Dynamics of a passively mode-locked semiconductor laser subject to dual-cavity optical feedback

We study the influence of dual-cavity optical feedback on the emission dynamics and timing stability of a passively mode-locked semiconductor laser using a delay differential equation model and verify the timing stability results by an initial experiment. By bifurcation analysis in dependence of the feedback delay times and feedback strength bistability, quasiperiodic and chaotic dynamics, as well as fundamental mode-locking are investigated, yielding a comprehensive overview on the nonlinear emission dynamics arising due to dual-cavity optical feedback. Optimum self-locking ranges for improving the timing stability by dual-cavity optical feedback are identified. A timing jitter reduction and an increase of the repetition rate tuning range of up to a factor of three, compared with single-cavity feedback, are predicted for the parameter ranges investigated. Improved timing stability on short and long timescales is predicted for dual-cavity feedback through the suppression of noise-induced fluctuations. Based on the numerical predictions, experimentally, a maximum timing jitter reduction up to a factor of 180 is found, accompanied by a side-band reduction by a factor of 58 dB, when both feedback cavities are resonant.

Figure 1

Schematic of the dual-cavity optical feedback configuration studied numerically. (a) Ring cavity laser subject to optical feedback from two external cavities (EC). (b) Fabry-Perot laser cavity subject to optical feedback from two external cavities.

Figure 2

(a)–(d) Calculated time traces of the electric-field amplitude (orange), the gain (dashed purple), the total losses $Q+\left|\ln \kappa \right|$ (dashed green), and the net gain $G-Q-|ln\kappa |$ (dash-dotted black) for ${\tau }_1$ and ${\tau }_2$ values indicated by the red crosses in subplot (e). (e) Number of pulses in the laser cavity as a function of ${\tau }_1$ and ${\tau }_2$. $\text{NP}$ indicates nonperiodic dynamics. Parameters: $J_g=0.08{\mathrm{ps}}^{-1},J_q=0.3{\mathrm{ps}}^{-1},K=0.2$, other parameters as given in Table 1.

Figure 3

Numerical bifurcation diagrams as a function of the total feedback strength $K$. Plotted on the $y$ axes are electric field maxima collected over a time period of $100T$ for up-sweeps (gray circles) and down-sweeps (blue diamonds) of $K$. ${\text{SN}}_1,{\text{SN}}_1$, and $\text{H}$ indicate the critical $K$ values where saddle-node bifurcations of limit cycles, saddle-node bifurcations of quasiperiodic solutions, and subcritical secondary Hopf bifurcations occur, respectively. Parameters: $J_g=0.08 \mathrm{ps}{}^{-1},J_q=0.3 \mathrm{ps}{}^{-1}$; other parameters as given in Table 1.

Figure 4

Number of pulses in the laser cavity as a function of ${\tau }_1$ and ${\tau }_2$. $\text{NP}$ indicates nonperiodic dynamics. Parameters: $K=0.1$, (a) $J_g=0.08 \mathrm{ps}{}^{-1}$ and $J_q=0.3 \mathrm{ps}{}^{-1}$, (d) $k=0.2$; all other parameters as given in Table 1.

Figure 5

(a) Calculated timing jitter as a function of ${\tau }_1$ and ${\tau }_2$. The timing jitter of the solitary laser is indicated by ${\sigma }_{\text{lt}}$ of the color bar. (b) Repetition rate as a function of ${\tau }_1$ and ${\tau }_2$. Parameters: $J_g=0.08 \mathrm{ps}{}^{-1},J_q=0.3 \mathrm{ps}{}^{-1},K=0.2,D=3.2 \mathrm{ns}{}^{-1}$; other parameters as given in Table 1.

Figure 6

(a) Calculated timing jitter (blue) and repetition rate (red) as a function of ${\tau }_1$ for dual (markers) and single (solid lines) cavity feedback. The gray dashed line indicates the timing jitter of the solitary laser. (b) Timing jitter as a function of ${\tau }_1$ and ${\tau }_2$ for the dual feedback case. The yellow dotted line indicates the ${\tau }_1$ and ${\tau }_2$ values used in (a). Parameters: $K=0.1,k=0.2,D=3.2 \mathrm{ns}{}^{-1}$; other parameters as given in Table 1.

Figure 7

(a) Calculated power spectrum for single-cavity (blue) and dual-cavity (orange) feedback. Parameters: $K=0.1,k=0.2,D=12.8 \mathrm{ns}{}^{-1}$; other parameters as given in Table 1. (b) Extracted phase noise spectra from (a).

Figure 8

Number of pulses in the laser cavity as a function of ${\tau }_1$ and ${\tau }_2$. $\text{NP}$ indicates nonperiodic dynamics. Parameters: $J_g=0.08 \mathrm{ps}{}^{-1},J_q=0.3 \mathrm{ps}{}^{-1},{\alpha }_g={\alpha }_q=1$; all other parameters as given in Table 1.

Figure 9

Long-term timing jitter as a function of the feedback phase of the short cavity $C_2$, for up- and down-sweeps of $C_2$. Parameters: $J_g=0.08 \mathrm{ps}{}^{-1},J_q=0.3 \mathrm{ps}{}^{-1},{\alpha }_g={\alpha }_q=1$, all other parameters as given in Table 1.

Figure 10

Sketch of the experimental setup used to study the influence of dual-cavity optical on the timing jitter and emission dynamics of a two-section quantum dot mode-locked laser. BS1 and BS2, $50/50$ beam splitters; M1 and M2, feedback mirrors; NDF, neutral density filters.

Figure 11

Experiment: (a) Radio-frequency spectra of the free-running laser (purple), short single-cavity feedback (green), long single-cavity feedback (blue), and dual-cavity feedback (orange). (b) Timing-phase-noise power density spectra corresponding to (a).