Self-organization in semiconductor lasers with ultrashort optical feedback

Numerical study of ultrashort-optical-feedback-enhanced photonic microwave generation using optically injected semiconductor lasers at period-one nonlinear dynamics

This study numerically investigates the enhancement of photonic microwave generation using an optically injected semiconductor laser operating at period-one (P1) nonlinear dynamics through ultrashort optical feedback. For the purpose of practical applications where system miniaturization is generally preferred, a feedback delay time that is one to two orders of magnitude shorter than the relaxation resonance period of a typical laser is emphasized. Various dynamical states that are more complicated than the P1 dynamics can be excited under a number of ultrashort optical feedback conditions. Within the range of the P1 dynamics, on one hand, the frequency of the P1 microwave oscillation can be greatly enhanced by up to more than three folds. Generally speaking, the microwave frequency enhances with the optical feedback power and phase, while it varies saw-wise with the optical feedback delay time. On the other hand, the purity of the P1 microwave oscillation can be highly improved by up to more than three orders of magnitude. In general, the microwave purity improves with the optical feedback power and delay time, while it only varies within an order of magnitude with the optical feedback phase. These results suggest that the ultrashort optical feedback provides the optically injected laser system with an extra degree of freedom to manipulate/improve the characteristics of the P1 microwave oscillation without changing the optical injection condition.

Mapping of external cavity modes for a laser diode subject to phase-conjugate feedback

We numerically investigate the dynamics of a semiconductor laser subject to phase-conjugate optical feedback. We explore the effects of the laser model and feedback parameters for the generation of time-periodic oscillations of the output power at harmonics of the external cavity frequency, i.e., dynamical solutions that have been named external cavity modes. We point out that both the experimentally tunable and other parameters have an influence on the frequency of such dynamics. Since the delay has to exist, it is not the relevant parameter as we show that the feedback rate fixes the frequency of the periodic self-pulsations. The interaction length of the crystal and the ratio between carrier and photon lifetimes tend to filter out high frequencies as they increase. Finally, the linewidth enhancement factor unlocks high frequencies as it increases. We conclude by providing a situation which leads to periodic solutions with higher frequencies using a set of realistic values of parameters.

Stability of steady and periodic states through the bifurcation bridge mechanism in semiconductor ring lasers subject to optical feedback

With the development of new applications using semiconductor ring lasers (SRLs) subject to optical feedback, the stability properties of their outputs becomes a crucial issue. We propose a systematic bifurcation analysis in order to properly identify the best parameter ranges for either steady or self-pulsating periodic regimes. Unlike conventional semiconductor lasers, we show that SRLs exhibit both types of outputs for large and well defined ranges of the feedback strength. We determine the stability domains in terms of the pump parameter and the feedback phase. We find that the feedback phase is a key parameter to achieve a stable steady output. We demonstrate that the self-pulsating regime results from a particular Hopf bifurcation mechanism referred to as bifurcation bridges. These bridges connect two distinct external cavity modes and are fully stable, a scenario that was not possible for diode lasers under the same conditions.

Photonic integrated circuits unveil crisis-induced intermittency

We experimentally investigate an intermittent route to chaos in a photonic integrated circuit consisting of a semiconductor laser with time-delayed optical feedback from a short external cavity. The transition from a period-doubling dynamics to a fully-developed chaos reveals a stage intermittently exhibiting these two dynamics. We unveil the bifurcation mechanism underlying this route to chaos by using the Lang-Kobayashi model and demonstrate that the process is based on a phenomenon of attractor expansion initiated by a particular distribution of the local Lyapunov exponents. We emphasize on the crucial importance of the distribution of the steady-state solutions introduced by the time-delayed feedback on the existence of this intermittent dynamics.

Generic stabilizability for time-delayed feedback control

Time-delayed feedback control is one of the most successful methods to discover dynamically unstable features of a dynamical system in an experiment. This approach feeds back only terms that depend on the difference between the current output and the output from a fixed time T ago. Thus, any periodic orbit of period T in the feedback-controlled system is also a periodic orbit of the uncontrolled system, independent of any modelling assumptions. It has been an open problem whether this approach can be successful in general, that is, under genericity conditions similar to those in linear control theory (controllability), or if there are fundamental restrictions to time-delayed feedback control. We show that, in principle, there are no restrictions. This paper proves the following: for every periodic orbit satisfying a genericity condition slightly stronger than classical linear controllability, one can find control gains that stabilize this orbit with extended time-delayed feedback control. While the paper's techniques are based on linear stability analysis, they exploit the specific properties of linearizations near autonomous periodic orbits in nonlinear systems, and are, thus, mostly relevant for the analysis of nonlinear experiments.

Cycles of self-pulsations in a photonic integrated circuit

We report experimentally on the bifurcation cascade leading to the appearance of self-pulsation in a photonic integrated circuit in which a laser diode is subjected to delayed optical feedback. We study the evolution of the self-pulsing frequency with the increase of both the feedback strength and the injection current. Experimental observations show good qualitative accordance with numerical results carried out with the Lang-Kobayashi rate equation model. We explain the mechanism underlying the self-pulsations by a phenomenon of beating between successive pairs of external cavity modes and antimodes.

Integrated semiconductor laser with optical feedback: transition from short to long cavity regime

A 4-section semiconductor laser with integrated optical feedback has been shown experimentally to be capable of operating in either the short- or long-cavity regime, by controlling the device relaxation oscillation frequency relative to the external cavity frequency. Systematic increase of the laser injection current, and the resulting increase in relaxation oscillation frequency, allowed the transition between the two regimes of operation to be observed. The system displayed a gradual transition from a dynamic dominated by regular pulse packages in the short-cavity regime to one dominated by broadband chaotic output when operating in the long-cavity regime. This suggests that the "short cavity" regular pulse packages continue to co-exist with the "long cavity" broadband chaotic dynamic in the system studied. It is the relative power associated with each of these dynamics that changes. This may occur more generally in similar systems.

Numerical study of extreme events in a laser diode with phase-conjugate optical feedback

Extreme intensity pulses sharing statistical properties similar to rogue waves have been recently observed in a laser diode with phase-conjugate feedback [A. Karsaklian Dal Bosco, D. Wolfersberger, and M. Sciamanna, Opt. Lett. 38, 703 (2013)], but remain unexplained. We demonstrate here that a rate equation model of a laser diode that includes an instantaneous phase-conjugate feedback field reproduces qualitatively well the statistical features of these extreme events as identified in the experiment, i.e., the deviation of the intensity statistics to a Gaussian-shape statistics and the statistics of the time separating extreme events. The numerical simulations confirm the importance of the feedback strength in increasing the number of such extreme events and allow us to explain how extreme events emerge from a sequence of bifurcations on self-pulsating solutions, the so-called external cavity modes.

Nonlinear dynamics in integrated coupled DFB lasers with ultra-short delay

We report rich nonlinear dynamics in integrated coupled lasers with ultra-short coupling delay. Mutually stable locking, period-1 oscillation, frequency locking, quasi-periodicity and chaos are observed experimentally. The dynamic behaviors are reproduced numerically by solving coupled delay differential equations that take the variation of both frequency detuning and coupling phase into account. Moreover, it is pointed out that the round-trip frequency is not involved in the above nonlinear dynamical behaviors. Instead, the relationship between the frequency detuning Δν and the relaxation oscillation frequency νr under mutual injection are found to be critical for the various observed dynamics in mutually coupled lasers with very short delay.

Odd-number theorem: optical feedback control at a subcritical Hopf bifurcation in a semiconductor laser

A subcritical Hopf bifurcation is prepared in a multisection semiconductor laser. In the free-running state, hysteresis is absent due to noise-induced escape processes. The missing branches are recovered by stabilizing them against noise through application of phase-sensitive noninvasive delayed optical feedback control. The same type of control is successfully used to stabilize the unstable pulsations born in the Hopf bifurcation. This experimental finding represents an optical counterexample to the so-called odd-number limitation of delayed feedback control. However, as a leftover of the limitation, the domains of control are extremely small.

Chaos-on-a-chip secures data transmission in optical fiber links

Security in information exchange plays a central role in the deployment of modern communication systems. Besides algorithms, chaos is exploited as a real-time high-speed data encryption technique which enhances the security at the hardware level of optical networks. In this work, compact, fully controllable and stably operating monolithic photonic integrated circuits (PICs) that generate broadband chaotic optical signals are incorporated in chaos-encoded optical transmission systems. Data sequences with rates up to 2.5 Gb/s with small amplitudes are completely encrypted within these chaotic carriers. Only authorized counterparts, supplied with identical chaos generating PICs that are able to synchronize and reproduce the same carriers, can benefit from data exchange with bit-rates up to 2.5Gb/s with error rates below 10(-12). Eavesdroppers with access to the communication link experience a 0.5 probability to detect correctly each bit by direct signal detection, while eavesdroppers supplied with even slightly unmatched hardware receivers are restricted to data extraction error rates well above 10(-3).

Tristability of a semiconductor laser due to time-delayed optical feedback

We present an experimental and theoretical study of multistability of a single-mode laser subject to feedback through phase tuning and amplifier sections integrated on the same chip. Closely above threshold, a regime of tristability of continuous-wave (CW) states is found for multiple ranges of amplifier and phase currents. The separation between the tristable wavelengths agrees with the channel spacing of dense wavelength multiplexing in the C band of optical communication making the device interesting for ternary logic applications. Complementary theoretical investigations in the framework of the paradigmatic Lang-Kobayashi model provide a consistent understanding of the experimental findings and additionally yield an analytic formula expressing the maximum number of coexisting stable CW states by the linewidth-enhancement factor alpha . Tristability belongs to the alpha range from 5 to 8 in good agreement with experiment.

Stabilizing continuous-wave output in semiconductor lasers by time-delayed feedback

The stabilization of steady states is studied in a modified Lang-Kobayashi model of a semiconductor laser. We show that multiple time-delayed feedback, realized by a Fabry-Perot resonator coupled to the laser, provides a valuable tool for the suppression of unwanted intensity pulsations, and leads to stable continuous-wave operation. The domains of control are calculated in dependence on the feedback strength, delay time (cavity round trip time), memory parameter (mirror reflectivity), latency time, feedback phase, and bandpass filtering. Due to the optical feedback, multistable behavior can also occur in the form of delay-induced intensity pulsations or other modes for certain choices of the control parameters. Control may then still be achieved by slowly ramping the injection current during turn-on.

Photonic integrated device for chaos applications in communications

A novel photonic monolithic integrated device consisting of a distributed feedback laser, a passive resonator, and active elements that control the optical feedback properties has been designed, fabricated, and evaluated as a compact potential chaotic emitter in optical communications. Under diverse operating parameters, the device behaves in different modes providing stable solutions, periodic states, and broadband chaotic dynamics. Chaos data analysis is performed in order to quantify the complexity and chaoticity of the experimental reconstructed attractors by applying nonlinear noise filtering.

Control of unstable steady states by extended time-delayed feedback

Time-delayed feedback methods can be used to control unstable periodic orbits as well as unstable steady states. We present an application of extended time delay autosynchronization introduced by Socolar [Phys. Rev. E 50, 3245 (1994)] to an unstable focus. This system represents a generic model of an unstable steady state which can be found, for instance, in Hopf bifurcation. In addition to the original controller design, we investigate effects of control loop latency and a bandpass filter on the domain of control. Furthermore, we consider coupling of the control force to the system via a rotational coupling matrix parametrized by a variable phase. We present an analysis of the domain of control and support our results by numerical calculations.

Correlation theory of delayed feedback in stochastic systems below Andronov-Hopf bifurcation

Here we address the effect of large delay on the statistical characteristics of noise-induced oscillations in a nonlinear system below Andronov-Hopf bifurcation. In particular, we introduce a theory of these oscillations that does not involve the eigenmode expansion, and can therefore be used for arbitrary delay time. In particular, we show that the correlation matrix (CM) oscillates on two different time scales: on the scale of the main period of noise-induced oscillations, and on the scale close to the delay time. At large values of the delay time, the CM is shown to decay exponentially only for large values of its argument, while for the arguments comparable with the value of the delay, the CM remains finite disregarding the delay time. The definition of the correlation time of the system with delay is discussed.

All-optical noninvasive control of unstable steady states in a semiconductor laser

All-optical noninvasive control of a multisection semiconductor laser by means of time-delayed feedback from an external Fabry-Perot cavity is realized experimentally. A theoretical analysis, in both a generic model as well as a device-specific simulation, points out the role of the optical phase. Using phase-dependent feedback we demonstrate stabilization of the continuous-wave laser output and noninvasive suppression of intensity pulsations.

Frequency versus relaxation oscillations in a semiconductor laser with coherent filtered optical feedback

We investigate the dynamics of a semiconductor laser subject to coherent delayed filtered optical feedback. A systematic bifurcation analysis reveals that this system supports two fundamentally different types of oscillations, namely relaxation oscillations and external roundtrip oscillations. Both occur stably in large domains under variation of the feedback conditions, where the feedback phase is identified as a key quantity for controlling this dynamical complexity. We identify two separate parameter regions of stable roundtrip oscillations, which occur throughout in the form of pure frequency oscillations.

Coherence resonance near a Hopf bifurcation

We report on the observation of coherence resonance for a semiconductor laser with short optical feedback close to Hopf bifurcations. Noise-induced self-pulsations are documented by distinct Lorentzian-like features in the power spectrum. The character of coherence is critically related to the type of the bifurcation. In the supercritical case, spectral width and height of the peak are monotonic functions of the noise level. In contrast, for the subcritical bifurcation, the width exhibits a minimum, translating into resonance behavior of the correlation time in the pulsation transients. A theoretical analysis based on the generic model of a self-sustained oscillator demonstrates that these observations are of general nature and are related to the fact that the damping depends qualitatively different on the noise intensity for the subcritical and supercritical case.

Frequency bistability of a semiconductor laser under a frequency-dependent feedback

The emission frequency of a diode laser submitted to a frequency-dependent optoelectronic feedback is observed to have more than one stable operation point together with a stable power emission. This is, to our knowledge, the first observation of bistability exclusively in the frequency of an optical system. The experiment was carried out with a semiconductor laser coupled to the cesium D2 line by an orthogonally polarized frequency-sensitive optical feedback.

Synchronization of delay-coupled oscillators: a study of semiconductor lasers

Two delay-coupled semiconductor lasers are studied in the regime where the coupling delay is comparable to the time scales of the internal laser oscillations. Detuning the optical frequency between the two lasers, novel delay-induced scenarios leading from optical frequency locking to successive states of periodic intensity pulsations are observed. We demonstrate and analyze these dynamical phenomena experimentally using two distinct laser configurations. A theoretical treatment reveals the universal character of our findings for delay-coupled systems.