Resonances between fundamental frequencies for lasers with large delayed feedbacks

High-order frequency locking phenomena were recently observed using semiconductor lasers subject to large delayed feedbacks. Specifically, the relaxation oscillation (RO) frequency and a harmonic of the feedback-loop round-trip frequency coincided with the ratios 1:5 to 1:11. By analyzing the rate equations for the dynamical degrees of freedom in a laser subject to a delayed optoelectronic feedback, we show that the onset of a two-frequency train of pulses occurs through two successive bifurcations. While the first bifurcation is a primary Hopf bifurcation to the ROs, a secondary Hopf bifurcation leads to a two-frequency regime where a low frequency, proportional to the inverse of the delay, is resonant with the RO frequency. We derive an amplitude equation, valid near the first Hopf bifurcation point, and numerically observe the frequency locking. We mathematically explain this phenomenon by formulating a closed system of ordinary differential equations from our amplitude equation. Our findings motivate experiments with particular attention to the first two bifurcations. We observe experimentally (1) the frequency locking phenomenon as we pass the secondary bifurcation point and (2) the nearly constant slow period as the two-frequency oscillations grow in amplitude. Our results analytically confirm previous observations of frequency locking phenomena for lasers subject to a delayed optical feedback.

Figure 1

Numerically obtained quasiperiodic oscillations. Panels (a) and (b) both show deviation $x$ of the intensity $I$ from its steady-state value in different timescales for $\eta =0.025,\theta =9.9\times 2\pi$, and $\varepsilon =0$. The two periods are $S_1=1.03$ and $S_2=19.81$.

Figure 2

Bifurcation diagram of the steady state (black) and periodic (blue) solutions of the slow-time amplitude equations (12) and (13). We use the decomposition $A=Rexp\left(i\varphi \right)$, where $\varphi =\nu r$ is introduced in the text. (a) The extrema of $R$ as a function of the scaled feedback strength $b$ for $\theta =2n\pi$, obtained by using the numerical continuation method [19]. Full and broken lines are stable and unstable solutions, respectively. Two stable branches of constant-$R$ solutions are shown emerging from $b=0.$ Circles are Hopf bifurcation points $b_H$ leading to stable oscillations up to new bifurcation points $b_{PD}$ denoted by squares. (b) The slow-time frequency correction $\nu$ for the first three branches of steady states: $\nu =-1,-2,$ and $-3$. (c) The period of the periodic solution bifurcating from $R=\sqrt{6(1+b)}$, $\nu =-1$ at $b_H=1/3$. The period remains constant as $b$ further increases from $b_H.$

Figure 3

Bifurcation diagram of the steady-state and periodic solutions of Eqs. (17) and (18). Scaled decomposion $A=Rexp\left(i\varphi \right)$, where $\varphi =\nu r$ is introduced in the text. (a) The extrema of $R$ as a function of the scaled feedback strength $b$. The parameter values are $\theta =9.9\times 2\pi$ and $P=0.5.\theta$ is close to ${\theta }_n=2n\pi$ with $n=10.$ This then implies that $\delta =1/\left(2n\right)=1/20,c=0.2$ if $\varepsilon =0.01$, and $\delta \theta =0.99\pi .$ They have been obtained by using the numerical continuation method [19]. Two stable branches of constant-$R$ solutions are shown emerging from limit points (triangles). The other notations and colors are the same as in Fig. 2. (b) The slow-time frequency correction $\nu$ for the first branch. (c) The period of the periodic solutions bifurcating from the two branches of constant-$R$ solutions. Note that only the first Hopf bifurcation leads to a constant period as we increase $b$.

Figure 4

Numerical bifurcation diagram of the extrema of $R$ as a function of the scaled feedback strength $b$ for $\theta =2n\pi .$ Scaled decomposion $A=Rexp\left(i\varphi \right)$, where $\varphi =\nu r$ is introduced in the text. Long-time simulations were obtained from Eq. (11) reformulated in terms of the real and imaginary parts of $A$. The original values of the parameters are $\varepsilon ={10}^{-2},\theta =9.9\times 2\pi ,$ and $P=0.5.\theta$ is close to ${\theta }_n=2n\pi$ with $n=10$ implying $\delta =1/20,c=0.2$, and $\delta \theta =0.99\pi .$ The red lines are the constant-$R$ solutions given by Eqs. (19) and (20).

Figure 5

(a) Domains of stable periodic solutions of Eqs. (17) and (18) in the $(\theta ,b)$ plane; (b) corresponding oscillation periods. Solid lines are the Hopf bifurcation lines leading to the periodic solutions. Dashed (dotted) lines stand for period-doubling (torus) bifurcation lines. The colored areas delimit the domains of stable oscillations. The red domain is for the periodic solutions that bifurcate from the first steady-state branch if ${\theta }_{c1}<\theta <{\theta }_{c2}$. The other colors correspond to periodic solutions bifurcating from other neighboring branches of steady states. The black dots denote codimension-2 bifurcations located at the cusps: $DH$ are the double Hopf bifurcation points of the zero solution at ${\theta }_{c1}$ and ${\theta }_{c2}$; $FT$ are the flip-torus bifurcation points of the periodic solutions. The parameters are the same as in Fig. 3.

Figure 6

(a) Numerical solution of the DDEs (12) and (13). Scaled decomposion $A=Rexp\left(i\varphi \right)$, where $\varphi =\nu r$ is introduced in the text. (b) Numerical solution of the ODEs (42) and (43) with $E=16.32$ and initial conditions $R_0\left(0\right)=3.61$, $\mathrm{\Phi }\left(0\right)=\pi .$ Parameter $b=0.5.$

Figure 7

Experimental arrangement of a laser diode subjected to optoelectronic feedback. The optical intensity is photodetected, amplified, and added to the DC bias current of the laser diode. An oscilloscope monitors the amplifier output. Various feedback levels are obtained by rotating the linear polarizer in the free space optical path. CL = collimating lens, OI = optical isolator, LP = linear polarizer, OL = objective lens, PD = photodetector.

Figure 8

Optical intensity (left column) and RF spectra (right column) at 68 mA of pump current for various feedback levels: (a) $\eta =0.56$, (b) $\eta =0.58$, (c) $\eta =0.59$.

Figure 9

Optical intensities (left) and RF spectra (right) for a range of pump currents $J$ and feedback levels: (a) $J=64\mathrm{mA}$, $\eta =0.56$, (b) $J=65\mathrm{mA}$, $\eta =0.57$, (c) $J=66\mathrm{mA}$, $\eta =0.58$, (d) $J=67\mathrm{mA}$, $\eta =0.59$.

Figure 10

(a) Primary Hopf bifurcation lines in the interval $8.5<\theta /\left(2\pi \right)<11$ as we increase $\eta$ from zero $(\varepsilon =0.01$ and $P=0.5).$ At double Hopf points $DH$ (black dots), ${\theta }_{c1}=8.94\times 2\pi$ and ${\theta }_{c2}=9.93\times 2\pi$, two distinct bifurcation lines are crossing. (b) Primary Hopf bifurcation lines if $\varepsilon =0.$ They are given by (B16) and (B17) with $n=9,10,11$ and by (B15). The shaded (white) areas in panels (a) and (b) denote the zones of unstable (stable) steady-state solutions.