Third Order Dispersion in Time-Delayed Systems

Time-delayed dynamical systems materialize in situations where distant, pointwise, nonlinear nodes exchange information that propagates at a finite speed. However, they are considered devoid of dispersive effects, which are known to play a leading role in pattern formation and wave dynamics. We show how dispersion may appear naturally in delayed systems and we exemplify our result by studying theoretically and experimentally the influence of third order dispersion in a system composed of coupled optical microcavities. Dispersion-induced pulse satellites emerge asymmetrically and destabilize the mode-locking regime.

Figure 1

(a) Eigenvalue spectrum of Eq. (1) for $A=-1$, $B=M=1$ (red circles) and for $\mathbf{A}=(-1,2;0,-1)$, $\mathbf{B}=(0,0;1,-1)$ and $\mathbf{M}=\mathrm{diag}(1,0)$ (blue crosses). We represent the real part of the eigenvalues ${\gamma }_m$ (normalized to the time delay $\tau$) as well as the imaginary part ${\omega }_m$ normalized to the nominal separation ${\omega }_0$. In the absence of chromatic dispersion, ${\omega }_m/{\omega }_0$ would be an integer. (b) Normalized separation of the imaginary part of the eigenvalue $\mathrm{\Delta }{\omega }_m/{\omega }_0$ as a function of the index $m$. Departure from unity indicates chromatic dispersion. For both cases, $\tau =100$.

Figure 2

A schematic of the coupled cavities configuration. $E_j$ denote the intracavity fields, $j=1$, 2. The output and input fields in the external cavity are represented by $O_j$ and $Y_j$, respectively. The time of flight in the cavity is $\tau$.

Figure 3

Cavity response for two Gaussian pulses of FWHM ${\tau }_p=1$ (blue line) and ${\tau }_p=5$ (red dotted line) after a variable number of round-trips $N_{\mathrm{rt}}$. At each round-trip, the total response consists of the superposition between the reflected and the filtered pulse inducing a train of satellites. The maximal intensity is normalized to unity for clarity.

Figure 4

Satellite instability regime found by direct numerical simulations of Eqs. (2), (7) for a single pulse. (a) Space-time diagram of the intensity. (b) Low-pass filtered @10 GHz detection. (c) Single round-trip optical spectrum. For sufficiently large gain the largest satellite is amplified, eventually replacing its parent pulse. Bias and parameters are $J_1=1.087J_{\mathrm{th}}$, $({\kappa }_1^{-1},{\kappa }_2^{-1},{\gamma }_1^{-1},{\gamma }_2^{-1},\tau )=(343\mathrm{fs},34.3\mathrm{fs},2.45\mathrm{ns},34\mathrm{ps},4.12\mathrm{ns})$, $(J_1,J_2,{\alpha }_1,{\alpha }_2,h_1,h_2)=(0.044,-0.03,2.5,1,2,1.9985)$, as well as $(t_{\mathrm{bs}},s,\delta )=(0.985,5.4,-0.5)$. (d),(e),(f) Deformed satellite instability for slightly different parameters $(J_1,{\alpha }_1,s)=(0.041,2.3,5.3)$. Here, $J_1=1.014J_{\mathrm{th}}$. In both cases $J_{\mathrm{th}}\simeq 0.0404$.

Figure 5

(a) Experimental evidence of the satellite instability for two pulses in the same round-trip. Spatiotemporal diagram for the pulses evolution as a function of the number of cavity round-trips is shown. $J=1.18J_{\mathrm{th}}$. (b) Time-averaged optical spectrum.