Changing Dynamical Complexity with Time Delay in Coupled Fiber Laser Oscillators

We investigate the complexity of the dynamics of two mutually coupled systems with internal delays and vary the coupling delay over 4 orders of magnitude. Karhunen-Loève decomposition of spatiotemporal representations of fiber laser intensity data is performed to examine the eigenvalue spectrum and significant orthogonal modes. We compute the Shannon information from the eigenvalue spectra to quantify the dynamical complexity. A reduction in complexity occurs for short coupling delays while a logarithmic growth is observed as the coupling delay is increased.

Figure 1

Schematic of experimental apparatus. The arrows show the direction of light propagation through the rings and the coupling lines. DL, delay line (determines ${\tau }_d$); VA, variable attenuator (adjusts $\kappa$).

Figure 2

(a) KL decomposition for an uncoupled laser. KL decompositions for mutually coupled lasers with (b) ${\tau }_d=0.050\mu \mathrm{s}$, (c) ${\tau }_d=0.218\mu \mathrm{s}$, and (d) ${\tau }_d=120\mu \mathrm{s}$. For (a)–(d) panel (I) is the spatiotemporal representation of the intensity dynamics, panel (II) shows some KL modes ${\psi }_i(x)$ associated with eigenvalues ${\lambda }_i$ (modes above the bottom mode are offset for clarity), panel (III) shows the corresponding expansion coefficients, ${\alpha }_i(n_t)$, for the KL modes in (II), and panel (IV) shows the 25 largest normalized eigenvalues, ${\lambda }_i$, on a logarithmic scale. The bar to the right maps the colors in panels (I) to normalized intensity values.

Figure 3

(a) Entropy $H$, for the uncoupled lasers. (b) Entropy $H$, for the lasers coupled with different delays, ${\tau }_d$. Blue ⋄: laser 1, red X: laser 2. The error bars in both plots are the statistical standard error based on a sample set of ten. The letters by four data points identify the set of panels in Fig. 2 that displays one of the time series represented by that point.