Using Disorder to Overcome Disorder: A Mechanism for Frequency and Phase Synchronization of Diode Laser Arrays

Noise and disorder are known, in certain circumstances and for certain systems, to improve the level of coherence over that of the noise-free system. Examples include cases in which disorder enhances response to periodic signals, and those where it suppresses chaotic behavior. We report a new type of disorder-enhancing mechanism, observed in a model that describes the dynamics of external cavity-coupled semiconductor laser arrays, where disorder of one type mitigates (and overcomes) the desynchronization effects due to a different disorder source. Here, we demonstrate stabilization of dynamical states due to frequency locking and subsequently frequency locking-induced phase locking. We have reduced the equations to a potential model that illustrates the mechanism behind the misalignment-induced frequency and phase synchronization.

Figure 1

(a),(c) $\cos {\varphi }_i$ (blue is $-1$ and yellow is $+1$) as a function of time ($x$ axis) and laser number ($y$ axis) for a two-dimensional array of $M=100$ lasers with $d_x=d_y=0.95$ and ${\kappa }^f=30{\mathrm{ns}}^{-1}$. (b),(d) The corresponding power spectra. The lasers are detuned with variance of $\sigma =3\mathrm{GHz}$. For the top figures, the facets are perfectly aligned $ϵ=0$, and for the bottom figures, the facet is misaligned so that $ϵ=0.1$.

Figure 2

Diagrams of the effective potential functions of two detuned lasers’ delay coordinates for $ϵ=0$ (a) and $ϵ\ne 0$ (b). The solid lines denote the potentials for a very small value of ${\kappa }^f$ and the dash-dotted lines denote the potentials for increased ${\kappa }^f$.

Figure 3

(Left panel) Values of ${\mathrm{\Omega }}_i$ are plotted for each laser for a potential model [Eq. (11)] simulation of an array of ten detuned lasers ($\sigma =3\mathrm{GHz}$) with $d_x=0.8$. (Right panel) Phase delay $[\varphi (t)-\varphi (t-\tau )]/2\pi$ is plotted for each laser in an array of ten lasers. This plot is generated from a single simulation with ten detuned lasers ($\sigma =3\mathrm{GHz}$) with $d_x=0.8$ and slowly increasing $ϵ$. The same realizations of ${\tau }_{ij}$ and ${\omega }_i$ are used for each simulation. The coupling strength is ${\kappa }^f=30{\mathrm{ns}}^{-1}$. For both figures color represents laser number.

Figure 4

Right panel (e) shows synchrony level $⟨S⟩$ for 100 realizations of disorder: (a) random normal distribution, (b) linear, (c) sinusoidal, and (d) constant. All disorders have average $ϵ=0.1$. Diodes are arranged as a $10\times 10$ array.