Spontaneous exchange of leader-laggard relationship in mutually coupled synchronized semiconductor lasers

We investigate the instantaneous behavior of synchronized temporal wave forms in two mutually coupled semiconductor lasers numerically and experimentally. The temporal wave forms of two lasers are synchronized with a propagation delay time, with one laser oscillating in advance of the other, known as the leader-laggard relationship. The leader-laggard relationship can be determined by measuring the cross-correlation between the two temporal wave forms with the propagation delay time. The leader can be identified when the optical carrier frequency of the leader laser is higher than that of the other laser. However, spontaneous exchange between the leader and laggard lasers can be observed in low-frequency fluctuations by short-term cross-correlation measurements, even for a fixed initial optical frequency detuning. The spontaneous exchange of the leader-laggard relationship originates from alternation of partial optical frequency locking between the two lasers. This observation is analyzed using a phase space trajectory on steady-state solutions for mutually coupled lasers with optical frequency detuning.

Figure 1

Model for mutually coupled semiconductor lasers with time delay. $\tau$ is the propagation delay time of the light.

Figure 2

(a) Temporal wave forms of the laser intensities $I_{1,2}\left(t\right)$ in the mutually coupled semiconductor lasers. The initial optical frequency detuning $\mathrm{\Delta }{f}_{\mathrm{ini}}$ is set to 2 GHz. (b) Low-pass-filtered temporal wave forms of (a) with a cutoff frequency of 0.1 GHz.

Figure 3

Cross-correlation function of the temporal wave forms of the two laser intensities. The initial optical frequency detuning is (a) 2 GHz and (b) $-2$ GHz. The propagation delay time is $\tau$ = 36.64 ns.

Figure 4

Averaged cross-correlation values ${\overline{C}}_1$ (black solid curve), ${\overline{C}}_2$ (blue dashed curve), and optical frequency detuning under the mutual coupling $\mathrm{\Delta }{\overline{f}}_{mc}$ (red dotted curve) as a function of the initial optical frequency detuning $\mathrm{\Delta }{f}_{\mathrm{ini}}$.

Figure 5

(a) Temporal wave forms of the laser 1 and 2 intensities at $\mathrm{\Delta }{f}_{\mathrm{ini}}=2$ GHz. (b)–(e) Correlation plots of the two temporal wave forms with a time shift. (b) and (c) are calculated from the temporal wave forms of the region X in (a). (d) and (e) are calculated from the temporal wave forms of the region Y in (a). The temporal wave form of laser 1 is delayed in (b) and (d). The temporal wave form of laser 2 is delayed in (c) and (e).

Figure 6

Temporal wave forms of the short-term cross-correlation values $C_{1,2}\left(t\right)$ and the short-term optical frequency detuning under the mutual coupling $\mathrm{\Delta }{f}_{mc}\left(t\right)$. The initial optical frequency detuning is set to $\mathrm{\Delta }{f}_{\mathrm{ini}}=2$ GHz. The black solid and blue dashed curves are $C_1\left(t\right)$ and $C_2\left(t\right)$, respectively.

Figure 7

Probability of being the local leader as a function of the initial optical frequency detuning $\mathrm{\Delta }{f}_{\mathrm{ini}}$. Laser 1 is the local leader when $C_1\left(t\right)-C_2\left(t\right)>C_{th}$, whereas laser 2 is the local leader when $C_2\left(t\right)-C_1\left(t\right)>C_{th}$, where $C_{th}=0.1$. The black solid and blue dashed curves represent the probabilities of the local leader being lasers 1 and 2, respectively.

Figure 8

(a), (b) Low-pass-filtered temporal wave forms of the laser intensities. The initial optical frequency detunings $\mathrm{\Delta }{f}_{\mathrm{ini}}$ are 5 GHz and 10 GHz for (a) and (b), respectively. (c), (d) Temporal wave forms of the short-term cross-correlation values $C_{1,2}\left(t\right)$ corresponding to (a) and (b). The black solid and blue dashed curves represent $C_1\left(t\right)$ and $C_2\left(t\right)$, respectively.

Figure 9

(a) Steady-state solutions calculated from Eqs. (13) and (14). The black and blue dots represent the steady-state solutions of lasers 1 and 2, respectively. (b) Averaged steady-state solutions between lasers 1 and 2.

Figure 10

(a) Averaged steady-state solutions (red dots) and averaged trajectory (black curve) in the phase space of the optical frequency shift and the carrier density. (b) Enlarged view of (a).

Figure 11

(a) Time evolution of the optical frequency shifts $f_{1,2,A}\left(t\right)$. The black, cyan blue (dark gray), and sky blue (light gray) curves represent $f_A\left(t\right)$, $f_1\left(t\right)$, and $f_2\left(t\right)$, respectively. The vertical dashed lines are drawn with the interval of the propagation delay time ($\tau =5$ ns) from 15 ns. (b) Time evolution of the short-term cross-correlation values $C_1\left(t\right)$ and $C_2\left(t\right)$. The black and cyan blue (dark gray) curves represent $C_1\left(t\right)$ and $C_2\left(t\right)$, respectively.

Figure 12

Enlarged view of Fig. 11. The black, cyan blue (dark gray), and sky blue (light gray) curves represent $f_A\left(t\right)$, $f_1\left(t\right)$, and $f_2\left(t\right)$, respectively. The horizontal dashed lines represent the averaged steady-state solutions of the optical frequency in the lower side of Fig. 10.

Figure 13

Experimental setup of mutually coupled semiconductor lasers. Amp, electric amplifier; ATT, attenuator; ISO, optical isolator; PD, photodetector.

Figure 14

Experimentally obtained optical spectra of lasers 1 and 2 when the two lasers are (a) not coupled and (b) mutually coupled. The initial optical frequency detuning is set to $\mathrm{\Delta }{f}_{\mathrm{ini}}=0$ GHz.

Figure 15

Experimentally obtained cross-correlation values ${\overline{C}}_{1,2}$ and optical frequency detuning under the mutual coupling $\mathrm{\Delta }{\overline{f}}_{mc}$ as a function of the initial optical frequency detuning $\mathrm{\Delta }{f}_{\mathrm{ini}}$. The black solid curve with circles and the blue dashed curve with triangles represent ${\overline{C}}_1$ and ${\overline{C}}_2$, respectively. The red dotted curve with squares represents $\mathrm{\Delta }{\overline{f}}_{mc}$.

Figure 16

(a) Experimentally obtained temporal wave forms of the laser intensities. (b) Temporal wave form of the short-term cross-correlation values $C_{1,2}\left(t\right)$, corresponding to (a). The initial optical frequency detuning is set to $\mathrm{\Delta }{f}_{\mathrm{ini}}=1$ GHz.

Figure 17

Experimentally obtained probability of being the local leader as a function of the initial optical frequency detuning $\mathrm{\Delta }{f}_{\mathrm{ini}}$. The black solid curve with circles and blue dashed curves with triangles represent lasers 1 and 2, respectively.