected to a PROF
4.3.2 Bifurcation scenario leading to sustained oscillations
4.3.3 Influence to the delay
4.3.4 Effect of the laser parameters Pump parameter P Carrier to photon lifetime ratio T Linewidth enhancement factor ® Gain coefficient ratio k and TM additional losses ¯
4.3.5 Effect of noise
4.4 Analytical investigation on the sustained oscillations frequency
4.4.1 Steady states
4.4.2 Hopf bifurcations
4.4.3 Approximations
4.5 Conclusion and perspectives
5 Optical chimera in light polarization 
5.1 Chimera state : a coexistence of coherence and incoherence
5.1.1 Demonstration of spatially extended chimera states Chemical oscillators Optolectronic oscillators Mechanical oscillators
5.1.2 Chimera states in optical spectrum and virtual-space In the optical spectrum of a mode-locked laser In a virtual space from a time-delay system
5.1.3 Discussion on optical chimera state
5.2 Chimera state in laser polarization
5.2.1 Experimental setup
5.2.2 Theoretical model
5.2.3 Observation of virtual chimera states
5.2.4 Stabilization of multi-headed chimeras with a 2nd delay
5.2.5 Multi-stability and chimera-heads mechanisms
5.2.6 Influence of the initialization and of the feedback strength Influence of the initialization Influence of the PROF strength
5.3 Conclusion
6 Conclusion 
6.1 Summary, contributions and perspectives
7 Résumé de la thèse 


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