Pulse Cluster Dynamics in Passively Mode-Locked Semiconductor Vertical-External-Cavity Surface-Emitting Lasers

In recent years different setups based on external cavities have been developed, in order to enhance the performance of pulsed semiconductor mode-locked lasers, namely the peak power and width of the emitted pulses. Depending on the operating conditions and the cavity configuration, pulse cluster solutions emerge with a nonidentical temporal interpulse spacing, which limit the performance of such devices. In this work we present a system of multidelay differential equations to describe the dynamics of a passively mode-locked vertical-external-cavity surface-emitting laser with V-shaped cavity geometry, that allows for an effective modeling and detailed studies of parameter dependencies. We apply numeric integration as well as path-continuation methods to understand the underlying bifurcation scenarios and hence the parameter regions of stable operation. Our investigations indicate that pulse cluster solutions emerge along the fundamental periodic solution branch with a critical influence of the cavity round-trip time on the number of pulses in the cluster. We find regions of multistability of higher-order pulse clusters and predict how different types of dynamics can be favored by tuning the cavity geometry.

Figure 1

Sketch of a passively mode-locked VECSEL with V-shaped cavity geometry and symmetric positioning of gain and absorber, i.e., ${\tau }_1={\tau }_2$, with ${\tau }_1$ and ${\tau }_2$ referring to the length of the cavity arms. The active gain and absorber chips are equipped with a highly reflective distributed Bragg reflector at the bottom side of the chip. The out-coupling facet has a reflectivity of 99% and is indicated by the lens shape. The propagating electric field is marked by the colored arrows indicating the propagation direction, with the forward direction in light red and backward direction in light blue.

Figure 2

(a) Maximum electric field amplitude along the $H_1$ periodic solution branch (solid line), emerging from the first Hopf bifurcation of the cw steady state (dashed line) for increasing pump power. The pump power is normalized to the lasing threshold $J_{\mathrm{th}}$, the pulse amplitude is dimensionless. Stable (unstable) solutions are indicated by thick (thin) lines. Red (orange) circles mark saddle-node bifurcations ${\mathrm{SN}}_{\mathrm{nu}}$ (${\mathrm{SN}}_{\mathrm{nl}}$) that change the stability at the top (bottom) of the loops. The first subscript refers to the number of pulses in the cluster. Torus bifurcations $T_n$ are indicated in green. (${\mathrm{b}}_1$) Electric field profile for one period of the stable fundamental mode-locking solution, marked by the gray diamond in (a). (${\mathrm{b}}_2$) Pulse profile of the unstable side pulse emerging shortly before the saddle-node point ${\mathrm{SN}}_{1u}$, marked by the blue diamond. (${\mathrm{b}}_3$) Stable pulse cluster solution (${\mathrm{PC}}_2$) marked with the green diamond. Round-trip time $T=193$ ps, all other parameters as given in Table 1.

Figure 3

(${\mathrm{a}}_{1-3}$): Normalized electric field amplitude profiles along the fundamental $H_1$ branch plotted as color code for round-trip times of $T=287$ ps (${\mathrm{a}}_1$), $412$ ps (${\mathrm{a}}_2$) and $537$ ps (${\mathrm{a}}_3$). The $x$-axis corresponds to one period of the periodic solution whereas the $y$-axis represents the position on the branch. (${\mathrm{b}}_{1-3}$): Maximum electric field amplitude along the solution branch emerging from the $H_1$ Hopf bifurcation for the same round-trip times as in (${\mathrm{a}}_{1-3}$). Torus bifurcations are marked in green, saddle-node bifurcations in red lines. Orange circles represent saddle-node bifurcations, which lead to the disappearance of one unstable EV, with a minimum of 2 unstable EVs after the bifurcation. Nomenclature as described in the text. The detached fundamental mode-locking solution is indicated in light blue, the detached ${\mathrm{PC}}_2$ region is shown in gray (${\mathrm{b}}_3$). All other parameters chosen as in Table 1.

Figure 4

Two-parameter bifurcation diagram in the ($J_g,T$) plane. (a) Path continuation of the saddle-node ${\mathrm{SN}}_{\mathrm{nu}}$ (${\mathrm{SN}}_{\mathrm{nl}}$) bifurcations, corresponding to the upper (lower) loop (see Fig. 3) plotted in red (orange). Torus-bifurcation lines $T_n$ are marked in green. The black line refers to the first Hopf bifurcation of the cw steady state. (b),(c) Numeric up and downsweep showing the underlying dynamics indicated by the color code, with FML corresponding to fundamental mode locking with low pulse width encoded in light blue and wide pulses repeating at a fundamental frequency in dark blue, ${\mathrm{PC}}_n$ refers to pulse clusters containing $n$ pulses, ${\mathrm{Irr}}_{1-5}$ corresponds to irregular pulse clusters, where light yellow represents one pulse and orange a minimum of five pulses in the irregular cluster (see Fig. 5). An enlargement of the blue rectangle in (a) is shown in Fig. 7.

Figure 5

Pseudo space-time plot of the electric field amplitude in the different dynamic regimes for pump powers along an upsweep at $T=625$ ps, marked by the black dotted line in Fig. 4. The color code is the same as in Fig. 3 on a logarithmic scale. Each plot shows 300 round trips after a transient time of 30000 round trips. The time is normalized to the cavity round-trip time. The pump power values $J_g/J_{\mathrm{th}}$ are as follows: (a) 4, (b) 6.5, (c) 10, (d) 12.5, (e) 15, (f) 20, (g) 22, (h) 25, (i) 28, and (j) 32.

Figure 6

(a) Close up of the two-parameter bifurcation diagram in the ($J_g$,$T$) plane, showing the torus connection point marked as a black rectangle in Fig. 4. Blue lines indicate the round-trip times at which the maximum pulse amplitude along the $H_1$ solution branch in the region of ${\mathrm{PC}}_3$ dynamics is shown in (c)–(e). The number of unstable EVs is indicated by the numbers and is color coded with (0) unstable EVs in black, (2) unstable EVs in green, and (3) unstable EVs in cyan. The ${\mathrm{SN}}_{3u}$ saddle-node point is marked as a red circle, torus bifurcations as green circles. Other parameters as given in Table 1.

Figure 7

Close up of the two-parameter ($J_g$,$T$) bifurcation diagram in the region of the blue square in Fig. 4. The orange line depicts the saddle-node bifurcations and C marks the cusp point. Red-shaded background corresponds to a detached branch, whereas in the yellow-shaded region the branch is still connected. (b)–(e) Corresponding plots of the solution branch for slices of the round-trip times marked by blue lines in (a), with the SN bifurcations marked with orange circles. Other parameters as given in Table 1.

Figure 8

Regions of pulse cluster emission in the (${\tau }_2,J_g$) plane, calculated by a numerical downsweep (a) and upsweep (b) in pump power. The gain-chip position ${\tau }_2$ is defined with respect to the absorber and normalized to the round-trip time. The color code is the same as in Fig. 4, the blue FML shading is proportional to the pulse width, which is $0.01T=6.25$ ps for the values chosen here. The red (orange) lines represent the upper (lower) saddle-node bifurcations, while green lines indicate torus bifurcations. The round-trip time is constant at $T=625$ ps. Other parameters as given in Table 1.

Figure 9

Sketch of the cavity containing the active sections of gain and absorber with the spatial width of the components denoted as $\mathrm{\Delta }{z}_g$ and $\mathrm{\Delta }{z}_q$.

Figure 10

Two-parameter bifurcation diagram (upsweep in $J_g$) in the ($J_g,T$) plane with color code and parameters as described in Fig. 4 except for ${\alpha }_g=2$ and ${\alpha }_q=1$ similar as given in Ref. [18].