$Q$-switched pulsing lasers subject to delayed feedback: A model comparison

We report theoretical and numerical results on the dynamics of pulsing lasers with gain and absorber sections subject to delayed optical self-feedback. We consider two modeling approaches, with the Yamada model, on the one hand, and a delay model for ring-cavity mode-locked lasers, on the other hand. We focus on the limit where a single mode is involved, and we show that, in the case without feedback, the dynamics of both models show very good quantitative agreement: $Q$-switched periodic pulsing regimes with similar properties are observed. A bifurcation analysis unveils the conditions for the dynamics to be similar in both systems when an additional delayed feedback term is considered: in particular, what is important is to match the period and the width of the pulsing dynamics of the solitary lasers. Our results show that the simple Yamada model can catch the essence of the dynamics of pulsing lasers with feedback in the limit of single-mode systems. This may considerably simplify, in future work, the investigation of the dynamics of certain self-pulsing lasers with feedback.

Figure 1

Sketch of a semiconductor micropillar laser with an integrated saturable absorber section, pumped optically, and subject to delayed optical feedback from an external mirror [10].

Figure 2

Sketch of a ring-cavity semiconductor laser with an integrated saturable absorber (LSA) section, spectral filtering, and one external feedback (FB) cavity.

Figure 3

Schematic of the relative contributions of the filter width $\gamma$ and the round-trip losses (modeled through the ratio $\kappa$) to the photon lifetime ${\tilde{\tau }}_{\mathrm{ph}}$ in the DDE MLL model, (3, 4, 5), for a fixed value of the round-trip time ${\tilde{\tau }}_c$. For a small filter width (left), ${\tilde{\tau }}_{\mathrm{ph}}$ depends on both the round-trip losses and the bandwidth of the spectral filter element, while for a large filter width (right), ${\tilde{\tau }}_{\mathrm{ph}}$ is constrained only by the round-trip losses.

Figure 4

Bifurcation diagrams of model (1) (top) and model (3, 4, 5) with $\gamma =1$ (bottom), with respect to the pump parameter. Left: Amplitude of the intensity variable, where the equilibrium and periodic solutions are in dark and light blue, respectively. Solutions are stable along thick parts of the curves. T and H, transcritical and Hopf bifurcations, respectively; SN, saddle-node bifurcations of periodic orbits. Right: Period $T_0$ in ps (solid blue line) and pulse width $w$ in ps (dashed red line) along the branch of periodic solutions. Inset: Evolution of $T_0$ near the leftmost Hopf bifurcation.

Figure 5

Bifurcation diagrams of models (1) (top) and (3, 4, 5) (bottom) in the $(A,B)$ plane and $(J_g,J_q)$ plane, respectively. Curves: S, saddle-node bifurcation; T, transcritical bifurcation; H, Hopf bifurcation; SN, saddle-node bifurcation of periodic orbits emerging from a degenerate Hopf bifurcation point, DH. Along thick parts of curve H the Hopf bifurcation is supercritical. In regions 4 and 5, the pulse width $w$ (in ps) is represented by color.

Figure 6

Hopf bifurcation curves of models (1) (left) and (3, 4, 5) (right) in the plane of feedback delay $\tau$ (in ps) and feedback strength $\eta$ and $K$. The parameter values $(A,B)=(9.34,3.36)$ and $(J_g,J_q)=(0.063,1.1)$ agree according to the conversion given in Appendix but yield different pulse profiles without feedback.

Figure 7

Intensity time series (left) and phase portrait in the physical space (right) of the pulsing solution of model (1) (top) and (3, 4, 5) (bottom) without feedback, for $(A,B)=(7.9,1.7)$ and $(J_g,J_q)=(0.063,1.1)$, chosen to yield a pulse width of $w\approx 24.5\mathrm{ps}$ and a pulse period of $T_0\approx 35\times w$ in both cases.

Figure 8

Hopf bifurcation curves of (1) (left) and (3, 4, 5) (right) in the plane of feedback delay and feedback strength, for ($A,B)=(7.9,1.7$) and ($J_g,J_q)=(0.063,1.1$), respectively, chosen such that without feedback ($\eta =0$ and $K=0$), both systems are pulsing with pulse width $w=24.5\mathrm{ps}$ and pulse period $T_0=850\mathrm{ps}$.

Figure 9

Hopf bifurcation curves of (1) (left) and (3, 4, 5) (right) in the plane of feedback parameters. From top to bottom: $B=1.9$ and $B=1.5$ (left); and $J_q=1.5$ and $J_q=0.8$ (right). Values of $A$ and $J_g$ are chosen so that, without feedback, the pulse period–to–pulse width ratio is 35.

Figure 10

Hopf bifurcation curves of (1) (left) and (3, 4, 5) (right) in the plane of feedback parameters. From top to bottom: $B=1.9, B=1.7$, and $B=1.5$ (left); $J_q=1.5, J_q=1.1$, and $J_q=0.8$ (right). Values of $A$ and $J_g$ are chosen so that, without feedback, the pulse period–to–pulse width ratio is 50.

Figure 11

Bifurcation diagram of (1) (top) and (3, 4, 5) (bottom) in the plane of feedback delay $\tau$ (in ps) and feedback strength, for $(A,B)=(7.9,1.7)$ and $(J_g,J_q)=(0.063,1.1)$, respectively. The solitary laser is in the $Q$-switched regime, with a pulse width of 24.5 ps and a pulse period 35 times as large. Curves: H, Hopf bifurcations; SN, saddle-node bifurcations of periodic orbits emerging from degenerate Hopf points, DH; TR, torus bifurcations emerging from Hopf-Hopf bifurcation points, HH; PD, period-doubling bifurcations.

Figure 12

(a) Bifurcation diagram of (3, 4, 5) in the ($J_g,J_q$) plane, for $\gamma =8$ and $\kappa =0.58$. Displayed are two curves, H and ${\mathrm{H}}_{\mathrm{f}}$, of Hopf bifurcations, associated with $Q$-switching and fundamental mode-locking, respectively, and two curves, SN, of saddle-node bifurcations of periodic orbits, emerging from degenerate Hopf bifurcation points, DH. (b) Intensity profile of the Q-switched periodic solution for values of $(J_g,J_q)$ indicated by the red and black $\mathsf{x}$'s, respectively, in (a).