Spatial instabilities of light bullets in passively-mode-locked lasers

Recently, the existence of robust three-dimensional light bullets (LBs) was predicted theoretically in the output of a laser coupled to a distant saturable absorber. In this paper, we analyze the stability and the range of existence of these dissipative localized structures and provide guidelines and realistic parameter sets for their experimental observation. In order to reduce the complexity of the analysis, we first approximate the three-dimensional problem by a reduced equation governing the dynamics of the transverse profile. This effective theory provides an intuitive picture of the LB formation mechanism. Moreover, it allows us to perform a detailed multiparameter bifurcation study and to identify the different mechanisms of instability. It is found that the LBs experience dominantly either homogeneous oscillation or symmetry-breaking transversal wave radiation. In addition, our analysis reveals several nonintuitive scaling behaviors as functions of the linewidth enhancement factors and the saturation parameters. Our results are confirmed by direct numerical simulations of the full system.

Figure 1

Exemplary solutions of Eqs. (1, 2, 3) showing the intensity profile of a stable LB. (a) Isosurface at $1\%$ of the maximal value. (b) Cross sections in the three orthogonal planes defined as $x=0, y=0$, and $z=0$. (c) and (d) The corresponding cross-section profiles. Parameters are $\left(\gamma ,\kappa ,\alpha ,\beta ,\mathrm{\Gamma },G_0,Q_0,s,d\right)=\left(40,0.8,1.5,0.5,0.04,0.425,0.354,30,{10}^{-2}\right)$.

Figure 2

Comparison between the behavior of the branch of the single LS solution (solid blue line) and that of the homogeneous solution (red dash-dotted line) calculated for $\alpha =1.5, \beta =0.5$. The evolution of (a) the spectral parameter $\omega$ and (b) the (peak) intensity $P$ as a function of the normalized gain $g$ is presented. (c) Three exemplary stationary LS profiles existing for different values of $g$. The LS is stable between the saddle-node bifurcation point (SN) and the Andronov-Hopf bifurcation point (${\mathrm{H}}_0$; thick blue line). (d) The real (solid red line) and imaginary (dashed blue line) parts of the corresponding critical eigenfunction ${\psi }_0$. Other parameters are $q=1.27, s=30, d=0.01$.

Figure 3

Bifurcation diagram for a two-dimensional LS obtained for $\alpha =1.7, \beta =0.5$ as a function of the gain $g$, showing the evolution of (a) the spectral parameter $\omega$ and (b) the peak intensity $P$. The LS is stable between the saddle-node bifurcation point (SN) and the AH bifurcation point (${\mathrm{H}}_2$; thick blue line). The secondary AH bifurcation point is indicated as ${\mathrm{H}}_0$. Two insets in (b) show intensity profiles, corresponding to unstable (labeled 1) and stable (labeled 2) LS solutions obtained at the same gain value $g=0.705$. (c) and (d) The real parts of both $n=0$ and $n=2$ critical modes calculated at $g=0.68$, respectively. Other parameters are the same as in Fig. 2.

Figure 4

(a) and (b) One-dimensional stability diagrams showing the threshold for the fold (blue triangles) and $n=0$ AH bifurcations (red circles) in terms of the dependence of $\alpha$ on the gain $g$ for (a) $\beta =0.5$ and (b) $\alpha =1.5$. (c) and (d). The same stability diagrams in two dimensions. Green squares depict the Andronov-Hopf line corresponding to the $n=2$ instability. The range of stability grows with $\alpha$ and shrinks with $\beta$.

Figure 5

One-dimensional stability diagrams showing the threshold for the fold (blue triangles) and the $n=0$ AH bifurcations (red circles) in terms of the dependence of $s$ on the gain $g$ for $\left(\alpha ,\beta \right)=\left(1.5,0.5\right)$. The range of stability shrinks for too large values of $s$. Similar trends are found in in two dimensions.

Figure 6

Two-dimensional bifurcation diagram showing the region of stable existence of the LBs by integrating the two-dimensional Haus equation in $\left(x,z\right)$ as a function of the gain $g$ and the parameters $\alpha$ (left) and $\beta$ (right). The color code represents the energy of the LB $\mathcal{E}$. The range of stability increases with $\alpha$ and decreases for small values of $\beta$, in agreement with Fig. 4. Similar trends were found in three dimensions. Parameters are $\left(\gamma ,\kappa ,\mathrm{\Gamma },Q_0,s,d\right)=\left(40,0.8,0.04,0.3,30,{10}^{-2}\right)$ and $\beta =0.5$ (left), $\alpha =1.5$ (right).

Figure 7

Two-dimensional bifurcation diagram showing the region of stable existence of the LBs by integrating the two-dimensional Haus equation as a function of the gain $g$ and the modulation of the absorption $q$. The color code represents the energy of the LB $\mathcal{E}$ for increasing values of the saturation parameter $s$. We find that the range of stability decreases with increasing values of $s$. Similar trends were found in three dimensions. Parameters are $\left(\gamma ,\kappa ,\alpha ,\beta ,\mathrm{\Gamma },d\right)=\left(40,0.8,1.5,0.5,0.04,{10}^{-2}\right)$.

Figure 8

(a) Evolution of the bistability regime for $s=10$ and increasing values of $q=0.05,0.75$, and 2 (black lines). (b) Approximation of the folding point $\left(P_m,g_m\right)$ for $s=10$ and $q=2.5$ (blue circle). In both cases the homogeneous solution is represented by a black line, and the asymptotic expansions for low and high power are represented by dashed red and dotted blue lines, respectively.

Figure 9

(a) Evolution of the approximation of the folding point as a function of $s$ and different values of $q$ that correspond to $q=0.5$ (solid black line), $q=1$ (dashed red line), $q=2.5$ (dotted blue line), and $q=10$ (dash-dotted green line). (b) Evolution of the approximation of the folding point as a function of $s$ and different values of $q$ that correspond to $q=0.5$ (solid black line), $q=1$ (dashed red line), $q=2.5$ (dotted blue line), and $q=10$ (dash-dotted green line).

Figure 10

(a) Continuation of the folding point $g_{SN}$ of a single LS branch as a function of the normalized absorption $q$ and different values of $s$ that correspond to $s=10$ (black lines), $s=20$ (red lines), $s=100$ (blue lines), and $s=1000$ (green lines). (b) Continuation of $g_{SN}$ of a single LS branch as a function of $s$ and different values of $q$ that correspond to $q=0.5$ (black lines), $q=1$ (red lines), and $q=2.5$ (blue lines). Dashed (solid) lines in both panels indicate one- (two-) dimensional continuation.

Figure 11

One-dimensional bifurcation diagram of Eq. (9) in the $(g,\omega )$ plane calculated for increasing values of $\alpha$ at fixed $\beta =0.5$. For small $\alpha$ the branch of a single LS forms a spiral which unfolds for increasing $\alpha$. Different colors correspond to different values of $\alpha$. Other parameters are the same as in Fig. 2.

Figure 12

Two-dimensional bifurcation diagrams showing the region of stable existence of the LBs by integrating the two-dimensional Haus equations (1, 2, 3) as a function of the gain $g$ and the parameters $\alpha$ (top) and $\beta$ (bottom). We represent (a) and (e) the FWHM in $x$, (b) and (f) the FWHM in $z$, (c) and (g) the frequency of the solution, and (d) and (h) the drift velocity. As mentioned in the main text, the range of stability increases with $\alpha$ and decreases for small values of $\beta$, in agreement with Fig. 4. Other parameters are $(s,\gamma ,\kappa ,\mathrm{\Gamma },Q_0,d)=(30,40,0.8,0.04,0.3,{10}^{-2})$ and (a)–(d) $\beta =0.5$ and (e)–(h) $\alpha =1.5$.

Figure 13

Two-dimensional bifurcation diagram showing the region of stable existence of the LBs by integrating the two-dimensional Haus equation Eqs. (1, 2, 3) as a function of the gain $g$ and the absorption $q$ parameters, for increasing values of $s$. The top, central and bottom lines correspond to $s=10, s=30$, and $s=90$, respectively. We represent (a,e,i) the FWHM in $x$, (b,f,j) the FWHM in $z$, (c,g,k) the frequency of the solution, and (d,h,l) the drift velocity. As mentioned in the main text, the range of stability decreases with larger values of $s$. Other parameters are $(\alpha ,\beta ,\gamma ,\kappa ,\mathrm{\Gamma },d)=(1.5,0.5,40,0.8,0.04,{10}^{-2})$.