Time-delayed Optical Feedback Research Articles

Experimental estimation of sample entropy in a semiconductor laser with optical feedback for random number generation

Estimating the entropy rate of physical random number generators with uncertainty is crucial for information security applications. We evaluate the sample entropy of chaotic temporal waveforms generated experimentally by a semiconductor laser with time-delayed optical feedback. We demonstrate random number generation with uncertainty using a quantitative measurement of the entropy rate.

Square waves and Bykov T-points in a delay algebraic model for the Kerr-Gires-Tournois interferometer.

We study theoretically the mechanisms of square wave formation of a vertically emitting micro-cavity operated in the Gires-Tournois regime that contains a Kerr medium and that is subjected to strong time-delayed optical feedback and detuned optical injection. We show that in the limit of large delay, square wave solutions of the time-delayed system can be treated as relative homoclinic solutions of an equation with an advanced argument. Based on this, we use concepts of classical homoclinic bifurcation theory to study different types of square wave solutions. In particular, we unveil the mechanisms behind the collapsed snaking scenario of square waves and explain the formation of complex-shaped multistable square wave solutions through a Bykov T-point. Finally, we relate the position of the T-point to the position of the Maxwell point in the original time-delayed system.

Complete and partial time-delay signature suppression in a laser array.

We model dynamics of a quantum dot based micropillar laser array subject to the time-delayed optical feedback. The global coupling provided by the feedback generates a rich set of various instabilities including chaotic regimes with strong time-delay signature in the autocorrelation function. We demonstrate that the dispersion of the array coupling phases leads to effective suppression of the time-delay signature due to the dispersion of the system's internal timescales. We find that the transition to the complete suppression of the time-delay signature appears via a chimera state where highly correlated and non-correlated laser outputs coexist. The degree of correlation in the chimera state depends on the coupling phase dispersion.

Temporal localized states and square-waves in semiconductor micro-resonators with strong time-delayed feedback.

In this paper, we study the dynamics of a vertically emitting micro-cavity operated in the Gires-Tournois regime that contains a semiconductor quantum-well and that is subjected to strong time-delayed optical feedback and detuned optical injection. Using a first principle time-delay model for the optical response, we disclose sets of multistable dark and bright temporal localized states coexisting on their respective bistable homogeneous backgrounds. In the case of anti-resonant optical feedback, we identify square-waves with a period of twice the round-trip in the external cavity. Finally, we perform a multiple time scale analysis in the good cavity limit. The resulting normal form is in good agreement with the original time-delayed model.

Optimizations of optical chaos in semiconductor lasers based on multiobjective genetic algorithms.

We investigated the frequency bandwidth, autocorrelation function, and complexity of chaotic temporal waveforms in unidirectionally coupled semiconductor lasers with time-delayed optical feedback. The effective bandwidth, peak value of autocorrelation function, and maximum Lyapunov exponent were simultaneously optimized by searching several control parameters of the laser systems based on multiobjective genetic algorithms. We found a conflicting relation between the effective bandwidth enhancement and the time-delay signature suppression, and a detailed relationship between the maximum Lyapunov exponent and the peak value of autocorrelation function.

Fast dynamics of low-frequency fluctuations in a quantum-dot laser with optical feedback.

We experimentally investigate the complex dynamics of a multi-mode quantum-dot semiconductor laser with time-delayed optical feedback. We examine a two-dimensional bifurcation diagram of the quantum-dot laser as a comprehensive dynamical map by changing the injection current and feedback strength. We found that the bifurcation diagram contains two different parameter regions of low-frequency fluctuations. The power-dropout dynamics of the low-frequency fluctuations are observed in the sub-GHz region, which is considerably faster than the conventional low-frequency fluctuations in the MHz region. Comparing the dynamics of quantum-dot laser with those of single- and multi-mode quantum-well semiconductor lasers reveals that the fast low-frequency fluctuation dynamics are unique characteristics of quantum-dot lasers with time-delayed optical feedback.

Delay-induced instability in phase-locked dual-polarization distributed-feedback fiber lasers

We study experimentally and theoretically a dual-polarization fiber laser submitted to time-delayed frequency-shifted optical feedback. In addition to the usual frequency-locking regime that is expected when the frequency shift is close to the polarization beat frequency, we observe at high pumping rates a dynamical pulsing regime inside the locking range. This regime is experimentally evidenced in a full-fibered experiment based on an erbium-doped distributed-feedback fiber laser in which polarization beat frequency is about 1 GHz. A rate-equation model including the frequency-shifted feedback term reproduces well the experimental bifurcation maps, provided that both the time delay and a phase-amplitude coupling parameter ($\ensuremath{\alpha}$ factor) are taken into account. The impact on microwave-photonics applications is discussed.

Tunable Kerr frequency combs and temporal localized states in time-delayed Gires-Tournois interferometers.

In this Letter, we study theoretically a new setup allowing for the generation of temporal localized states (TLSs) and frequency combs. The setup is compact (a few centimeters) and can be implemented using established technologies, while offering tunable repetition rates and potentially high power operation. It consists of a vertically emitting micro-cavity, operated in the Gires-Tournois regime, containing a Kerr medium strong time-delayed optical feedback, and detuned optical injection. We disclose sets of multistable dark and bright TLSs coexisting on their respective bistable homogeneous backgrounds.

Class-C semiconductor lasers with time-delayed optical feedback.

We perform a linear stability analysis and numerical bifurcation diagrams of a class-C laser with time-delayed optical feedback. We employ a rate equation system based on the Maxwell-Bloch equations, and study the influence of the dephasing time on the laser dynamics. We find a stabilizing effect of an intermediate dephasing time, i.e. when moving from a class-B to a class-C laser. At long dephasing times, a destabilization of the laser solution occurs by a feedback-induced unlocking of Rabi oscillations at the second laser threshold. We predict an optimum resistance to time-delayed optical feedback for dephasing times close to the photon cavity lifetime. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Quantum-dot micropillar lasers subject to coherent time-delayed optical feedback from a short external cavity

We investigate the mode-switching dynamics of an electrically driven bimodal quantum-dot micropillar laser when subject to delayed coherent optical feedback from a short external cavity. We experimentally characterize how the external cavity length, being on the same order than the microlaser’s coherence length, influences the spectral and dynamical properties of the micropillar laser. Moreover, we determine the relaxation oscillation frequency of the micropillar by superimposing optical pulse injection to a dc current. It is found that the optical pulse can be used to disturb the feedback-coupled laser within one roundtrip time in such a way that it reaches the same output power as if no feedback was present. Our results do not only expand the understanding of microlasers when subject to optical feedback from short external cavities, but pave the way towards tailoring the properties of this key nanophotonic system for studies in the quantum regime of self-feedback and its implementation to integrated photonic circuits.

Passively mode-locked quantum dot laser subject to dual long cavity time-delayed optical feedback

In this contribution, we study experimentally the impact of two long external passive optical cavities on the repetition rate and timing jitter of a monolithic passively mode-locked InAs/InGaAs quantum dot laser. The dual cavity scheme allows to fully suppress feedback induced sidebands in the radio-frequency spectrum. By fine delay control of the optical self-feedback delay lengths, a repetition rate tunability of 9.4 MHz is achieved. The lowest repetition rate line width amounts to less than 50 Hz, corresponding to a timing jitter of 1.7 fs. We explain the experimentally observed timing jitter and repetition rate trends qualitatively by adapting a versatile simple time-domain model. Both experiment and simulations indicate the strongest pulse train timing jitter reduction at feedback lengths that yield the lowest repetition rate frequency deviation

Effect of Bias Current on Complexity and Time Delay Signature of Chaos in Semiconductor Laser With Time-Delayed Optical Feedback

The effect of bias current on the complexity and time-delay signature of chaotic signals in semiconductor lasers with polarization preserved optical feedback has been studied experimentally and theoretically. The peak value of the autocorrelation coefficient and the normalized permutation entropy at the feedback round trip time are used to quantify the time delay signature and complexity, respectively. The results show that the time-delay signature is approximately in an inverse relationship with the complexity of chaos when the semiconductor laser is subject to low or strong optical feedback. However, the inverse relationship disappears when the laser operates at higher bias currents with intermediate feedback strength. The simulation results are qualitatively agreed with the experimental results.

Experimental characterization of the transition to coherence collapse in a semiconductor laser with optical feedback.

Semiconductor lasers with time-delayed optical feedback display a wide range of dynamical regimes, which have found various practical applications. They also provide excellent testbeds for data analysis tools for characterizing complex signals. Recently, several of us have analyzed experimental intensity time-traces and quantitatively identified the onset of different dynamical regimes, as the laser current increases. Specifically, we identified the onset of low-frequency fluctuations (LFFs), where the laser intensity displays abrupt dropouts, and the onset of coherence collapse (CC), where the intensity fluctuations are highly irregular. Here we map these regimes when both, the laser current and the feedback strength vary. We show that the shape of the distribution of intensity fluctuations (characterized by the standard deviation, the skewness, and the kurtosis) allows to distinguish among noise, LFFs and CC, and to quantitatively determine (in spite of the gradual nature of the transitions) the boundaries of the three regimes. Ordinal analysis of the inter-dropout time intervals consistently identifies the three regimes occurring in the same parameter regions as the analysis of the intensity distribution. Simulations of the well-known time-delayed Lang-Kobayashi model are in good qualitative agreement with the observations.

Role of dynamical injection locking and characteristic pulse events for low frequency fluctuations in semiconductor lasers.

We investigate the dynamics of semiconductor lasers subject to time-delayed optical feedback from the perspective of dynamical self-injection locking. Based on the Lang-Kobayashi model, we perform an analysis of the well-known Low Frequency Fluctuations (LFFs) in the frequency-intensity plane. Moreover, we investigate a recently found dynamical regime of fragmented LFFs by means of a locking-range analysis, spectral comparison and precursor pulse identification. We show that LFF dynamics can be explained by dynamical optical injection locking due to the delayed optical feedback. Moreover, the fragmented LFFs occur due to a re-injection locking induced by a particular optical pulse structure in the chaotic feedback dynamics. This is corroborated by experiments with a semiconductor laser experiencing delayed feedback from an optical fiber loop. The dynamical nature of the feedback injection results in an eventual loss, but also possible regaining, of the locking, explaining the recently observed phenomenon of fragmented LFFs.

Delayed feedback control of self-mobile cavity solitons in a wide-aperture laser with a saturable absorber.

We investigate the spatiotemporal dynamics of cavity solitons in a broad area vertical-cavity surface-emitting laser with saturable absorption subjected to time-delayed optical feedback. Using a combination of analytical, numerical, and path continuation methods, we analyze the bifurcation structure of stationary and moving cavity solitons and identify two different types of traveling localized solutions, corresponding to slow and fast motion. We show that the delay impacts both stationary and moving solutions either causing drifting and wiggling dynamics of initially stationary cavity solitons or leading to stabilization of intrinsically moving solutions. Finally, we demonstrate that the fast cavity solitons can be associated with a lateral mode-locking regime in a broad-area laser with a single longitudinal mode.

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.

Energy exchange between modes in a multimode two-color quantum dot laser with optical feedback.

We investigate experimentally and theoretically the multimode dynamics of a two-color quantum dot laser subject to time-delayed optical feedback. We unveil energy exchanges between the longitudinal modes of the excited state triggered by variations of the feedback phase, and observe that the modal competition between longitudinal modes appears independently within the ground state and excited state emission. These features are accurately reproduced with a quantum dot laser model extended to take into account multiple modes for both ground and excited states. Finally, we discuss the significant impact of such behavior on feedback-based control of two-color quantum dot lasers.

Complexity and bandwidth enhancement in unidirectionally coupled semiconductor lasers with time-delayed optical feedback.

We numerically investigate the frequency bandwidth and the autocorrelation characteristics of chaotic temporal wave forms in unidirectionally coupled semiconductor lasers with time-delayed optical feedback. We evaluate the complexity of the chaotic temporal wave forms by using Lyapunov exponents. We found that larger maximum Lyapunov exponents can be obtained for smaller peak values of the autocorrelation function at the delay time of the optical feedback. On the contrary, the maximum Lyapunov exponent is independent from the frequency bandwidth of the chaotic temporal wave forms.

Statistical Properties of an External-Cavity Semiconductor Laser: Experiment and Theory

We experimentally and numerically study the statistical properties of a semiconductor laser with time-delayed optical feedback. Our systematic analyses show that the statistical distribution of the raw intensity obtained from experiments is better fitted by a Laplacian distribution; both in experiments and simulations, the distribution pattern of the differential signal obtained from two independent intensities, as well as of that computed from its high-order finite differences, converges to a well-fitted Gaussian profile for high pump currents. This helps us to understand the changes in statistical properties of the intensity with varying control parameters and achieve desired entropy sources for random number generators. Furthermore, in numerical simulations, we find the distribution of the differential signal undergoes a marked transition from a Gaussian to Laplacian profile as the feedback rate increases at low to moderate pump currents.

Cavity solitons in vertical-cavity surface-emitting lasers

We investigate a control of the motion of localized structures (LSs) of light by means of delay feedback in the transverse section of a broad area nonlinear optical system. The delayed feedback is found to induce a spontaneous motion of a solitary LS that is stationary and stable in the absence of feedback. We focus our analysis on an experimentally relevant system, namely the vertical-cavity surface-emitting laser (VCSEL). We first present an experimental demonstration of the appearance of LSs in a 80 μm aperture VCSEL. Then, we theoretically investigate the self-mobility properties of the LSs in the presence of a time-delayed optical feedback and analyse the effect of the feedback phase and the carrier lifetime on the delay-induced spontaneous drift instability of these structures. We show that these two parameters affect strongly the space-time dynamics of two-dimensional LSs. We derive an analytical formula for the threshold associated with drift instability of LSs and a normal form equation describing the slow time evolution of the speed of the moving structure.