Polarization Message Encoding through Vectorial Chaos Synchronization in Vertical-Cavity Surface-Emitting Lasers

We show that self-pulsating vertical-cavity surface-emitting lasers can exhibit vectorial chaos, i.e., chaos in both intensity and polarization. The achievable synchronization degree of two such lasers is high when using a continuous control scheme and unidirectional coupling. We propose a novel encryption scheme, where the phase of the vectorial field is modulated. Therefore, the total intensity of these lasers remains synchronized while the intensities in the polarization modes (de)synchronize following the phase modulation at a ps time scale. This technique allows for transmission of secure data at high bit rates that are not limited by the relaxation oscillation frequency.

Figure 1

Sketch of the continuous control synchronization scheme needed to achieve synchronization between two vectorially chaotic VCSELs. The passive components are meant to be polarization insensitive.

Figure 2

The upper panels show the normalized Stokes parameter phase space, while the lower ones show the time evolution of the total intensity (left) and the power spectrum (right). Parameter values are the same as in Ref. 13, except for ${\gamma }_p=25{\mathrm{n}\mathrm{s}}^{-1}$.

Figure 3

Synchronization error (${\sigma }_{+}$, circles and ${\sigma }_{-}$, dots) as a function of the coupling strength $\Gamma$. The transition from an unsynchronized (${\sigma }_{\pm }\sim 1$) state to synchronization takes place at $\Gamma \sim 0.08$. Noise effects were included in the simulations as in 16 setting the lower limit of the synchronization error (noise floor).

Figure 4

Bit recovery: as the phase between the two polarization components in the input signal is modulated between $0$ and $\pi$ at the times shown by the vertical dashed lines, the information is recovered at the “Output” shown in Fig. 1, in both $x$ and $y$ polarizations. For $\pi$ phase shift both polarizations are desynchronized, while they resynchronize very rapidly as the phase is set back to $0$. The total intensity (not shown) remains synchronized all the time. $\Gamma =0.9$, and the modulation bit frequency shown here is $3\mathrm{G}\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{s}/\mathrm{s}$, but can be increased up to $100\mathrm{G}\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{s}/\mathrm{s}$.

Figure 5

Synchronization diagrams corresponding to Fig. 4: (a) the total intensities of the Master and Slave sources ($T_1$ and $T_2$, respectively) keep synchronized all the time except for small bursts and a small variation in the cross correlation between $T_1$ and $T_2$: $0.998$ ($0.996$) when 0 (1) is transmitted. The respective $x$ (b) and $y$ (c) polarizations (de)synchronize when a 0 (1) is received. The units are dimensionless, $\Gamma =0.9$, and the bit frequency is $3\mathrm{G}\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{s}/\mathrm{s}$.

Figure 6

Probability distribution of the synchronization time calculated as the time at which the difference between the two fields drop below the 10% of the mean emitted power. Noise effects were included in the simulations as in 16.