Semiconductor laser under resonant feedback from a Fabry-Perot resonator: Stability of continuous-wave operation

We study the continuous-wave (cw) operation of a semiconductor laser subject to optical feedback from a Fabry-Perot resonator in a case where the emission is resonant to a reflection minimum of the resonator. This configuration is treated in the framework of Lang-Kobayashi equations. The nature of bifurcations and the stability of steady state solutions is analyzed in terms of the dependence on magnitude and phase of the feedback. In contrast to conventional optical feedback from a single mirror, the locus of external cavity modes is not elliptic but represents a tilted eight with possible satellite bubbles. Below a critical feedback strength, which is analytically given, only one single mode exists representing the completely unchanged cw emission of the laser. In this weak-feedback regime, the feedback phase allows noninvasive control of the cw emission and a tailoring of its small-signal response within wide limits. The results obtained are a prototype for all-optical realizations of delayed feedback control.

Figure 1

Schematic view of a single-mode DFB laser coupled to a Fabry-Perot resonator. $R$, mirror reflectivity; $L$, resonator length; $l$, distance between laser and resonator. Due to the finite group velocity $c_g$, the geometry causes two time delays, the round trip times ${\tau }_{\mathrm{FP}}=2L∕c_g$ and ${\tau }_l=2l∕c_g$ within the FP and between laser and FP, respectively.

Figure 2

Curves of ECMs in the $(\omega ,N)$ plane for $K=0.05$. (a) Conventional optical feedback. (b) Resonant feedback from Fabry-Perot cavity. SN, Saddle-node bifurcation. ECM, External cavity mode at a particular feedback phase. SLM, Solitary laser mode. Solid line, Node. Dashedline, Saddle. Arrows indicate how the ECM move with increasing feedback phase. SLM keeps its position independent of phase.

Figure 3

Top panel, Reflection spectrum of a Fabry Perot resonator with mirror reflectivity $R=0.1$. Solid line, $n=0$ contribution of Eq. (2). Dashedline, including all multiple round-trips in the resonator. Arrow, resonant position of the solitary laser frequency ${\omega }_0$. Lower panel, Location of ECMs of systems (3, 4, 5) for $K=0.1$. Circles, ECMs for fixed $\phi =0$. Dottedline, Saddle-node curves (SN) separating between modes and antimodes.

Figure 4

Lines of saddle-node (SN) and transcritical (TR) bifurcations at different locations; number of ECMs in corresponding regions

Figure 5

Hopf bifurcations of stable ECMs at different locations (thick solid): Solitary laser mode $\left(H_S\right)$, ECMs on the right $\left(H_{+A}\right)$ and left $\left(H_{-A}\right)$ part of the figure eight, and on satellites $(H_{+B},H_{-B})$. Stability of the central mode. White area, CW operation. Light-gray region, unstable regime of central mode. Dark-gray regions, bistability between central and first modes. Thick dotted line, Transcritical bifurcation. Thin solid line, saddle-node bifurcation.

Figure 6

Region of stable ECMs in the $(N_S,{\omega }_S)$ plane (shaded), bounded by lines of Hopf bifurcations (thin solid) and saddle-node bifurcations (dotted). Upper panel, With figure of eight for $K=0.025$ and $K=0.05$. Lower panel, with figure eight and satellite bubles for $K=0.1$. Thick solid line Location of stable ECMs. Thick dashed line, location of unstable ECMs. Full circles, Hopf bifurcation points for fixed values of $K$. Labels as in Fig. 3.

Figure 7

Bifurcation diagrams for $K=0.05$. Upper panel, $({\omega }_S,\phi )$ plane. Lower panel, $(\mid E\mid ,\phi )$ plane. Transcritical and Hopf bifurcations are marked by triangles and circles, respectively. Solid lines, stable stationary solutions; dashed lines, unstable ones. Squares, Saddle-node bifurcations.

Figure 8

Bifurcation diagram of the steady states in the plane $\mid E\mid$ vs the feedback strength $K$ for $\phi =0$ and $\phi =-0.65\pi$. The solid lines show the stable stationary solutions. Circles indicate the Hopf bifurcation points. The empty circles denote the Hopf-bifurcation which appears on unstable branches. The triangle denotes transcritical bifurcation.

Figure 9

Solutions $\lambda$ of the characteristic equation (15) for the central mode vs feedback strength $K$. Only the respective solution with highest $\mathrm{Re}\lambda <0$ is shown. (a) $\gamma =-\mathrm{Re}\lambda$ and (b) $\omega =\mid \mathrm{Im}\lambda \mid$ correspond to damping and frequency of relaxation oscillations, respectively. H and TR label Hopf and transcritical bifurcations, respectively. The curves are normalized to the values without feedback. Parameters, Eq. (13).