Optoelectronic oscillators with time-delayed feedback

Time-delayed optoelectronic oscillators are at the center of a large body of scientific literature. The complex behavior of these nonlinear oscillators has been thoroughly explored both theoretically and experimentally, leading to a better understanding of their dynamical properties. Beyond fundamental research, these systems have also inspired a wide and diverse set of applications, such as optical chaos communications, pseudorandom number generation, optoelectronic machine learning based on reservoir computing, ultrapure microwave generation, optical pulse-train synthesis, and sensing. The aim of this review is to provide a comprehensive survey of this field, to outline the latest achievements, and discuss the main challenges ahead.

Figure 1

Generic representation of an optoelectronic oscillator with time-delayed feedback. Energy flows alternatively in the optical and electrical forms, and conversion between both paths is performed by electrical-to-optical (E/O) and optical-to-electrical (O/E) converters. This closed-loop system is always nonlinear, dissipative, and infinite dimensional because of the time delay. The interplay between these three properties is the main source of dynamical complexity in OEOs. For metrological applications, a fourth property, stochasticity, becomes relevant as well.

Figure 2

One of the earliest architectures of the optoelectronic oscillator, consisting of an He-Ne laser with optoelectronic feedback. Analogously to the generic configuration displayed in Fig. 1, this system has an optical path (from the modulator to the photomultiplier) concatenated to an electrical path (from the photomultiplier back to the modulator). This feedback scheme permitted one to achieve stable mode locking between the longitudinal modes of the laser. From [141].

Figure 3

Illustrative example of an application of OEOs (Doppler radars). In their simplest configuration, pulsed radars allow the determination of distances by monitoring the round-trip time of flight of a microwave pulse bouncing back from a target, while continuous-wave (or Doppler) radars are used instead to monitor the velocity of targets via the Doppler frequency shift of the reflected microwave (note that both distance and radial velocity can be measured simultaneously using pulse-Doppler radars). When a noiseless microwave (Dirac peak) of frequency $f_0$ is used as a radar probe, the target generates a Doppler echo of frequency $f_0+\mathrm{\Delta }f$, where the Doppler shift $\mathrm{\Delta }f=[2v/c]f_0$ is proportional to the radial velocity $v$ of the target, with $c$ being the velocity of light in vacuum. However, radar probe microwaves are necessarily noisy (Lorentzian profile), so that targets with low reflectivity and/or radial velocity have a spectral signature like Echo 1 that might not be detected for being indiscernible from the microwave phase noise background (gray area). For this reason, ultralow phase noise microwaves are critically needed in aerospace engineering applications, and narrow-band OEOs provide a mature technological platform to generate such ultrapure signals, as discussed in Sec. 5.

Figure 4

Time-domain block diagram for an Ikeda-like OEO. The variable $x(t)$ circulates in the clockwise direction and is subjected to the four main elements of the loop, namely, the nonlinearity $f_{\mathrm{NL}}$, the time delay $T$, the gain $\beta$, and the linear filter $\widehat{\mathsf{H}}$. The closure condition of the feedback loop yields Eq. (3) that governs the OEO dynamics.

Figure 5

Experimental setup for one of the most common implementations of Ikeda-like OEOs. The dynamical variable here is the voltage $V_{\mathrm{r}\mathrm{f}}$ at the radio-frequency input of the Mach-Zehnder modulator, which induces a sinusoidal nonlinearity in the feedback loop with a constant offset phase proportional to $V_{\mathrm{dc}}$. The filter can be of the low-pass or bandpass kind. In the latter case, it can feature either a narrow or a broad bandwidth. PC: polarization controller; MZM: Mach-Zehnder modulator; DL: delay line; PD: photodiode; Amp: rf amplification; MC: microwave coupler.

Figure 6

Comparison between experimental measurements and numerical simulations with Eq. (5) for varying feedback strength in a broad bandpass Ikeda-like OEO. In the experiments, the feedback strength is controlled via the laser power $P$, while in the simulations it is controlled via $\beta$. (a) Measured and simulated time traces for five different values of the feedback strength. (b) Measured and simulated bifurcation diagrams. (c) Lyapunov dimension as a function of feedback strength. The vertical lines in (b) and open circles in (c) indicate the values of feedback corresponding to the five cases shown in (a). From [57].

Figure 7

Time-domain dynamics of an Ikeda-like OEO with broad bandpass filtering, as the feedback gain is increased beyond the oscillation threshold. The sinusoidal nonlinearity is provided here by an integrated lithium niobate Mach-Zehnder modulator. The numerical simulations are performed using a suitably normalized version of Eq. (5). (a), (b) Slow-fast (or relaxation) oscillations ($\beta \simeq 1.5$). (c), (d) Periodic breathers ($\beta \simeq 2$). (e), (f) Chaotic breathers ($\beta \simeq 3$). (g), (h) Fully developed chaos ($\beta \simeq 3.5$). Adapted from [358].

Figure 8

Bifurcation transition at the border of a multistable chimera domain. When starting with a chimera state having the maximum number $N$ of heads in a given domain, the crossing of the border toward the next domain leads to the vanishing of at least one head. Spacetime plots show the capture of such bifurcation events, both numerically and experimentally, as $N$-headed chimera becomes unstable being replaced by $N-1$ heads. (a) 2-to-1 transition. (b) 6-to-5 transition. From [187].

Figure 9

(Top) Experimental setup to investigate the transition from flow (cw pump) to map (pulsed pump) in the broad bandpass Ikeda-like OEO. (Bottom) Numerical (left) and experimental (right) bifurcation diagrams of the map behavior as a function of the feedback gain $\lambda$; the insets show the bifurcation diagrams obtained when seeding the oscillator with a cw laser. From [185].

Figure 10

Schematic view of an experimental setup for a semiconductor laser with optoelectronic feedback. The dynamics of this OEO essentially depends on whether or not the bandwidth of the feedback loop overlaps the relaxation oscillation frequency $F_{\mathrm{RO}}$ ($\sim \mathrm{GHz}$) of the laser. DL: delay line; PD: photodiode; Amp: rf amplification; MC: microwave coupler.

Figure 11

Experimentally measured power spectra, phase portraits (with $P$ and $N$ being the normalized photon and carrier densities, respectively), and time series of different pulsing states for a semiconductor laser with optoelectronic feedback. (a) Regular pulsing (RP); (b) two-frequency quasiperiodic pulsing (Q2); (c) quasiperiodic frequency-locked pulsing (Q-FL); (d) frequency-locked pulsing (FL); (e) chaos (C). From [217].

Figure 12

Experimental setup for an analog video feedback experiment. Information flows counterclockwise through the electronic and optical pathways. Depending on the camera axis and the seed image, the monitor can display a wide variety of complex spatiotemporal behaviors, including periodic, labyrinthic, fractal, or chaotically turbulent patterns. Modern technology allows an easy implementation of such experiments using digital cameras and monitors. From [61].

Figure 13

Optoelectronic oscillator with spatiotemporal and time-delayed photochemical coupling. (a) Experimental setup. The camera records fluorescent light emitted by a large population of nonlocally coupled Belousov-Zhabotinsky (BZ) chemical oscillators. (b) Evidence of spiral-wave chimera in an array of $40\times 40$ BZ oscillators. (c) Oscillator phases obtained from the gray-scale values in (b). From [379].

Figure 14

Quantum behavior of an Ikeda-like OEO with feedback delay time $T_d$. The variable attenuation in the optical branch can be set to be so high that photons arrive sparsely in the photodetector following a Poisson process of rate $\lambda (t)={\lambda }_{0}I(t)$, where ${\lambda }_0$ is a proportionality coefficient and $I(t)$ is the output intensity of the Mach-Zehnder modulator. (a)–(c) Experimental data, and (d) the result of a deterministic simulation, with $N_w(t)$ being the number of photon arrivals in the interval [$t-w$, $t$]. These figures display the Poincaré sections as the OEO transitions from the quantum (Poissonian process) to the classical regime (chaotic attractor) when the normalized photon rate ${\lambda }_0{T}_d$ is increased. The time-delay embedding is performed with a delay $\mathrm{\Delta }=T_d/4$. From [119].

Figure 15

OEO in the cavity quantum electrodynamics regime. (a) Sample layout. (b) Sketch of the setup. The extracted photocurrent is amplified by the transimpedance amplifier (TA) and is then passed through a 2 MHz bandwidth low-pass filter (LP) before being amplified at the main voltage amplifier (VA). The output voltage of the VA is coupled into the laser channel via a bias-T (BT) and monitored on an oscilloscope (OSC). The feedback delay is induced by the round-trip time ${\tau }_{\mathrm{FB}}$. The whispering-gallery mode (WGM) laser and the photodetector are helium cooled down to 15 K. From [267].

Figure 16

Field experiment of chaos communication using an Ikeda-based optoelectronic oscillator. (a) Emitter-receiver system. The emitter uses a laser diode (LD) at 1550 nm to seed an integrated Mach-Zehnder modulator (MZ). An erbium-doped fiber amplifier (EDFA) boosts the power of the signal to be launched into the transmission line. (b) Time traces of the input, transmitted, and decoded signals after 120 km (at $1\mathrm{Gbit}/\mathrm{s}$). (c) Metropolitan optical fiber network of Athens, Greece, where the experiment was performed (total length of 120 km). Adapted from [9].

Figure 17

Schematic of an experiment consisting of three coupled optoelectronic oscillators. LD: laser diodes; PC: polarization controllers; MZM: Mach-Zehnder modulators; C: optical circulators; OS: optical splitters; $\alpha$: adjustable optical attenuators; $\tau$: adjustable optical fiber delay lines; D: photodetectors; S: electronic splitters; MD: modulator drivers. (Inset) Coupling architecture. From [146].

Figure 18

An early architecture for analog optoelectronic computing, emulating the Hopfield neural model. From [92].

Figure 19

Schematic illustration of the individual processes involved in a delay-reservoir computer, highlighting the input, reservoir, and output layers. The input layer is implemented by modulating the input $u(n)$ with the temporal mask $W^{\mathrm{in}}(l)$ to create the input $u^{\mathrm{in}}(t)$, with $l\delta \tau$ being the temporal position of node $l$ within one delay. The reservoir response autonomously creates the state $x(n)$ within one delay ${\tau }_D$. The readout weights correspond to temporal modulations of $x(n)$ according to $W^{\mathrm{out}}$, and summing the resulting sequence over the delay length creates output $y^{\mathrm{out}}(n)$. Some of the steps for OEO-based RC (such as training and testing, for example) are performed off-line. From [22].

Figure 20

Photograph of an Ikeda-like OEO reservoir computer built at the FEMTO-ST Institute, in Besançon, France. Multiple delay lines with random weights have been implemented using a FPGA board, thereby increasing the dimensionality and connectivity of the reservoir. Courtesy of R. Martinenghi.

Figure 21

Schematic of an experimental OEO reservoir computer. The reservoir layer is a delay dynamical system in which a Mach-Zehnder (M-Z) modulator acts as nonlinearity. The information is mixed to the signal using an arbitrary waveform generator (AWG). From [81].

Figure 22

Neuron-inspired regenerative memory with autapic connection. (a) Diagram of a neuron with a self-feedback effect due to the presence of an autapse. (b) Schematic of the equivalent time-delayed neuromorphic photonic resonator in which the optical output is reinjected after a time delay due to the propagation into an optical fiber. RTD-PD: resonant tunneling diode photodetector; LD: laser diode; EDFA: erbium-doped fiber amplifier. From [315].

Figure 23

Basic single-loop OEO architecture for ultrapure microwave generation as initially proposed by [447]. They used this OEO to generate a 9.22 GHz signal with a laser emitting at 1310 nm.

Figure 24

Bifurcation sequence in single-loop OEOs as the feedback gain $|\gamma |$ is increased in Eq. (11), in the conventional configuration where $F_0\gg \mathrm{\Delta }F\gg 1/T$. (a) Bifurcation diagram of the microwave amplitude as a function of gain $|\gamma |$. (b)–(d) Topological changes in the state space. Further increase of the gain beyond (d) leads to a chaotic torus [$\text{Gain}\gtrsim 3$, as shown in the bifurcation diagram (a)]. Note that this full bifurcation sequence is totally deterministic and unrelated to phase or amplitude noise.

Figure 25

Experimental evidence of the Neimark-Sacker bifurcation as the gain is increased in an OEO with a carrier at ${\mathrm{\Omega }}_0/2\pi =3\mathrm{GHz}$, a bandwidth $\mathrm{\Delta }\mathrm{\Omega }/2\pi =20\mathrm{MHz}$, and a delay $T=20\mu \mathrm{s}$. (a), (c), and (d) are time traces, while (b), (d), and (f) are the Fourier spectra of the corresponding reconstructed envelopes (relatively to the carrier). (a), (b) Before the bifurcation, the signal has a constant amplitude and an angular frequency ${\mathrm{\Omega }}_0$ [see Fig. 24]. (c), (d) At the onset of the Neimark-Sacker bifurcation, the amplitude starts to be sinusoidally modulated at angular frequency ${\mathrm{\Omega }}_T/2=\pi /T=2\pi \times 25\mathrm{kHz}$. In the Fourier spectrum, deterministic modulation side peaks appear at $\pm 25\mathrm{kHz}$ around the carrier; they are unrelated to the noisy peaks that appear instead at multiples of $1/T=50\mathrm{kHz}$. The normalized gain value is 2.42, to be compared to the theoretical prediction of 2.31 [see Fig. 24]. (e), (f) Beyond the bifurcation, the small sinusoidal modulation morphs into large square-wave relaxation oscillations with the same frequency. The torus in Fig. 24 now has a rectangularlike cross section. Adapted from [37].

Figure 26

Phase noise comparison between the best OEO performance ([84]) and other oscillators operating at room temperature in the X band (8–12 GHz). The OEO oscillates at 10 GHz and the other oscillators considered here are the following: a commercial dielectric resonator oscillator at 10.8 GHz (Miteq PLDRO-10-10800-5P), a commercial multiplied ($\times 80$) quartz oscillator at 10.24 GHz (Wenzel GMXO-FR), a 10 GHz optical frequency division oscillator ([98]), a 9 GHz sapphire oscillator ([149]), and a 12 GHz fiber-based optical frequency comb oscillator ([428]).

Figure 27

Phase noise performance at 10 kHz offset [${\mathcal{L}}_{10\mathrm{k}}\equiv \mathcal{L}(f=10\mathrm{kHz})$] for all the OEOs reported in this review with an oscillation frequency in the 0–50 GHz range. Two frequency bands of particularly high technological interest are highlighted, namely, the X band (8–12 GHz) and mm waves (30–300 GHz).

Figure 28

Typical single-sideband phase noise spectrum for a single-loop OEO oscillating at 10 GHz with a 4-km-long fiber delay line (corresponding to a time delay $T=20\mu \mathrm{s}$) in the optical branch, and a narrow bandpass filter with bandwidth $\mathrm{\Delta }F=50\mathrm{MHz}$. (a) Experimental phase noise spectrum for a microwave power of 10.5 dBm. The phase noise floor at $-148\mathrm{dBc}/\mathrm{Hz}$ is identical to the model prediction. The peak at 50 Hz is a parasite signal originating from the electric mains supply, while the spurious peak around $1/T=50\mathrm{kHz}$ (in red with a gray background) is a ring-cavity mode induced by the fiber delay line. (b) Enlargement of the spectrum around the spurious ring-cavity peak at the offset frequency 50 kHz. Note that its height in (a) is not accurate because of insufficient resolution, while this enlargement provides a more precise measurement. The maximum of this peak is at $-28.5\mathrm{dBc}/\mathrm{Hz}$, corresponding to a height of 119.5 dB from the phase noise floor (the theoretical prediction was 120 dB). Its $-3\mathrm{dB}$ linewidth is around 40 mHz (with a theoretical prediction of 32 mHz). Adapted from [38].

Figure 29

Common configurations of a dual-loop OEO. CPL: coupler; MZM: Mach-Zehnder modulator; PD: photodiode. (a) OEO with a dual-output Mach-Zehnder modulator; (b) OEO with an optical coupler. From [200].

Figure 30

Tunable OEO for mm-wave generation (10–46 GHz), based on a semiconductor laser subject to optical injection and optoelectronic feedback. The blue (yellow) lines depict (non)polarization-maintaining single-mode optical fiber; the black lines represent rf cables; the dashed lines highlight optional changes to the system’s configuration. VOA: variable optical attenuator; PC: polarization controller; ESA, OSA: electrical or optical spectrum analyzer. From [351].

Figure 31

Schematic diagram of the experimental setup for the OEO pulse generation using an optically injected semiconductor laser. The repetition can be coarsely tuned in the 6.5–15 GHz range. The full width at half maximum of the generated optical pulses can be narrowed down to less than 15 ps. ML: master laser; SL: slave laser; PC: polarization controller; DCM: dispersion compensation module; SMF: single-mode fiber; PD: photodetector; PS: phase shifter; Amp: amplifier; EC: electrical coupler. From [476].

Figure 32

Experimental setup to investigate the time-domain behavior and dynamical stability of a WGM-based OEO. The optical path is in thin red, while the electric path is in thick black. The insets show the spectra of the optical fields $\mathcal{E}(t)$, $\mathcal{F}(t)$, and $\mathcal{G}(t)$ at the input, interior, and output of the WGM resonator, respectively. The spectral lines are separated by a free-spectral range of the resonator (10.4 GHz for a ${\mathrm{CaF}}_2$ disk resonator with 3.2 mm radius), which is also the OEO output frequency. In the open-loop configuration (no oscillations), the “through” port can also be used to perform the cavity-ring down measurement, thereby enabling the determination of the intrinsic and coupling loss factors (characterized by ${\tau }_{\mathrm{i},\mathrm{e},\mathrm{d}}$). In the electric branch, a fast oscilloscope enables one to resolve the temporal dynamics of the microwave $V(t)$. EDFA: erbium-doped fiber amplifier; ESA: electrical spectrum analyzer; OSA: optical spectrum analyzer. From [58].

Figure 33

Miniature OEO based on a lithium niobate WGM resonator. From [247].

Figure 34

Architecture of a 10 GHz OEO using a 860-m-long hollow-core photonic band-gap fiber delay line. The temperature stability of the OEO was improved by a factor $\simeq 15$, as compared to a standard optical fiber delay line. (a) Cross sectional scanning electron microscope image of the hollow-core fiber. (b) Experimental OEO setup. MZM: Mach-Zehnder modulator; PC: polarization controller; DE: delay element; PD: photodiode; RFA: rf amplifier; EDFA: erbium-doped fiber amplifier; Att: attenuator; BPF: bandpass filter; GPS: global positioning system. From [270].

Figure 35

Schematic representation of an OEO strain sensor. LD: laser diode; PC: polarization controller; OC: optical coupler; PM: phase modulator; PSFBG: phase-shifted fiber Bragg gratings; Cir: circulator; OSA: optical spectrum analyzer; EDFA: erbium-doped fiber amplifier; PD: photodiode: MA: microwave amplifier; ESA: electrical spectrum analyzer. From [432].

Figure 36

Schematic representation of an OEO for temperature sensing. ASE: amplified spontaneous emission; MZM: Mach-Zehnder modulator; DCF: dispersion compensating fiber; EDFA: erbium-doped fiber amplifier; PD: photodetector; EA: electrical amplifier; ESA: electrical spectrum analyzer. From [411].

Figure 37

Schematic representation of an OEO pressure sensor. TLS: tunable laser source; PM: phase modulator; OC: optical circulator; EDFA: erbium-doped fiber amplifier; PD: photodetector; EA: electrical amplifier; ESA: electrical spectrum analyzer; PC: polarization controller; SHF: side-hole fiber; PSFBG: phase-shifted fiber Bragg gratings. From [410].

Figure 38

An artistic view of a fully integrated OEO based on stimulated Brillouin scattering (SBS). A partially integrated version of this OEO has permitted the demonstration of ultrapure microwave generation with a close-in phase noise of only $-80\mathrm{dBc}/\mathrm{Hz}$ at 10 Hz offset from a signal in the 5–40 GHz band. From [261].