Autocorrelation properties of chaotic delay dynamical systems: A study on semiconductor lasers

We present a detailed experimental characterization of the autocorrelation properties of a delayed feedback semiconductor laser for different dynamical regimes. We show that in many cases the autocorrelation function of laser intensity dynamics can be approximated by the analytically derived autocorrelation function obtained from a linear stochastic model with delay. We extract a set of dynamic parameters from the fit with the analytic solutions and discuss the limits of validity of our approximation. The linear model captures multiple fundamental properties of delay systems, such as the shift and asymmetric broadening of the different delay echoes. Thus, our analysis provides significant additional insight into the relevant physical and dynamical properties of delayed feedback lasers.

Figure 1

(a) AC function showing multiple delay echoes. The current and attenuation are 13 mA and 6.1 dB, respectively. Panel (b) is a zoom into the highlighted central (black), first (dark color), and second (light color) delay echoes. The AC (full lines) and envelope (dashed lines) echoes have been shifted by a temporal offset equal to their respective multiples of the delay time.

Figure 2

Envelope (dashed lines) and real part (full lines) of the central peak and the first two delay peaks of the autocorrelation function of a linear stochastic model [Eq. (1)]. Parameters are $\alpha =3,\beta =1.8,\omega =5,\varphi =0$, and $\tau \to \infty$. The broadening of the consecutive peaks and the shift of the maximum are clearly visible.

Figure 3

Scheme of the experimental setup. LD, laser diode; Circ, optical circulator; PC, polarization controller; Att, optical attenuator; 50/50, two-by-two 3-dB coupler; $\to$, optical isolator; and PD, photodiode. The secondary incoming port of the two-by-two coupler is used to inject a train of short pulses to precisely measure the delay time (i.e., external cavity round-trip time, here indicated by a dashed arrow).

Figure 4

Maximum of the first delay echo of the AC versus the rescaled attenuation for four different operating currents: circles $(I=13$ mA), squares $(I=14$ mA), triangles $(I=16$ mA), and crosses $(I=18$ mA).

Figure 5

The central peak (a) and the first two delay echoes [(b) and (c), respectively] of the AC of a semiconductor laser subject to delayed feedback for a pump current of $I=16$ mA and a rescaled attenuation of 27.5 dB with respect to the maximum feedback. The solid lines represent the experimental data and the dashed lines correspond to the fitted analytic expressions for the linear stochastic model.

Figure 6

Plot of the rescaled fitted parameters for the different experimental conditions. Panel (a) depicts the internal frequency $\left[\frac{\omega }{2\pi }\right]$, (b) the decay that includes the effect of the feedback term $\left[\Lambda \right]$, (c) the feedback strength $\left[\beta \right]$, (d) the feedback phase $\left[\varphi \right]$, and (e) the internal decay $\left[\alpha \right]$. The injection currents associated with each curve are as follows: circles $(I=13$ mA), squares $(I=14$ mA), triangles $(I=16$ mA), and crosses $(I=18$ mA).

Figure 7

Comparison of $\alpha$ from the AC (green crosses) and $\tilde{\alpha }$ from the linear response approximation (blue circles). The parameters for the simulations are listed in Table 1.

Figure 8

Phase shift $\varphi$ extracted from the AC fit (black circles). The dashed line corresponds to the prediction of the phase $(\phi$, Appendix pp2), using a pseudocontinuous spectrum approach at the undamping point.

Figure 9

Rescaled shift of the maximum of the AC at the first delay echo for the full set of our experimental conditions. Dots correspond to the shift directly measured from the AC of the experimental time series, and crosses correspond to the shift calculated via the extracted parameters.

Figure 10

Normalized fitting error of the experimental AC function for a current of 16 mA. The solid line and the dashed line represent the errors for the AC center and for the first delay echo, respectively.

Figure 11

Panels (a) to (c) depict the successive delay echoes. Panel (d) depicts the corresponding power spectral densities. Pump current and rescaled attenuation are $I=16$ mA and 19.1 dB, respectively.

Figure 12

The central peak (a) and the first two delay echoes [(b) and (c), respectively] of the AC calculated from simulations of the LK model at a pump current of $p=1.32$ and a rescaled feedback strength of 46.29 ${\mathrm{ns}}^{-1}$. The solid lines represent the numeric data and the dashed lines correspond to the fitted analytic expressions for the linear stochastic model.

Figure 13

Solutions of the characteristic equation (dots) together with the pseudocontinuous spectrum (lines). Left panel: $\gamma =\Re \left(\lambda \right)\tau$ touches the imaginary axis at the bifurcation point, which is indicated with a vertical dashed line. Right panel: $\phi =\Im \left(\lambda \right)\tau$. Parameters are as follows: ${\alpha }_H=2.5,\tau =75.25\mathrm{ns},p=1.32,$ and ${\kappa }_c=0.99{\mathrm{ns}}^{-1}$.