Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers

Complex phenomena in photonics, in particular, dynamical properties of semiconductor lasers due to delayed coupling, are reviewed. Although considered a nuisance for a long time, these phenomena now open interesting perspectives. Semiconductor laser systems represent excellent test beds for the study of nonlinear delay-coupled systems, which are of fundamental relevance in various areas. At the same time delay-coupled lasers provide opportunities for photonic applications. In this review an introduction into the properties of single and two delay-coupled lasers is followed by an extension to network motifs and small networks. A particular emphasis is put on emerging complex behavior, deterministic chaos, synchronization phenomena, and application of these properties that range from encrypted communication and fast random bit sequence generators to bioinspired information processing.

Figure 1

Scheme of a semiconductor laser (SL) in an external cavity. The external cavity is characterized by the cavity length $L_{\mathrm{EC}}$, the corresponding round trip delay ${\tau }_f$, the feedback strength ${\kappa }_f$, and the feedback phase ${\varphi }_f$.

Figure 2

Low-frequency intensity pulsations of a laser in an external cavity. From 261.

Figure 3

Ellipse structure of fixed points in the $\Delta \omega {\tau }_f=\eta$ and $n$ space. From 126.

Figure 4

Numerically obtained LFF for moderate feedback, $p=2$, ${\tau }_f=10\mathrm{ns}$, and strong field noise. For $t>100\mathrm{ns}$ the noise has been switched off. From 221.

Figure 5

Experimentally obtained LFF for moderate feedback, $p=2.03$, and ${\tau }_f=3.6\mathrm{ns}$. (a) Oscilloscope time trace, and (b) streak camera trace. From 83.

Figure 6

Experimentally probability distribution functions of LFF events for (a) intensity dynamics and (b) carrier dynamics. Parameters were $p=2.01$ and ${\tau }_f=14.3\mathrm{ns}$. From 257.

Figure 7

Experimentally obtained phase dependent scenario for a monolithic integrated laser structure. Experimental phase plots (left column) and corresponding intensity power spectra (right column) of the device output for varying the phase current (a)–(e), $I_{\mathrm{ph}}=3.3$, 4.8, 5.5, 5.9, and 6.9 mA, respectively. From 14.

Figure 8

Experimental bifurcation diagrams for optoelectronic feedback scheme showing the transition to wavelength chaos in dependence on the feedback strength ${\beta }_{\lambda }$. From 103.

Figure 9

Comparison of LFF dynamics, detected with 1 GHz bandwidth (upper panel) vs 16 GHz bandwidth detection (lower panel). Courtesy of Daniel Brunner, IFISC, Spain.

Figure 10

Schematic representation of the (a) auxiliary system approach to detect generalized synchronization, and (b) consistency test.

Figure 11

Scheme of a setup for the unidirectionally coupled laser system.

Figure 12

Low-frequency intensity pulsations of a laser in an external cavity. From 4.

Figure 13

Experimental results of the correlation function vs the detuning frequency. From 195.

Figure 14

Experimental optical spectra of the two lasers demonstrating localized synchronization. (a) and (b) depict the spectra without coupling, and (c) and (d) the ones with weak coupling. From 133.

Figure 15

Comparison of the maximum degree of correlation achieved as a function of the coupling strength for experiment and modeling. Both lasers are pumped at the solitary laser threshold. From 224.

Figure 16

Probability density function of the time shift between power dropouts of the two mutually coupled lasers. From 224.

Figure 17

Upper panel: Experimental intensity time series of the two lasers. The lower trace shows the inverted time series. Both lasers are pumped at $I=I_{\mathrm{th}}^{\mathrm{sol}}$. Lower panel: Subnanosecond synchronized dynamics between two consecutive power dropouts with the delay-coupling time compensated for. From 124.

Figure 18

(a) Numerical and (b) experimental cross-correlation functions. From 224.

Figure 19

Characteristics of “death by delay”. (a), (b) Time series and power spectra, respectively, of the two lasers before the mutual coupling. (c), (d) Corresponding plots after the mutual coupling. From 308.

Figure 20

Parameter space of coupling strength and self-feedback strength indicating the synchronization regions. From 162.

Figure 21

Maximum of the cross-correlation coefficient between two mutually coupled lasers subject to an external common driving. From 147.

Figure 22

Different schemes for stabilizing the isochronal solution. (a) Bidirectionally delay-coupled self-oscillating lasers. (b) Bidirectionally delay-coupled lasers driven by a third laser.

Figure 23

Chain of three mutually coupled semiconductor lasers interacting bidirectionally with a time delay.

Figure 24

(a)–(c) Time series (in pairs) of the output intensity of the lasers; (d)–(f) cross-correlation functions of the corresponding time series. The time series of the central laser have been shifted by the coupling delay to allow an easier comparison. Adapted from 85.

Figure 25

Maximum transversal (dashed) and parallel (solid) Lyapunov exponents as a function of the feedback strength $\kappa$ for (a) a passive relay and (b) an active relay. At the two blowout bifurcations $B1$ and $B2$ the maximum transversal Lyapunov exponent of the chaotic attractor changes sign. From 86.

Figure 26

(a) Experimental time series of synchronized fast intensity dynamics in the coherence collapse regime. A short desynchronization event is highlighted. The pump current corresponds to 1.25, the threshold current of the solitary laser. (b) Corresponding normalized intensity difference (synchronization error). From 314.

Figure 27

Scheme of all possible modules containing three nodes. Standard modules and those most frequently discussed in this review are named. An asterisk denotes that the module is a motif, namely, that it occurs more frequently than randomly in selected real-life networks, as calculated by 217: module 3 is a motif in food webs, 7, 12, and 13 in the Internet, 10 in electronic circuits, and 11 in transcriptional, neuronal, and electronic networks.

Figure 28

Experimental observation of zero-lag synchronization in a bidirectional ring of three lasers. (a) Experimental setup, (b) scheme of the resulting laser motif, and (c) average cross-correlation function of the three laser pairs. From 17.

Figure 29

Experimental observation of sublattice synchronization in a bidirectional ring of four lasers. (a) Experimental setup, (b) scheme of the resulting laser motif, and (c) average cross-correlation function of the two laser pairs. From 17.

Figure 30

Experimental observation of sublattice phase locking in different networks of delay-coupled solid-state lasers. Top panel: experimental setup. Bottom panel: (a) Linear chain of seven lasers (center), showing the far-field intensity profiles of lasers 1 and 7 (left) and 4 and 5 (right); (b) ring of six lasers; and (c) ring of seven lasers. Panels b1, b2, c1, and c2 show the cross section of the far-field intensity profiles of laser pairs (given in the plots). In all the network schemes, lasers with the same level of blue are phase locked with each other. From 232.

Figure 31

Crowd synchronization in a star-coupled laser network. The normalized pump current of the hub laser is fixed to 0.4 in all three plots (1 corresponding to the threshold current in the solitary case), while that of the outer lasers decreases from left to right: (a) 1.02, (b) 0.99, and (c) 0.7. Three values of the coupling strength (assumed equal for all links) are considered: $10{\mathrm{ns}}^{-1}$ is represented in light gray, $20{\mathrm{ns}}^{-1}$ in dark gray, and $30{\mathrm{ns}}^{-1}$ in black. From 369.

Figure 32

Contour line of the master stability function, corresponding to a modified LK model of delay-coupled semiconductor lasers for increasing delay times: $\tau =1$ (solid line), $\tau =8$ (dashed line), $\tau =20$ (dot-dashed line), and $\tau =1000$ (dotted line). The light gray circle denotes the line $|\gamma |\equiv r=\sigma$, which separates regions of different dynamics in the synchronization manifold: chaotic when the critical radius $r_0$ is inside the circle, and periodic or stationary otherwise. From 89.

Figure 33

Average convergence time (here denoted by $\overline{\mu }$, bottom panel) as a function of the number of links in a four-node network of optoelectronic oscillators. The top panel shows the different modules labeled with $m$, which denotes the number of links in each case. From 256.

Figure 34

Time evolution of the synchronization error [here denoted by $\theta (t)$, bottom panel] for two networks of four elements with different geometric degeneracy $g_d$. Dark gray lines correspond to nonsensitive networks with nondegenerate eigenvectors ($g_d=1$), and light gray lines correspond to sensitive networks with $g_d=3$. The topologies of those two network motifs and of two other sensitive configurations are given in the top panel. From 256.

Figure 35

A ring configuration of $N$ unidirectionally delay-coupled lasers which can optionally be coupled to a chain of lasers. From 326.

Figure 36

Maximum value of the cross correlation (CCmax) between the semiconductor laser $N/2$ and 0 (squares) and between $N/2$ and $N^{\prime }/2$ (circles) vs the number of lasers $N$. The solid line is an exponential fit. From 326.

Figure 37

Measured optical spectra of a semiconductor laser (light gray) without feedback and (black) with intermediate feedback strength. Details of the experimental setup can be found in 289.

Figure 38

Coherence of a semiconductor laser with delayed optical feedback as a function of the feedback strength. From 343.

Figure 39

Schematic setup of the LIDAR system with a chaotic semicondutor laser (CSL) as a source. D1 and D2 denote the photodetectors.

Figure 40

Cross-correlation traces of a target moving 50.0 cm in the line of sight. From 188.

Figure 41

Schematic setup of a remote sensing system, exploiting the self-mixing effect of a semicondutor laser (SL). D1 denotes the photodetector.

Figure 42

Schematic representation of the generation of random bits (RBG) with a semicondutor laser (SL) subject to delayed feedback. PD and ADC denote the photodetector and the analog-to-digital converter, respectively.

Figure 43

Bin distribution of the amplitude fluctuations of a semiconductor laser with optical feedback with (a) 8 bits ADC resolution and (b) keeping the four least significant bits (LSBs). Details of the experimental setup can be found in 234.

Figure 44

Schematic arrangement of chaos communication in a setup with two unidirectionally coupled semiconductor lasers. The external cavity of the receiver laser is enclosed by a dotted box to distinguish between the open-loop scheme (no external cavity) and the closed-loop scheme (with external cavity for the receiver laser). The message $M(t)$ is encoded into the chaotic carrier at the transmitter and recovered at the receiver side.

Figure 45

Dynamical evolution of the output intensities of unidirectionally coupled transmitter and receiver lasers. (a) Typical laser output from the transmitter, (b) encoded digital message at a bit rate of $4\mathrm{Gb}/\mathrm{s}$, (c) transmitter output after encoding the digital signal, (d) output of the receiver, (e) decoded message, and (f) decoded message after filtering. From 273.

Figure 46

Illustration of the key exchange process in a scheme with two mutually delay-coupled lasers mediated via a semitransparent mirror. $m_{1,2}$ are the bit sequences encoded by ${\mathrm{SL}}_{1,2}$, $m_1(t)–m_2(t+\Delta t)$ is the subtraction of the bit sequences with a time lag $\Delta t$. The synchronization error $P_1(t)–P_2(t+\Delta t)$, with $P_1\mathrm{\text{−}}{P}_2$ being the output intensities of the corresponding delay-coupled lasers, provides information about the modulated sequences. The synchronization error was low-pass filtered to improve the signal-to-noise ratio. Details of the numerical simulations can be found in 340.

Figure 47

Three neuron populations (top panel) interacting through dynamical relaying. (a) Raster plots of a selection of 300 neurons from each population, 1–3 from bottom to top. (b) Firing histogram of 100 neurons from each population; black, solid gray, and dashed gray for populations 1, 2, and 3, respectively. (c) Cross correlogram between neurons of populations 1 and 2; (d) populations 2 and 3; and (e) populations 1 and 3. Adapted from 339.

Figure 48

Transients and reverberating activity patterns in six neuron circuits, using a conditioned stimulation protocol following the circuit connectivity (arrows in the left column). The right column shows spike trains of the six recording electrodes. (a) A directed loop with six activity groups of one neuron each. (b) $\mathrm{GCD}(5,6)=1$ group of reverberating activity. (c) $\mathrm{GCD}(4,6)=2$ groups of reverberating activity. (d) $\mathrm{GCD}(3,6)=3$ groups of reverberating activity. (e) A single directed loop with one bidirectional coupling forming a loop of size 2, $\mathrm{GCD}(2,6)=2$ groups of reverberating activity as in (c). Adapted from 332.

Figure 49

Role of higher order thalamic relays in corticocortical communication. The suggested route of much of this communication involves a projection from layer 5 of the cortex to a higher order thalamic relay to another cortical area. In question is the function, driver or modulator, of the direct corticocortical projections. From 280.

Figure 50

Left panel: Chip of semiconductor ring lasers, emitting at $\lambda =1.55\mu \mathrm{m}$. The top ring is connected to a bus waveguide and the bottom ring to a spiral, implementing delayed feedback. Right panel: Chip of six coupled ring lasers (cavity length $1250\mu \mathrm{m}$), illustrating the integration perspectives for coupled lasers. Courtesy of Marc Sorel, University of Glasgow.