Estimation of time delay for laser diodes with feedback in chaotic regime

Semiconductor lasers with feedback have been considered as potential devices for optical communication systems due to the possibility of obtaining high-dimensional chaotic dynamics. In order to test the privacy of these systems, the first step is to extract the delay time from the data. To this end, two different methods have been considered. One method uses the forecast error of linear model with input data for the laser chaotic intensity: I(t Δ),..., I(t - mΔ), I(t - τ),..., I(t - (m - 1 )τ), where Δ corresponds to the embedding delay and τ to feedback delay. The second method is based on the statistics of the number of extrema of the chaotic signal separated by a time τ, N(τ). In the absence of inertia in the system N(τ) shows a maximum when τ=τ₀, where τ₀ is the delay time. When inertia is taken into account the maximum of N(τ) shifts from τ₀ to larger values and a minimum of N(τ) is obtained at τ=τ₀. This study shows that the feedback delay of the system τ₀ can be obtained by applying these methods to time series generated by numerical simulations of a chaotic laser diode with both optoelectronic and optical feedback and different codification schemes for the message.