Optical networks as complex lasers

We discuss the main features of a recently introduced system capable of laser action: the complex active optical network, or lasing network (LANER) [Lepri et al., Phys. Rev. Lett. 118, 123901 (2017)]. The system is experimentally realized with optical fibers linked each other with couplers and with one or more coherently amplifying sections. The LANER displays a standard laser behavior: When the gain provided by the active sections is high enough to overcome the losses, a coherent emission is produced, with a complicated intensity spectrum. A linear theoretical description is discussed in detail, showing how the LANER can be considered as a generalization of the laser with the physical network acting as a complicated cavity. Among its main aspects, the system can be represented by directed graphs disclosing the analogies with the problem of quantum chaos on graphs. Moreover, when the links' lengths are all integer multiples of the same value, the LANER framework corresponds to a lattice problem, with the equivalence of the Brillouin zone with the cavity free spectral range. Experiments in simple configurations are also performed, reporting the evidence of lasing action and its characterization. Examples of spectra of the detected emitted intensity are obtained in different cases, in a phenomenological agreement with the numerical findings of the theory.

Figure 1

The LANER concept. A physical network, built with optical fibers (blue) connected though standard components such as power splitters (green dots), acts as a laser with a complex cavity. The coherent gains are provided by active regions (red thick segments), employing, e.g., Er-doped fibers, semiconductor amplifiers, etc.

Figure 2

The observables: the propagating fields at the extrema of the links (a) and at the ports of the splitters (b). The indexing shown is chosen only to distinguish each component of the fields for the two cases; in the general approach, a further index for the links and the splitters will be required.

Figure 3

A possible LANER configuration and its correspondent directed graph as described in the text. (a) The physical network: the thick, red segments indicate the gain sections and the black, numbered arrows are the propagating fields along the fiber links. The insertion of isolators imposes a single propagation direction in the active links (indicated by the red arrows on top of the gain sections). [(b), (c)] The graphs represent the fields as vertices and the connections induced by the splitters as bonds: A targeted mode is linearly dependent on the sourced one. The red and black dots indicate that the links are respectively active or passive. (c) Change in the connectivity induced by the insertion of the isolators in the physical network (a): The graph (b) is pruned as the opaque parts are removed.

Figure 4

LANER examples: (a) 1PARA, (b) 2PARA, and (c) 2PERP. The arrows indicate the (arbitrarily chosen) reference propagation directions. Red blocks depict possible position choices for the active gains.

Figure 5

(Numerics) Threshold behavior of various LANER systems upon increasing gain on one link. Results for 1PARA [(a), (b)], 2PARA [(c), (d)], and 2PERP [(e), (f)]. Panels (a), (c), and (e) are the beating spectra (in arbitrary units) computed as described in the text; panels (b), (d), and (f) are poles in the complex plane. The chosen lengths are $L_1=9.16\mathrm{m},L_2=18.12$ m (1PARA) and $L_1=9.16\mathrm{m},L_2=18.12\mathrm{m},L_3=5.24\mathrm{m},L_4=10.0$ m (2PARA, 2PERP); the loss factors are $g_j=0.9$ on all the links except the one specified in the figures. The spectra are computed considering roughly ${10}^3–{10}^4$ poles in each case.

Figure 6

The poles in the complex plane for the 1PARA configuration with lengths having rational rations; see Eq. (23) and text. Only the data in the range $\left|k\right|<\pi /L_0$ are reported (corresponding to first Brillouin zone); the gains are $g_1=1.3,g_2=0.7$. [(a)–(c)] Case $n_2=N{n}_1$, solid blue line is the analytical curve, Eq. (26); [(d)–(e)] $n_1$ and $n_2$ are successive values of the Fibonacci series. For comparison, the real parts have been multiplied by the total length $(n_1+n_2)$.

Figure 7

Block setup for the case of Fig. 4 (2PARA). Left: (Oriented) active section composed of a 980-nm pumped, ${\text{Er}}^{3+}$ fiber, WDMs combining and separating pump and $1.55-\mu \mathrm{m}$ signals and an optical isolator. Center: The rest of the system. Right: The frequency and time-domain detection and acquisition section.

Figure 8

Experimental setup for the 1PARA and 2PARA configurations. The unconnected ports of the splitters allow us to expand the LANER without modifications to the existing parts (see the top ring), at the expenses of an higher amount of losses in the simpler arrangements.

Figure 9

Measurements of the LANER emitter power as a function of the laser pump current (black) for the 1PARA configuration. An hysteresis cycle is visible for intermediate pumping power. Inset: the same curve in semilog scale showing clearly the lasing transition. Red curve: laser pump power measured at the unused port of the WDM-out (see Fig. 7); the curve has been rescaled for comparison. The slope change indicates the LANER threshold at $I=43.1$ mA as well.

Figure 10

Measurement of the laser pump power detected at the unused port of the WDM-out (see Fig. 7) for the 1PARA and 2PARA configurations. The corner points indicating the thresholds are marked.

Figure 11

Measurements of the relaxation oscillations frequency for the 1PARA and 2PARA configurations as a function of the pump laser current. Inset: rescaled curves for the two configurations (see text). The dashed line is a power law with exponent $1/2$.

Figure 12

Experimental power spectra of the LANER emitted intensity in the 1PARA configuration for increasing values of the pump laser current. The threshold is $I_{\mathrm{th}}=43.1$ mA. Each spectrum is shifted vertically by $100\mathrm{dB}$ from the previous one. The inset is an enlargement of the spectrum at $I=44$ mA.

Figure 13

Experimental power spectra of the LANER emitted intensity in the 2PARA configuration for increasing values of the pump laser current. The threshold is $I_{\mathrm{th}}=37$ mA. Each spectrum is shifted vertically by $50\mathrm{dB}$ from the previous one. The inset is a larger bandwidth spectrum at $I=43$ mA.

Figure 14

Experimental setup for the 2PERP configuration.

Figure 15

Experimental power spectra for the 2PERP configuration when scanning the two pump laser currents. The inset shows a larger bandwidth spectrum for $I_1=40$ mA, $I_2=60$ mA.