Phase instabilities in semiconductor lasers: A codimension-2 analysis

We compute the normal form description of a semiconductor laser near its threshold, with the sole assumption that the polarization be related to the electric field through the susceptibility (dependent on laser frequency and population inversion). We prove both analytically and numerically the possible existence of a phase-unstable regime, characterized by a periodic oscillation of the optical frequency and a constant intensity. This regime bears close resemblance to the mode-switching behavior, with constant total output power, experimentally observed in semiconductor lasers. In addition, our model predicts the appearance of a phase- and amplitude-turbulent regime, compatible with experimental observations. Both regimes are well known in fluid dynamics under the name Benjamin-Feir instability.

Figure 1

Experimental data of the band-gap renormalization of a 103 $\text{Å}$ $\text{GaAs}/{\text{Ga}}_{1-x}{\text{Al}}_x\text{As}$ quantum well at 300 K extracted from [45] and replotted to provide the required information. The plot displays $\text{ln}\left(\Delta E\right)$ versus $\text{ln}\left(N\right)$, where $\Delta E=E_t\left(0\right)-E_t\left(N\right)$ is expressed in Joule and $N$, the surface charge density, in ${\text{m}}^{-2}$ [Eq. (53)]. The continuous line is the linear best fit. The fitting parameters are $\text{ln}\left(a\right)=-70\pm 1$ and $b=0.62\pm 0.03$.

Figure 2

Experimental data of the band-gap renormalization of an $80\text{Å}$ (white squares) and a $150\text{Å}$ (black diamonds) ${\text{In}}_x{\text{Ga}}_{1-x}\text{As}/\text{InP}$ quantum well at 77 $\text{K}$ extracted from [46] and replotted to provide the required information. The plot displays $\text{ln}\left(\Delta E\right)$ versus $\text{ln}\left(N\right)$, where $\Delta E=E_t\left(0\right)-E_t\left(N\right)$ is expressed in joule and $N$, the surface charge density, in ${\text{m}}^{-2}$ [Eq. (53)]. The continuous lines are the linear best fits. The fitting parameters are $\text{ln}\left(a\right)=-60.93\pm 0.06$ and $b=0.386\pm 0.002$ (black diamonds), and $\text{ln}\left(a\right)=-60.4\pm 0.2$ and $b=0.367\pm 0.006$ (white squares).

Figure 3

Log-linear power spectrum of $F$, from Eqs. (40) at different times, in the phase-stable regime. Some parameters are common to all our simulations: $\tilde{\mu }=0.1$, $\tilde{\sigma }=2$, ${\chi }_r=3$. Specific to this simulation $c_0=0.01255+i0.0025$, $c_1=1.1955$, $c_2=0$. The time increment is $0.05$, the space increment $0.125$, and the discretization is taken on 1024 points. The spectral components are computed in the time intervals ($\times {10}^4$): (a) $[1,4]$, (b) $[5,8]$, (c) $[10,13]$, and (d) $[57,60]$.

Figure 4

Numerical simulation of Eq. (40) in the amplitude-turbulent regime. 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, and the dashed one for the real part of $F$ versus time. The dynamics is clearly turbulent. The power spectrum is shown in Fig. 5.

Figure 5

Typical power spectra of $F$ in a log-linear scale obtained from the integration of Eqs. (40). (a) Phase-stable regime with no added noise corresponding to Fig. 3. (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. 6) and (d) to amplitude turbulence (cf. Fig. 4).

Figure 6

Numerical simulation of Eq. (40) 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 solid, top line (black) represents the temporal evolution of the field amplitude, at a fixed spatial position; the dashed, bottom line (red) shows the time derivative of the phase of $F$ [i.e., $\text{Im}\left(\frac{F^{*}{\partial }_{t}F}{{\left|F\right|}^2}\right)$]. As observed experimentally, the total intensity is constant and the electric field frequency oscillation is not symmetric. The power spectrum is shown in Fig. 5.

Figure 7

Numerical simulation of Eq. (A1) with exactly the same parameter values used in [20] for Fig 4. The plots show the power spectra of $F$ at different times: $2\times {10}^4$, ${10}^5$, $3\times {10}^5$, and $8\times {10}^5$. The power spectra are computed by integration over a ${10}^3$ time interval in rescaled units. For the first plots, the spectrum is still slowly evolving with time, but it becomes rigorously stationary for the last ones.