Phase Instability in Semiconductor Lasers

For many years, the apparent absence of a phase instability has characterized lasers as peculiar nonlinear oscillators. We show that this unusual feature is solely due to the approximations used in writing the standard models. A new, careful derivation of the fundamental equations, based on codimension 2 bifurcation theory, shows the possible existence of dynamical regimes displaying either a pure phase instability, or mixed phase-amplitude turbulence. A comparison to existing experimental results convincingly shows that the Benjamin-Feir instability, common to all nonlinear wave problems, is a fundamental, satisfactory interpretation for their deterministic multimode dynamics.

Figure 1

Benjamin-Feir stability boundary vs $\tilde{\sigma }$. The susceptibility is taken as in [24], without (solid line) and with a band-gap renormalization estimated from the experimental measurements of [25] (dashed line) or [26] (dash-dotted line).

Figure 2

Numerical simulation of Eq. (12) in the amplitude turbulence regime. Some parameters are common to all our simulations: $\mu =0.1$, $\sigma =2$, $D=1$, ${\chi }_r=3$. Specific to this simulation, the length $L$ of the numerical box is 512, $c_0=0.0050+i0.0065$, $c_1=0.25-i0.225$, and $c_2=-i2.5006$. The continuous line stands for the amplitude evolution with time, at a fixed spatial position, the dashed one for the real part of $F$ versus time. The dynamics is clearly turbulent.

Figure 3

Numerical simulation of Eq. (12) in the pure phase-unstable regime with $c_0=0.0125+i0.0025$, $c_1=1.1955-i11.9552$, and $c_2=-i2.5006$. The length $L$ of the numerical box is 119.3984. The top line (black online) represents the temporal evolution of the field amplitude, at a fixed spatial position, the bottom line (red online) the time derivative of the phase of $F$ [i.e., $\mathrm{Im}[(F^{*}{\partial }_{t}F)/(|F{|}^2)]$]. As observed experimentally, the total intensity is constant and the electric field frequency oscillation is not symmetric.

Figure 4

Typical power spectra of $F$ in log-linear scales obtained from the integration of Eq. (12). (a) Phase-stable regime with no added noise (aside from that of the numerical scheme): $c_0=0.01255+i0.0025$, $c_1=1.1955$, $c_2=0$. (b) White noise added on space and time in the regime shown in (a). (c) and (d) Phase-unstable regime (no added noise)—(c) corresponds to the pure phase instability (cf. Fig. 3) and (d) to amplitude turbulence (cf. Fig. 2). The horizontal scale in panel (d) is approximately 10 times larger than that of the other three panels.