Predicting the evolutionary dynamic behavior of a laser with injected signal using Lyapunov exponents

We show that the role of the Lyapunov exponents can be extended beyond the customary local instability, such as limit cycle behavior, to include its use as an evolutionary predictor of the dynamics of a laser with injected signal (LIS). Numerical studies of LIS reveal that as a function of the input-signal strength the evolution of two nonzero Lyapunov exponents (generally equal) distinctively predicts the evolutionary trend of the fundamental frequency of the laser output signal (an important dynamic characteristic of the LIS) even with the presence of some noise. This globally predictive behavior of the Lyapunov exponents includes also the dynamic behavior of the individual coexisting attractors. Different coexisting attractors of LIS and configurations of Lyapunov exponents for both individual attractors and the global system are reported. Two LIS case studies are considered: (I) a high-gain system with a rich history of nonlinear behavior but not experimentally accessible, and (II) a low-gain system that has complex dynamics and is experimentally accessible for Class B lasers. Universality arguments support the thesis that these different configurations and the extended role of the Lyapunov exponents as an evolutionary predictor of the dynamics will be observed in other nonlinear, dynamic dissipative systems as well.

Figure 1

Schematic representation of ring laser cavity with two partially transmitting mirrors and two perfectly reflecting mirrors. The length of the ring cavity is $\mathrm{\Lambda }$, while the length of the active medium is L. Y and $\left|X\right|$ represent the scaled injected-field amplitude and the modulus of the output field, respectively.

Figure 2

Two state equations for a laser with injected signal corresponding to Case I marked $A$: $C=20, \tilde{\mathrm{\Delta }}=1, \frac{\mathrm{\Phi }}{\tilde{\kappa }}=-2, \tilde{\kappa }=0.5$, and $\tilde{\gamma }=0.05$; to Case II marked $B$: $C=3, \tilde{\mathrm{\Delta }}=0.5, \frac{\mathrm{\Phi }}{\tilde{\kappa }}=-0.5, \tilde{\kappa }=0.1$, and $\tilde{\gamma }=0.01$.

Figure 3

Schematic of the global behavior of the LIS system in which the relative portion of the fundamental frequency ${\mathrm{\Omega }}_0$ as a function of the injected-field amplitude over a domain of $5<Y<15$ is represented. The parameters are consistent with Case I. $1P$ denotes a singly periodic signal while $2P, 3P, 6P$, and $12P$ denote limit cycles with subharmonics of the fundamental ${\mathrm{\Omega }}_0/2,{\mathrm{\Omega }}_0/3,{\mathrm{\Omega }}_0/6$, and ${\mathrm{\Omega }}_0/12$, respectively; $2QP$ denotes quasiperiodic motion with two incommensurate frequencies (two-dimensional torus); $2P^{+}$ denotes an inverse Feigenbaum cascade; $3P^{*}$ denotes an inverse period-doubling sequence of the type $3\times 2^n$; I–VIII represents attractors of the system; $V_1$ and $V_2$ are the different parts of attractor V; $C1, C2$, and $C3$ represent chaotic regions; the numbers, for example, 5.88, 11.12, etc., represent approximate threshold of transitions. Attractors VII and VIII in red are attractors identified in this study.

Figure 4

3D phase-space plots for Case I with $Y=9.6$ of (a) attractor I and (b) attractor II.

Figure 5

Case I output-field amplitude $\left|X\right|$ as a function of dimensionless time $\tau$ for $Y=9.6$: (a) attractor I and (b) attractor II.

Figure 6

Behavior of the fundamental frequency ${\mathrm{\Omega }}_0$ generated from the Fourier transform of a given time-dependent trajectory as a function of the driving-field strength $Y$ for parameters of Case I. I–VIII are the known attractors that evolve under the influence of distinct domains of attraction. Refer to Fig. 3 for the specific characteristics of these attractors. Open space denotes broadband behavior in the power spectrum where chaos is identified.

Figure 7

Parameters for Case I attractor VII for injected-field strength of $Y=7.27$; (a) 3D phase-space plot; (b) 2D Poincaré surface of section.

Figure 8

Power spectrum of torus dynamics for Case I and injected-field strength of $Y=7.39$.

Figure 9

Case I parameters provide a temporal (a) and phase-space plot; (b) 3D of the torus behavior of attractor VII at an injected-signal strength of $Y=7.397$.

Figure 10

The evolution of the torus behavior of attractor VII with Case I parameters using a 2D Poincaré surface of section: red at $Y=7.397$, green at $Y=7.390$, and blue at $Y=7.380$.

Figure 11

The Lyapunov exponents are shown for the global view in black and attractor I in colors. The fundamental frequencies ${\mathrm{\Omega }}_0$ of attractors I and II are shown overlaid on the original members as a function of the injected-field strength $Y$ for Case I. Black circles with enclosed dots means that two equal $E^L$'s are represented for a given value of $Y$ [17]. (a) First three $E^L$'s of attractor I are represented as solid lines in purple, orange, and green. Letter A marks $E_2^L$ and $E_3^L.$ (b) The ${\mathrm{\Omega }}_0$ of attractors I and II in blue are shown overlaid on the smallest two $E_{}^{L}s$ marked $E_4^L$ and $E_5^L$ in brown for attractor I.

Figure 12

Global view of $E^L$'s and the fundamental frequencies ${\mathrm{\Omega }}_0$ (in blue online) are shown overlaid as a function of the injected-field strength $Y$ for Case I. (a) ${\mathrm{\Omega }}_0$ of attractor IV overlay the first three $E^L$'s of the global view, marked AIV. (b) ${\mathrm{\Omega }}_0$ of attractor VI overlaid on the first three $E^L$'s of the global view, marked with AVI.

Figure 13

Global view for Case II of the Lyapunov exponents for the forward and backward adiabatic scans as a function of the injected-field strength $Y$. All five $E^L$'s are in blue for the forward scan and in red for the backward scan; (a) the first three $E^L$'s and (b) the last two $E^L$'s.

Figure 14

Output-field amplitude $\left|X\right|$ as a function of dimensionless time $\tau$ for parameters of Case II for $Y=1.801$: (a) attractor I and (b) attractor II.

Figure 15

3D phase-space plots of (a) attractor I and (b) attractor II for Case II with $Y=1.801$.

Figure 16

Case II global view of $E^L$'s for a backward adiabatic scan as a function of the injected field $Y$. The coexistence domain is identified by a dashed rectangle: (a) the first three $E^L$'s and ${\mathrm{\Omega }}_0, {\mathrm{\Omega }}_{01}$, and ${\mathrm{\Omega }}_{02}$; (b) the last two $E^L$'s.