Bifurcation to square-wave switching in orthogonally delay-coupled semiconductor lasers: Theory and experiment

We analyze the dynamics of two semiconductor lasers with so-called orthogonal time-delayed mutual coupling: the dominant TE ($x$) modes of each laser are rotated by 90${}^{\circ }$ (therefore, TM polarization or $y$) before being coupled to the other laser. Although this laser system allows for steady-state emission in either one or in both polarization modes, it may also exhibit stable time-periodic dynamics including square waveforms. A theoretical mapping of the switching dynamics unveils the region in parameter space where one expects to observe long-term time-periodic mode switching. Detailed numerical simulations illustrate the role played by the coupling strength, the mode frequency detuning, or the mode gain to loss difference. We complement our theoretical study with several experiments and measurements. We present time series and intensity spectra associated with the characteristics of the square waves and other waveforms observed as a function of the strength of the delay coupling. The experimental observations are in very good agreement with the analysis and the numerical results.

Figure 1

Intensity of the $I_y$ polarization of the pure-mode solution (red online) and of the mixed-mode solution (green online), and $I_{1y}$, $I_{2y}$ calculated numerically by integrating the rate equations (dots, see text for details) with the following parameters: $k=300$ ns${}^{-1}$, $\mu =2$, $\alpha =3$, ${\gamma }_N=0.5$ ns${}^{-1}$, $\tau =3$ ns, $\beta =0.04$, ${\epsilon }_{xx}=0.01$, ${\epsilon }_{xy}=0.015$, ${\epsilon }_{yx}=0.02$, ${\epsilon }_{yy}=0.025$, $\delta =-3$ rad GHz, and ${\beta }_{sp}={10}^{-5}$ ns${}^{-1}$. In (a), the coupling strength $\eta$ gradually increases while in (b), it gradually decreases.

Figure 2

Numerically stable square-wave switching. (a), (b) Time traces of the $x$ (thick line, red online) and $y$ (thin line, blue online) polarizations of laser 1 (a) and of laser 2 (b). Notice that the lasers emit the same polarization and switch simultaneously. In (a) and (b), the circles indicate the switching times $t_i$. In (c), the time intervals spend in each polarization $\Delta t=t_i-t_{i-1}$ are plotted vs time for laser 1. The switchings are regular, and the switching time is slightly larger than $\tau =3$ ns. (d), (e) Intensity power spectrum of the $x$ and $y$ polarizations of laser 1. The coupling strength is $\eta =50$ ns${}^{-1}$, and other parameters are as indicated in the text.

Figure 3

Transient dynamics toward a mixed-mode solution. Parameters are as in Fig. 2 but with weaker coupling strength $\eta =30$ ns${}^{-1}$.

Figure 4

Symmetric antiphased sinusoidal oscillations $\eta =40$ ns${}^{-1}$ and other parameters are as in Fig. 2. (a), (b) Intensity of the $x$ and $y$ polarizations in lasers 1 and 2; (c), (d) intensity power spectrum of the $x$ and $y$ polarizations of laser 1.

Figure 5

Nonsymmetric antiphased pulselike oscillations $\eta =48$ ns${}^{-1}$ and other parameters are as in Fig. 2. (a), (b) Intensity of the $x$ and $y$ polarizations in lasers 1 and 2; (c), (d) intensity power spectrum of the $x$ and $y$ polarizations of laser 1.

Figure 6

Nonsymmetric and irregular squarelike switching $\eta =49.8$ ns${}^{-1}$ and other parameters are as in Fig. 2. (a), (b) Intensity of the $x$ and $y$ polarizations in lasers 1 and 2. In (a) and (b), the circles indicate the switching times $t_i$. In (c), the time intervals spent in each polarization, $\Delta t=t_i-t_{i-1}$, are plotted vs time for laser 1. (d), (e) Intensity power spectrum of the $x$ and $y$ polarizations of laser 1.

Figure 7

Nonsymmetric regular pulsing behavior $\eta =50.2$ ns${}^{-1}$ and other parameters are as in Fig. 2. (a), (b) Intensity of the $x$ and $y$ polarizations in lasers 1 and 2. In (a) and (b), the circles indicate the switching times $t_i$. In (c), the time intervals spent in each polarization $\Delta t=t_i-t_{i-1}$ are plotted vs time for laser 1.

Figure 8

Transient square waves. Parameters are as in Fig. 2, but with larger coupling strength $\eta =51$ ns${}^{-1}$. (a), (b) Intensity of the $x$ and $y$ polarizations in lasers 1 and 2. (c) Switching time intervals of laser 1. It can be noticed that toward the end of the transient, the intervals spent in one polarization ($x$) gradually increase while the intervals spent in the orthogonal polarization gradually decrease. (d) Time variation of the normalized standard deviation ${\sigma }_i/{\langle I_{x,i}\rangle }_t$ of the intensities of the $x$ polarization in the two lasers (laser 1: thin black line, laser 2: thick line, green online). ${\sigma }_i/{\langle I_{x,i}\rangle }_t$ are calculated over a moving time window of length $5\tau$. One can notice that when the transient finishes, laser 1 reaches stable continuous wave emission on the $x$ polarization, and ${\sigma }_1/{\langle I_{x,1}\rangle }_t\sim 0$.

Figure 9

Phase diagram in the parameter space (cavity-loss difference $\beta$, coupling strength $\eta$). Logarithm of the average transient time (a), normalized dispersion of the transient time (b), the average largest intensity of the $x$ polarization (c), and of the $y$ polarization (d) (see text for details). In each caption, the dots (blue online) indicate parameters where we have seen regular square-wave switching, as that shown in Fig. 2.

Figure 10

Phase diagram in the parameter space (mode frequency detuning $\delta$, coupling strength $\eta$). Logarithm of the average transient time (a), normalized dispersion of the transient time (b), the average largest intensity of the $x$ polarization (c), and of the $y$ polarization (d) (see text for details). In each caption, the dots (blue online) indicate parameters where we have seen regular square-wave switching, as that shown in Fig. 2.

Figure 11

Schematic diagram of experimental apparatus. Heavy line (red online) shows beam path of dynamical system; thin line (blue online) indicates beam paths for detection. All definitions of abbreviations are in the text.

Figure 12

Experimentally observed square-wave dynamics. (a), (b) Time traces of the $x$ (thick line, red online) and $y$ (thin line, blue online) polarizations of laser 1 (a) and laser 2 (b). (c), (d) Intensity power spectra of the $x$ and $y$ polarizations of laser 1. Operating conditions are noted in the text.

Figure 13

Experimentally observed asymmetric and nonsymmetric square-wave dynamics resulting from change in coupling strength. (a), (b) Time traces of the $x$ (thick line, red online) and $y$ (thin line, blue online) polarizations of laser 1 (a) and laser 2 (b). (c), (d) Intensity power spectra of the $x$ and $y$ polarizations of laser 1. Operating conditions are noted in the text.

Figure 14

Experimental bifurcation sequence as a function of coupling strength, shown as time traces of $x$ polarization of laser 2. Coupling strength increases from top (a) to bottom (e). Operating conditions are noted in the text.

Figure 15

Experimental set of coexisting solutions, shown as time traces of $x$ polarization of laser 2. Operating conditions are noted in the text, and are constant for all traces displayed. Pure-mode steady states (not shown) also coexist with these dynamical solutions.