How Lasing Localized Structures Evolve out of Passive Mode Locking

We investigate the relationship between passive mode locking and the formation of time-localized structures in the output intensity of a laser. We show how the mode-locked pulses transform into lasing localized structures, allowing for individual addressing and arbitrary low repetition rates. Our analysis reveals that this occurs when (i) the cavity round-trip is much larger than the slowest medium time scale, namely the gain recovery time, and (ii) the mode-locked solution coexists with the zero intensity (off) solution. These conditions enable the coexistence of a large quantity of stable solutions, each of them being characterized by a different number of pulses per round-trip and with different arrangements. Then, each mode-locked pulse becomes localized, i.e., individually addressable.

Figure 1

Intensity of the field (a),(c), gain and total losses (b),(d). For short delay $\tau =10{\mathrm{\Gamma }}^{-1}$ and $G_0=1.1G_{\text{th}}$ (a),(b) the intensity and the absorption ($I$, $Q$) reach an equilibrium between pulses but not the gain that remains far from the equilibrium value $G_0$. For $\tau =100{\mathrm{\Gamma }}^{-1}$ and $G_0=0.98G_{\text{th}}$ (c),(d) all the variables reach equilibrium and the solution becomes localized. The parameters are $\alpha =\beta =0$, $\kappa =0.8$, $s=3$, $q_0=0.3$, ${\mathrm{\Gamma }}^{-1}=25$ and $\gamma =10$.

Figure 2

Panels (a)–(c) depict the bifurcation scenario as a function of the gain for different values of the delay: (a) $\tau =1.2{\mathrm{\Gamma }}^{-1}$, (b) $\tau =2{\mathrm{\Gamma }}^{-1}$, and (c) $\tau =4{\mathrm{\Gamma }}^{-1}$. For short delays, the PML periodic solution (in color) appears as a supercritical Andronov-Hopf bifurcation of the dominant cw mode (thin black line). The stability of the PML solution is indicated by thick lines while the stability of the cw mode is omitted. Panel (d) shows, for $\tau =16{\mathrm{\Gamma }}^{-1}$, the folding of several PML solutions having a different number of equally separated pulses per round-trip while the folding point $g_{\text{sn}}$ is represented by a circle. The other parameters are like in Fig. 1.

Figure 3

Evolution over $N=100$ round-trips of a bit pattern written optically by injecting 1 ps pulses in the cavity (top) and detail over a single period (bottom). Parameters as in Fig. 1 with $\tau =200{\mathrm{\Gamma }}^{-1}$. The bit sequence is 10101010010001001110011000111001100.

Figure 4

Experimental setup: temperature-stabilized VCSEL and RSAM. Coll.: aspheric lens, BS: beam splitter, M: mirror, and D1: detector.

Figure 5

Panels (a), (b), (c), and (d): coexisting time output traces ($J=290\mathrm{mA}$). Panel (e): experimentally obtained bifurcation diagram for the number of pulses per round-trip. The stability of each solution is indicated by the solid horizontal lines.