Phase-Incoherent Photonic Molecules in V-Shaped Mode-Locked Vertical-External-Cavity Surface-Emitting Semiconductor Lasers

Passively mode-locked vertical external-cavity surface-emitting semiconductor lasers (VECSELs) composed of a gain chip and a semiconductor saturable absorber have been drawing much attention due to their excellent performance figures. In this work we investigate how localized structures and incoherent, nonlocally bound, pulse molecules emerge in a long cavity VECSEL using a $\mathsf{V}$-shaped cavity geometry. We show that these states are bistable with the laser off state and that they are individually addressable. Using a model based upon delay differential equations, we demonstrate that pulse clusters result from the cavity geometry and from the nonlocal coupling with the gain medium; in the long cavity regime this leads to locally independent, yet globally bound, phase incoherent photonic molecules. Using a multiple timescale analysis, we derive an amplitude equation for the field that allows us to predict analytically the distance between the elements of a cluster.

Figure 1

Setup of a passively mode-locked VECSEL with $\mathsf{V}$-shaped cavity geometry. The main constituents are a semiconductor saturable absorber mirror, an out-coupling facet with high reflectivity ($\kappa \approx 0.99$), and a semiconductor gain chip that can be optically or electrically pumped. The time of flights in the two cavity arms are ${\tau }_1$ and ${\tau }_2$. (b),(c) Photonic molecules in the long cavity limit for fixed ${\tau }_2=2.2$ ns but at varied cavity length $T$, with $t_m$ referring to the size of the molecule.

Figure 2

Localized structures (colored pulses) and the corresponding evolution of the gain $G$ (black lines) in the intermediate and long symmetric cavity regime found by direct numerical integration of Eqs. (1)–(4). (a) Single pulse regime; a secondary ghost gain depletion corresponding to the second gain pass is clearly visible. (b) Pulse cluster with two elements (${\text{PC}}_2$). (c) Phase difference between the two elements of the ${\text{PC}}_2$ bound state $\mathrm{\Delta }\varphi$ (in red). It is freely drifting while the relative distance (${\mathrm{\Delta }}_{{\text{PC}}_2}$ in blue) between the two bits remains constant, up to the influence of noise. In all cases presented in (a)–(c) $T=1.875$ ns. (d) Evolution of the ${\text{PC}}_2$ state in the long cavity regime $T=12.5$ ns and (e) ${\text{PC}}_3$ dynamics with $T=25$ ns. Other parameters are ${\gamma }_g=5{\text{ns}}^{-1}$, ${\gamma }_q=180{\text{ns}}^{-1}$, $Q_0=0.177$, $\gamma =240{\text{ns}}^{-1}$, $G_0/G_{\text{th}}=0.8$, $\kappa =0.99$, $s=20$, and ${\alpha }_{g,q}=0$. For (c), ${\alpha }_g=1.5$, ${\alpha }_q=0.5$, and the noise is added as $\sqrt{(\eta /dt)}D$ in each integration step for Eq. (1), with Gaussian white noise term $D$ and noise strength $\eta =25$ (normalized to $T$).

Figure 3

(a) Evolution of the maximum pulse intensity with respect to the normalized pump power $G_0/G_{\text{th}}$ of the fundamental mode-locking (FML) and harmonic mode-locking (${\text{HML}}_n$) solutions, where subscript $n$ denotes the number of equally spaced pulses. Thick and thin lines indicate stable and unstable dynamics, respectively. The solutions are born in subsequent Andronov-Hopf (AH) bifurcations along the steady-state cw branch (intensity stretched by a factor of 40). The FML branch is found utilizing the path continuation of the DDE model (1)–(3) (light blue) and the Haus master equation (4) (dark blue). The branches all fold back in saddle-node bifurcations (squares) and become multistable to the off solution depending on the cavity geometry. The ${\text{HML}}_3$ solution stabilizes in a torus (tr) bifurcations (circle). (${\mathrm{b}}_{1\text{--}4}$) Gain and field intensity profiles at $G_0/G_{\text{th}}=0.85$ [marked by a dashed line in (a)] and symmetric cavity ${\tau }_1={\tau }_2=0.25T$. Other parameters as in Fig. 2.

Figure 4

(a) Maximum pulse intensity of bound PC states with different numbers of pulses with respect to the normalized pump power $G_0/G_{\text{th}}$. Thick and thin lines indicate stable and unstable solutions, respectively. The branches of the PC solutions with two and three pulses (${\text{PC}}_2$ and ${\text{PC}}_3$) as well as the irregular four pulses solution (${\text{PC}}_4$ irreg) all become stable below the threshold $G_{\text{th}}$. The stabilizing torus bifurcations (tr) are marked by black circles. The corresponding electric field intensity and gain profiles in the stable regions at $G_0=0.85G_{\text{th}}$ (dashed line) are displayed in (${\mathrm{b}}_{1\text{--}3}$). The PC solutions are shown for a symmetric cavity ${\tau }_2=0.25T$; all other parameters are as given in Fig. 2.

Figure 5

(${\mathrm{a}}_1$) Saddle-node (folding) and (${\mathrm{a}}_2$) torus (stabilizing) bifurcations in the $(G_0,{\tau }_2)$ plane; the colors indicate FML, HML, and PC dynamics. Subscripts label the number of pulses in the cavity, ${\alpha }_{g,q}=0$. For comparison, the numerically found saddle-node line of the FML solution resulting from the PDE model is plotted in dark blue in (${\mathrm{a}}_1$). (b),(c) Stable localized pulsations, evolving after injecting different PC solutions into the empty cavity as initial conditions (stable at the filled black circles). The injected solutions are found to be stable after ${10}^4$ round trips at $G_0=0.7G_{\text{th}}$. The injected solutions are (b) ${\text{PC}}_2$ and (c) ${\text{PC}}_3$. The color coding is such that the pulse distance ${\mathrm{\Delta }}_{{\text{PC}}_n}$ increases for darker shading with a seperation of 0.5T at the maximum. Here $({\alpha }_g,{\alpha }_q)=(1.5,0.5)$; all other parameters are as given in Fig. 2.

Figure 6

(a) Change of the pulse distances ${\mathrm{\Delta }}_{{\text{PC}}_n}$ within the ${\text{PC}}_2$ (green) and ${\text{PC}}_3$ (red) solutions for varied cavity configurations ${\tau }_2$ calculated utilizing path continuation of the DDEs (1)–(4) and the analytical expressions (9) and (10) (black lines). Thick lines indicate stable regions; thin lines correspond to unstable solutions. (b) Maximum pulse intensity along the branch shown in (a). Here $G_0/G_{\text{th}}=0.67$; all other parameters are as given in Fig. 2.

Figure 7

Electric field (green) at the output, gain (G in black) and absorber (A in red) dynamics in the untransformed DDE model. The arrow indicates the interaction of the pulse with the active sections during one round-trip, starting and ending at the ouput coupler (O).

Figure 8

Space-time representation of pulse trains that are excited below the threshold $G_{\text{th}}$. The $y$ axis corresponds to the fast timescale normalized to one period of the fundamental solution $(T_{\mathrm{RT}})$ and the $x$ axis to the evolution over several periods (round trips in the cavity). Additional pulses can be written by applying a Gaussian-shaped pump pulse to the constant pump $G_0=0.85G_{\text{th}}$. The amplitudes and FWHMs of the first two and third excitations are $\mathrm{\Delta }{G}_0=2.2G_{\text{th}}$, $T_{\mathrm{FWHM}}=47T$ and $\mathrm{\Delta }{G}_0=2.75G_{\text{th}}$, $T_{\mathrm{FWHM}}=59T$, respectively. Here ${\tau }_2=0.15$ with all other parameters as given in the text. The Gaussian pulse is centered at 100, 700, and 1500 round trips, indicated by the white lines.

Figure 9

(a) Real and imaginary parts of the two maximum Floquet multipliers $\mu$ of the ${\text{PC}}_2$ solution at different round-trip times (denoted by colors). At each $T$, the maximum Floquet multiplier is located at $\mu =1$, indicated by the large green circle. The second largest Floquet multiplier comes closer to $\mu =1$ as the round-trip time is increased. The corresponding gain dynamics is shown in (b)–(e), with the equilibrium value indicated by the dashed line. (f) Distance of the absolute value of the second largest Floquet multiplier to unity at different round trips found using DDE-Biftool (red) and exponential fit (blue). Parameters as in Fig. 2 and $G_0/G_{\text{th}}=0.65$.

Figure 10

Evolution of the ${\text{PC}}_2$ solution born in a period doubling bifurcation (PD) of the ${\text{HML}}_2$ branch and then stabilizing in a tours bifurcation (tr). (a) Solution branches continued in $G_0$ of the ${\text{PC}}_2$ and ${\text{HML}}_2$ solutions. (${\mathrm{b}}_{1\text{--}4}$) Gain and electric field profiles at the points of the branch marked by the colored diamond.

Figure 11

(a) Intensity dynamics of the ${\text{PC}}_2$ solution. (b) Gain depletions of the ${\text{PC}}_2$ solution, with points described in Eqs. (F1)–(F9) marked by black circles. (c) Absorber dynamics, with points described by Eqs. (F10)–(F13) marked by red circles.

Figure 12

(a) Relationship between the pulse power $P$ and pulse distance $\mathrm{\Delta }$ given by Eq. (F15), plotted for $T=1.875$ ns, ${\tau }_2=0.474T$, and ${\gamma }_g=5{\text{ns}}^{-1}$. (b) Solution for the pulse distance ${\mathrm{\Delta }}_{{\text{PC}}_2}$ for the ${\text{PC}}_2$ solution plotted in green, and the asymptotic behavior for ${\tau }_2\to 0,T/2$ in orange.