Synchronization of complex spatio-temporal dynamics with lasers

Abstract

Synchronization is the spontaneous alignment of the dynamics of weakly-coupled oscillators. In addition to temporal dynamics like periodic and chaotic oscillations, also the spatio-temporal dynamics of spatially-extended systems like wildlife populations can synchronize. We exploit here the intrinsic spatio-temporal complex dynamics of broad area lasers to demonstrate such synchronization at lab-scale. Broad-area vertical-cavity surface-emitting lasers (BA-VCSELs) exhibit chaos from the nonlinear coupling between laser modes with different spatial profiles and polarization. When coupling two BA-VCSELs, several synchronization and anti-synchronization regimes are observed, highlighting the complex interplay between oscillating modes with different frequencies and spatial patterns. The correlation coefficient varies between 0.2 and 0.9 depending on the dynamics and on the time scale under analysis. Besides its fundamental interest, our experiment with commercial devices marks the first step towards real-world spatial multiplexing in multiple user physical-layer secure communication based on chaos synchronization.

Figures (14)

• Figure 1: Setup and free-running master laser. (a) Schematic representation of the synchronization experiment setup. This configuration allows both the analysis of the dynamics of the master laser, and the response of the slave laser under injection. It is possible to study both of the lasers in the spatio-spectral domain, using an imaging spectrometer (with a half-wave plate (HWP) to control the polarization orientation at the spectrometer input), or in the temporal domain, using two high-bandwidth photodetectors connected to a high-bandwidth real-time oscilloscope. (b$_1$) Spatio-spectral images of the free-running master laser for different pump currents in $u$-polarization, which is selected for injection; (b$_2$) Spatio-spectral image in both $u$- and $v$-polarizations for $I = 6.5$ mA, with the strong modes $M_{u,v}(3, 1)$ highlighted. The symmetry axis of the transverse mode is given by the blue dashed line to indicate the + (-) orientation of the modes. Master laser dynamics for $I_M$= 6 mA (c$_1$) and $I_M$= 9 mA (c$_2$) in the time and frequency domains. The vertical dashed lines in the RF-spectra indicate the birefringence $\Delta \nu_b$. The red time trace in (c$_1$) is low-pass filtered at $0.1$ GHz.
• Figure 2: Synchronization of polarization-hopping dynamics. Spatio-spectral images in $u$-polarization of master and slave laser for $\Delta \nu =$ 0 GHz (a$_1$), 80 GHz (a$_2$), and 133 GHz (a$_3$). The camera field of view corresponds to 2600 $\mu m$$\times$ 740 $\mu m$ in the object plane. (b$_1$) Correlation between the unfiltered and (b$_2$) low-pass filtered (cutoff at 1 GHz) time traces in $u-$polarization of master and slave laser as a function of the detuning or pump current $I_S$. (c) Low-pass filtered time traces (cutoff at 0.1 GHz) of the master (black) and slave (red) lasers for $\Delta \nu = 0$ GHz (top) and $\Delta \nu = 1.7$ GHz (bottom), corresponding to a correlation of approximately $\pm$80%, respectively. (d) Absolute value of correlation for $\Delta \nu = 0$ GHz (blue), $\Delta \nu = 1.7$ GHz (green), and $\Delta \nu = 180$ GHz (red) as a function of the low-pass filter cutoff frequency. The vertical blue dashed lines indicate the cutoff value chosen to compute the correlation coefficient shown in (b$_2$).
• Figure 3: Synchronization of broadband chaotic dynamics. (a,b) Spatio-spectral images of $u$-polarized emission of the slave laser under injection (top) and the master laser (bottom) for $I_M = 8.8\ \mathrm{mA}$ and $I_S = 4.8\ \mathrm{mA}$. The camera field of view corresponds to 2470 $\mu m$$\times$ 950 $\mu m$ in the object plan. The detunings are $\Delta \nu =$ -3.50 GHz (a$_1$), 0 GHz (a$_2$), 4.14 GHz (a$_3$), 40 GHz (b$_1$), 44.3 GHz (b$_2$), and 48.0 GHz (b$_3$). The symmetry axis of the transverse modes is indicated with the blue dashed lines. (c) Correlation between the $u-$polarized time traces of master and slave laser as a function of the detuning, without filter (green) and with low-pass filter at 1 GHz cutoff (black). The magnification on the right shows the three-peak scenario around $\Delta \nu =$ 44.3 GHz.
• Figure S1: Experimental setup. Obj1: 40x objective (NA = 0.6); Obj2: 20x objective (NA = 0.5); OI: optical isolator; NDF: neutral density filter; BS: beam splitter; HWP: half-wave plate; FC: fiber coupler; PD: Photodetector; Amp: RF-Amplifier; OSC: oscilloscope; L1 (L2): lens with 200 mm (100 mm) focal length; BD: beam displacer.
• Figure S2: LI-curves of master (a) and slave laser (b) for 19 polarizer angles ranging from 0$^\circ$ to 180$^\circ$. The color corresponds to the polarizer angle, where $u$-polarization (at 90°) is shown in red and $v$-polarization (at 0$^\circ$ and 180$^\circ$) in blue. The dashed green lines indicate the polarization switching points (PSPs).