Emeric Mercier

Physical random bit generation from chaotic solitary laser diode

We demonstrate the physical generation of random bits at high bit rates (> 100 Gb/s) using optical chaos from a solitary laser diode and therefore without the complex addition of either external optical feedback or injection. This striking result is obtained despite the low dimension and relatively small bandwidth of the laser chaos, i.e. two characteristics that have been so far considered as limiting the performances of optical chaos-based applications. We unambiguously attribute the successful randomness at high speed to the physics of the laser chaotic polarization dynamics and the resulting growth rate of the dynamical entropy.

Enhanced complexity of optical chaos in a laser diode with phase-conjugate feedback

We demonstrate numerically that a semiconductor laser subjected to phase-conjugate feedback (PCF) can exhibit an enhancement in the complexity of chaos by comparison to conventional optical feedback. Using quantifiers from spectral analysis and information theory, we demonstrate that under similar parametric conditions, PCF exhibits a larger chaotic bandwidth and higher spectral flatness and statistical complexity. These properties are of utmost importance for applications in secure communications and random number generation.

Bifurcation to chaotic low-frequency fluctuations in a laser diode with phase-conjugate feedback

We unveil theoretically the bifurcations to chaotic low-frequency fluctuations (LFF) in a laser diode with phase-conjugate feedback (PCF). LFF occur from a chaotic itinerancy among destabilized limit-cycle attractors that correspond to the external-cavity modes (ECMs) of the PCF laser system and with a directional motion toward a self-pulsating dynamics of increasing frequency and larger output power. When increasing the feedback strength, the frequency of the fast-pulsing dynamics changes about a multiple of the external-cavity frequency, which is a unique feature of LFF in a laser diode with PCF.

Bistability of time-periodic polarization dynamics in a free-running VCSEL

We report experimentally a bistability between two limit cycles (i.e. time-periodic dynamics) in a free-running vertical-cavity surface-emitting laser. The two limit cycles originate from a bifurcation on two elliptically polarized states which exhibit a small frequency difference and whose main axes are symmetrical with respect to the linear polarization eigenaxes at threshold. We demonstrate theoretically that this peculiar behavior can be explained in the framework of the spin-flip model model by taking into account a small misalignment between the phase and amplitude anisotropies.

High-frequency chaotic dynamics enabled by optical phase-conjugation

Wideband chaos is of interest for applications such as random number generation or encrypted communications, which typically use optical feedback in a semiconductor laser. Here, we show that replacing conventional optical feedback with phase-conjugate feedback improves the chaos bandwidth. In the range of achievable phase-conjugate mirror reflectivities, the bandwidth increase reaches 27% when compared with feedback from a conventional mirror. Experimental measurements of the time-resolved frequency dynamics on nanosecond time-scales show that the bandwidth enhancement is related to the onset of self-pulsing solutions at harmonics of the external-cavity frequency. In the observed regime, the system follows a chaotic itinerancy among these destabilized high-frequency external-cavity modes. The recorded features are unique to phase-conjugate feedback and distinguish it from the long-standing problem of time-delayed feedback dynamics.

High-order external cavity modes and restabilization of a laser diode subject to a phase-conjugate feedback

We experimentally report the sequence of bifurcations destabilizing and restabilizing a laser diode with phase-conjugate feedback when the feedback rate is increased. Specifically, we successively observe the initial steady state, undamped relaxation oscillations, quasi-periodicity, chaos, and oscillating solutions at harmonics up to 13 times the external cavity frequency but also the restabilization to a steady state. The experimental results are qualitatively well reproduced by a model that accounts for the time the light takes to penetrate the phase-conjugate mirror. The theory points out that the system restabilizes through a Hopf bifurcation whose frequency is a harmonic of the external cavity frequency.

Improving the chaos bandwidth of a semiconductor laser with phase-conjugate feedback

Common applications using optical chaos in a semiconductor laser include, among others, random number generation and chaos-encrypted communications. They rely on chaos of high dimension with a large bandwidth and a high entropy growth rate to achieve good results. Optical chaos from a semiconductor laser with conventional optical feedback (COF) is typically used as the primary source of chaos. Additional enhancing techniques are used to enlarge the chaos bandwidth. In this contribution, we show experimentally how using phase-conjugate feedback (PCF) can naturally produce a chaos of higher bandwidth than COF. PCF is an alternative to COF which consists of feeding the conjugate of the optical output back into the laser cavity, with a time-delay. Thanks to an oscilloscope with a fast sampling rate, and a large bandwidth, we were able to measure and observe the time-resolved frequency dynamics with a good precision. In the regime of low-frequency fluctuations (LFF), where dropouts of optical power occur randomly, we were able to compare the difference in dynamics before and after a dropout, for PCF and COF. In the range of attainable reflectivities, we measured a bandwidth increase of up to 27 % with PCF when compared to COF. Interestingly, we found that high-frequency dynamics are enabled before dropouts in PCF, where it was theoretically shown that the system jumps between destabilized self-pulsing states at harmonics of the external-cavity frequency, the so-called external-cavity modes (ECMs). This observation tends to confirm that ECMs in PCF are indeed fundamentally different than ECMs in COF, where they are simple steady-states. Finally, we believe that the enhancing techniques used with COF could also be used with PCF to obtain even wider chaotic bandwidths. These results could lead to studies about the dimension and the entropy growth rate of chaos from a laser diode with PCF.

Harmonic fundamental self-pulsations from a laser diode using phase-conjugate optical feedback

Thanks to the band-gap engineering of quantum confined semiconductor materials and the development of semiconductor-based saturable absorber mirrors, recent years have seen the development of compact and low-cost external-cavity laser diodes generating pulses at several tens of GHz. The physics of the bifurcation leading to selfpulsation leads however to an intrinsic limitation: the fundamental repetition rate is fixed to and limited by the externalcavity round-trip time. By contrast, we demonstrate here that an external-cavity diode laser may generate fundamental self-pulsating dynamics at harmonics of the external-cavity frequency, when a phase conjugate mirror replaces the conventional mirror. As is known from theory, a laser diode with phase conjugate external feedback supports a single stationary solution that bifurcates to self-pulsating dynamics of increasing frequency when increasing the amount of light reflected back to the laser diode. The self-pulsation frequency then increases in step of the external-cavity frequency as one increases the feedback strength. We provide here the first experimental evidence of such harmonic external-cavity fundamental self-pulsation. As a proof-of-concept, we generate experimentally a self-pulsating dynamics at twice and three times the fundamental external-cavity frequency using an edge-emitting laser with a self-pumped ring-cavity photorefractive phase conjugator. Numerical simulations also predict stable higher harmonics.

Experimental and theoretical analysis of limit cycle bistability in a free-running VCSEL

Laser diodes typically behave like damped oscillators: they are generally expected to only show damped relaxation oscillations toward a stable fixed point. In vertical-cavity surface-emitting lasers (VCSELs), the picture appears to be quite different as polarization dynamics can be experimentally observed including bifurcations to self- pulsation and even chaos. Physically, the circular geometry of VCSELs makes the polarization selection very weak and, thus, the additional degree of freedom can enable complex dynamical behavior in the laser diode. Here we report on a new dynamical behavior in a free-running VCSEL: we observe a bistability between two limit cycles oscillating around two distinct elliptical polarization states whose main axes are symmetrical with respect to the polarization at threshold. Although the existence of two symmetric elliptical polarizations and the associated limit cycles are predicted by the San Miguel, Feng and Moloney (SFM) model, the hysteresis cycle observed experimentally highlights the importance of asymmetry in the dynamics from the elliptically polarized states. We demonstrate that this behavior can be accurately reproduced in theory within the SFM framework when taking into account a small misalignment between the phase and amplitude anisotropies of the laser cavity. Our results bring new light into VCSEL polarization dynamics and provide a very good qualitative agreement with the bifurcation scenario predicted by the SFM model.

Random bit generation using polarization chaos from free-running laser diode

During the last five years, optical chaos-based random bit generators (RBGs) attracted a lot of attention and demonstrated impressive performances with bit rates up to hundreds of Gbps. However all the suggested schemes use external injection schemes (optical injection or feedback) to turn the lasers into chaos, hence strongly increasing setup complexity. On the other hand, we reported that a laser diode can generate a chaotic output without the need for external perturbation or forcing, hence unveiling a highly simplified way to generate an optical chaos at high frequency. However the low dimension and limited number of positive Lyapunov exponent casted doubts about its direct use for chaos-based applications. Here we make a proof-of-concept demonstration for a Random Bit Generator based on polarization chaos. We therefore suggest a highly simplified RBG scheme using only a free-running laser and small-bandwidth acquisition electronics and demonstrate convincing performances with bit rates up to 100 Gbps without unusual or complex post-processing methods. We link these performances to the double-scroll structure of the chaotic attractor rather than the bandwidth of the dynamics, hence bringing new light on the importance of chaos topology for chaos-based applications. In addition our scheme exhibit a strong potential as it enables a low-cost and/or integrated in parallel on-chip scheme.