Complex Active Optical Networks as a New Laser Concept

Complex optical networks containing one or more gain sections are investigated, and the evidence of lasing action is reported; the emission spectrum reflects the topological disorder induced by the connections. A theoretical description compares well with the measurements, mapping the networks to directed graphs and showing the analogies with the problem of quantum chaos on graphs. We show that the interplay of chaotic diffusion and amplification leads to an emission statistic with characteristic heavy tails: for different topologies, an unprecedented experimental demonstration of Lévy statistics expected for random lasers is here provided for a continuous-wave pumped system. This result is also supported by a Monte Carlo simulation based on the ray random walk on the graph.

Figure 1

Simpler LANERs (white) and their equivalent graphs. Arrowed (red) segments represent oriented gain sections, numbered (black) arrows the modes in the links. Graphs: (red) circumferences denote active links modes.

Figure 2

Results for the LANER in Fig. 1, with $L_1=9.16\mathrm{m}$, $L_2=18.12\mathrm{m}$, $L_3=5.24\mathrm{m}$, $L_4=10.0\mathrm{m}$. (a) Experimental intensity spectrum increasing the pump current. Numerics: (b) beatings for increasing $g_3$ at fixed $g_1=g_2=g_4=0.9$, (c) poles in the complex plane, (d) level spacing distribution, averaged over an ensemble of 30 graphs ($10^4$ poles each) with the same topology, $g_1=g_2=g_4=0.9$, $g_3=1.4$ and uniformly distributed random $L_j$ with average 10. Wigner (dashed) and Poisson (solid) distributions are plotted for comparison. (e) Experimental length spectrum [see Eq. (4) in the Supplemental Material [9] ] at $J=80\mathrm{mA}$; the vertical lines represent the combinations $m{L}_1+n(L_2+L_3)+l{L}_4$ with $m$, $l$, $n$ integers.

Figure 3

LANER emission dynamics and statistics. (a)–(d): Experiment. Setup of Fig. 1. (a) Time-resolved intensity at $\nu =163.57\mathrm{MHz}$. (b) Boxed event in (a). (c) Distributions of intensity for $J=42.2\mathrm{mA}$: left, $\nu =191.94\mathrm{MHz}$, (blue) dashed line, log-normal fit; right, $\nu =163.57\mathrm{MHz}$, (magenta) dashed line, Lévy stable fit with $\alpha =1.4$; inset, setup of Fig. 1, distribution at $\nu =327.13\mathrm{MHz}$ for $J=37.0\mathrm{mA}$, (magenta) dashed line, Lévy stable fit with $\alpha =1.4$. (d) Setup of Fig. 1: distribution of the intensity of the mode 3+ at $\nu =380.25\mathrm{MHz}$ for $J_1=10\mathrm{mA}$ (link with losses) and $J_4=55\mathrm{mA}$ (link with gain); (red) dashed line, Lévy stable fit with $\alpha =1.32$; inset, bistable distribution at $\nu =380.34\mathrm{MHz}$ for $J_1=40\mathrm{mA}$ and $J_4=60\mathrm{mA}$ (both links have gain). Monte Carlo simulation, geometry of Fig. 1 (see Supplemental Material [9]). (e) Time series. (f) Distribution of (e), (yellow) dashed line, Lévy stable fit with $\alpha =0.995$.