Dynamical properties of two delay-coupled lasers: on spectra, correlations and synchronization

We numerically and analytically study the dynamics of two semiconductor lasers which are delay-coupled via a semitransparent mirror. We vary the transmission and reflection of the mirror, while keeping their sum constant. For equal transmission and reflection of the mirror, the lasers show identical chaos synchronisation. If the reflection is zero, the lasers show generalised synchronisation of leader-laggard type. Setting the transmission to zero results in uncoupled delay dynamics. We study the transition between these types of dynamics via autocorrelation and spectral properties. As the system evolves from uncoupled dynamics to identical synchronisation, the dynamics of the individual elements does not change significantly, but the crosscorrelation function increases with crosscoupling. As the lasers evolve from identical to generalised synchronisation, some extrema disappear in the correlation functions, while new maxima appear in the spectral density. To interpret this dynamical behaviour, we replace the lasers by delay-coupled linear stochastic maps. In this case, we are able to compute the correlation functions and spectral densities analytically. Surprisingly, we find that the correlations and spectra of delay-coupled stochastic maps are generally a good approximation for the laser dynamics, even becoming exact in the limit of face-to-face coupling.