Harvesting entropy and quantifying the transition from noise to chaos in a photon-counting feedback loop

Dark current and single photon detection by 1550 nm avalanche photodiodes: dead time corrected probability distributions and entropy rates

Single photon detectors have dark count rates that depend strongly on the bias level for detector operation. In the case of weak light sources such as novel lasers or single-photon emitters, the rate of counts due to the light source can be comparable to that of the detector dark counts. In such cases, a characterization of the statistical properties of the dark counts is necessary. The dark counts are often assumed to follow a Poisson process that is statistically independent of the incident photon counts. This assumption must be validated for specific types of photodetectors. In this work, we focus on single-photon avalanche photodiodes (SPADs) made for 1550 nm. For the InGaAs detectors used, we find the measured distributions often differ significantly from Poisson due to the presence of dead time and afterpulsing with the difference increasing with the bias level used for obtaining higher quantum efficiencies. We find that when the dead time is increased to remove the effects of afterpulsing, it is necessary to correct the measured distributions for the effects of the dead time. To this end, we apply an iterative algorithm to remove dead time effects from the probability distribution for dark counts as well as for the case where light from an external weak laser source (known to be Poisson) is detected together with the dark counts. We believe this to be the first instance of the comprehensive application of this algorithm to real data and find that the dead time corrected probability distributions are Poisson distributions in both cases. We additionally use the Grassberger-Procaccia algorithm to estimate the entropy production rates of the dark count processes, which provides a single metric that characterizes the temporal correlations between dark counts as well as the shape of the distribution. We have thus developed a systematic procedure for taking data with 1550 nm SPADs and obtaining accurate photocount statistics to examine novel light sources.

Batch and stream entropy with fixed partitions for chaos-based random bit generators

Measures are proposed for reliably estimating the entropy of bits produced in an entropy source using a chaotic physical system. The measures are reliable with respect to a "guessing" attack and depend on the end-to-end method of transfer of entropy from the chaotic physical system to the bit entropy source. Fixed partitions are considered to correspond with practical methods for fast digital sampling of analog signals. We propose two different measures corresponding to the batch and streaming modes of entropy transfer. Numerical examples are provided to demonstrate features of dependence of the batch and stream entropy on fixed partitions with uniform or nonuniform types of chaos.

Entropy rate of chaos in an optically injected semiconductor laser for physical random number generation

We evaluate the (ɛ, τ) entropy of chaotic laser outputs generated by an optically injected semiconductor laser for physical random number generation. The vertical resolution ɛ and sampling time τ are numerically optimized by comparing the (ɛ, τ) entropy with the Kolmogorov-Sinai entropy, which is estimated from the Lyapunov exponents using linearized model equations. We then investigate the dependence of the (ɛ, τ) entropy on the optical injection strength of the laser system. In addition, we evaluate the (ɛ, τ) entropy from the experimentally obtained chaotic temporal waveforms in an optically injected semiconductor laser. Random bits with an entropy close to one bit per sampling point are extracted to satisfy the conditions of physical random number generation. We find that the extraction of the third-most significant bit from eight-bit experimental chaotic data results in an entropy of one bit per sample for certified physical random number generation.

Nonlinear dynamics in an optoelectronic feedback delay oscillator with piecewise linear transfer functions from the laser diode and photodiode

We investigate the nonlinear dynamics of a recent architecture of an optoelectronic oscillator, where the emitting laser and the receiving diode are connected in a head-to-tail configuration via an optical fiber delay line. The resulting nonlinear transfer function is a piecewise linear profile, and its interplay with the delay leads to many complex behaviors such as relaxation oscillations and deterministic chaos. This system belongs to a recent class of optoelectronic oscillators where the nonlinearity does not originate from the sinusoidal transfer function of an imbalanced interferometer, and, in particular, it is a simple optoelectronic oscillator configuration that is capable of displaying a chaotic behavior. The results of the analytic study are confirmed by numerical simulations and experimental measurements.

Nonlinear dynamics of miniature optoelectronic oscillators based on whispering-gallery mode electrooptical modulators

We propose a time-domain model to analyze the dynamical behavior of miniature optoelectronic oscillators (OEOs) based on whispering-gallery mode resonators. In these systems, the whispering-gallery mode resonator features a quadratic nonlinearity and operates as an electrooptical modulator, thereby eliminating the need for an integrated Mach-Zehnder modulator. The narrow optical resonances also eliminate the need for both an optical fiber delay line and an electric bandpass filter in the optoelectronic feedback loop. The architecture of miniature OEOs therefore appears as significantly simpler than the one of their traditional counterparts and permits us to achieve competitive metrics in terms of size, weight, and power. Our theoretical approach is based on the closed-loop coupling between the optical intracavity modes and the microwave signal generated via the photodetection of the output electrooptical comb. The resulting nonlinear oscillator model involves the slowly-varying envelopes of the microwave and optical fields, and its stability analysis permits the analytical determination the critical value of the feedback gain needed to trigger self-sustained oscillations. This stability analysis also allows us to understand how key parameters of the system such as cavity detuning or coupling efficiency influence the onset of the radiofrequency oscillation. Our study is complemented by time-domain simulations for the microwave and optical signals, which are in excellent agreement with the analytical predictions.

Entropy evaluation of white chaos generated by optical heterodyne for certifying physical random number generators

The entropy of white chaos is evaluated to certify physical random number generators. White chaos is generated from the electric subtraction of two optical heterodyne signals of two chaotic outputs in semiconductor lasers with optical feedback. We use the statistical test suites of NIST Special Publication 800-90B for the evaluation of physical entropy sources of white chaos with an eight-bit resolution. The minimum value of entropy is 2.1 for eight most significant bits data. The entropy of white chaos is enhanced from that of the chaotic output of the semiconductor lasers. We evaluate the effect of detection noise and distinguish between the entropy that originates from the white chaos and the detection noise. It is found that the entropy of five most significant bits originates from white chaos. The minimum value of entropy is 1.1 for five most significant bits data, and it is considered that the entropy can be obtained at at least one bit per sample.

Evaluating entropy rate of laser chaos and shot noise

Evaluating entropy rate of high-dimensional chaos and shot noise from analog raw signals remains elusive and important in information security. We experimentally present an accurate assessment of entropy rate for physical process randomness. The entropy generation of optical-feedback laser chaos and physical randomness limit from shot noise are quantified and unambiguously discriminated using the growth rate of average permutation entropy value in memory time. The permutation entropy difference of filtered laser chaos with varying embedding delay time is investigated experimentally and theoretically. High-resolution maps of the entropy difference are observed over the range of the injection-feedback parameter space. We also clarify an inverse relationship between the entropy rate and time delay signature of laser chaos over a wide range of parameters. Compared to the original chaos, the time delay signature is suppressed up to 95% with the minimum of 0.015 via frequency-band extractor, and the experiment agrees well with the theory. Our system provides a commendable entropy evaluation and source for physical random number generation.

Delayed dynamical systems: networks, chimeras and reservoir computing

We present a systematic approach to reveal the correspondence between time delay dynamics and networks of coupled oscillators. After early demonstrations of the usefulness of spatio-temporal representations of time-delay system dynamics, extensive research on optoelectronic feedback loops has revealed their immense potential for realizing complex system dynamics such as chimeras in rings of coupled oscillators and applications to reservoir computing. Delayed dynamical systems have been enriched in recent years through the application of digital signal processing techniques. Very recently, we have showed that one can significantly extend the capabilities and implement networks with arbitrary topologies through the use of field programmable gate arrays. This architecture allows the design of appropriate filters and multiple time delays, and greatly extends the possibilities for exploring synchronization patterns in arbitrary network topologies. This has enabled us to explore complex dynamics on networks with nodes that can be perfectly identical, introduce parameter heterogeneities and multiple time delays, as well as change network topologies to control the formation and evolution of patterns of synchrony. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Microchaos in human postural balance: Sensory dead zones and sampled time-delayed feedback

Models for the stabilization of an inverted pendulum figure prominently in studies of human balance control. Surprisingly, fluctuations in measures related to the vertical displacement angle for quietly standing adults with eyes closed exhibit chaos. Here we show that small-amplitude chaotic fluctuations ("microchaos") can be generated by the interplay between three essential components of human neural balance control, namely time-delayed feedback, a sensory dead zone, and frequency-dependent encoding of force. When the sampling frequency of the force encoding is decreased, the sensitivity of the balance control to changes in the initial conditions increases. The sampled, time-delayed nature of the balance control may provide insights into why falls are more common in the very young and the elderly.

Multi-Gbit/s optical phase chaos communications using a time-delayed optoelectronic oscillator with a three-wave interferometer nonlinearity

We propose a chaos communication scheme based on a chaotic optical phase carrier generated with an optoelectronic oscillator with nonlinear time-delay feedback. The system includes a dedicated non-local nonlinearity, which is a customized three-wave imbalanced interferometer. This particular feature increases the complexity of the chaotic waveform and thus the security of the transmitted information, as these interferometers are characterized by four independent parameters which are part of the secret key for the chaos encryption scheme. We first analyze the route to chaos in the system, and evidence a sequence of period doubling bifurcations from the steady-state to fully developed chaos. Then, in the chaotic regime, we study the synchronization between the emitter and the receiver, and achieve chaotic carrier cancellation with a signal-to-noise ratio up to 20 dB. We finally demonstrate error-free chaos communications at a data rate of 3 Gbit/s.

Laser-based optoelectronic generation of narrowband microwave chaos for radars and radio-communication scrambling

This Letter proposes an optoelectronic oscillator architecture for narrowband microwave chaos generation. In the time domain, the microwave signal features a slowly varying envelope with amplitude and phase chaos while, in the frequency domain, it is quasi-indistinguishable from a band-limited white noise. A full theoretical analysis is performed to investigate the stability properties and route to chaos for the microwave oscillations. We experimentally generate the narrowband microwave chaos with a central frequency of 3 GHz and a bandwidth of only 16 MHz, and we discuss the applications for radar engineering and radio-communication scrambling.

Intermittent chaos for ergodic light trapping in a photonic fiber plate

Extracting the light trapped in a waveguide, or the opposite effect of trapping light in a thin region and guiding it perpendicular to its incident propagation direction, is essential for optimal energetic performance in illumination, display or light harvesting devices. Here we demonstrate that the paradoxical goal of letting as much light in or out while maintaining the wave effectively trapped can be achieved with a periodic array of interpenetrated fibers forming a photonic fiber plate. Photons entering perpendicular to that plate may be trapped in an intermittent chaotic trajectory, leading to an optically ergodic system. We fabricated such a photonic fiber plate and showed that for a solar cell incorporated on one of the plate surfaces, light absorption is greatly enhanced. Confirming this, we found the unexpected result that a more chaotic photon trajectory reduces the production of photon scattering entropy.

Randomness evaluation for an optically injected chaotic semiconductor laser by attractor reconstruction

State-space reconstruction is investigated for evaluating the randomness generated by an optically injected semiconductor laser in chaos. The reconstruction of the attractor requires only the emission intensity time series, allowing both experimental and numerical evaluations with good qualitative agreement. The randomness generation is evaluated by the divergence of neighboring states, which is quantified by the time-dependent exponents (TDEs) as well as the associated entropies. Averaged over the entire attractor, the mean TDE is observed to be positive as it increases with the evolution time through chaotic mixing. At a constant laser noise strength, the mean TDE for chaos is observed to be greater than that for periodic dynamics, as attributed to the effect of noise amplification by chaos. After discretization, the Shannon entropies continually generated by the laser for the output bits are estimated in providing a fundamental basis for random bit generation, where a combined output bit rate reaching 200 Gb/s is illustrated using practical tests. Overall, based on the reconstructed states, the TDEs and entropies offer a direct experimental verification of the randomness generated in the chaotic laser.