Analysis of temporal structure of laser chaos by Allan variance

Allan variance has been widely utilized for evaluating the stability of the time series generated by atomic clocks and lasers, in time regimes ranging from short to extremely long. This multiscale examination capability of the Allan variance may also be beneficial in evaluating the chaotic oscillating dynamics of semiconductor lasers— not just for conventional phase stability analysis. In the present study, we demonstrated Allan variance analysis of the complex time series generated by a semiconductor laser with delayed feedback, including low-frequency fluctuations (LFFs), which exhibit both fast and slow dynamics. While the detection of LFFs is difficult with the conventional power spectrum analysis method in the low-frequency regime, the Allan variance approach clearly captured the appearance of multiple time-scale dynamics, such as LFFs. This study demonstrates that Allan variance can help in understanding and characterizing diverse laser dynamics, including LFFs, spanning a wide range of timescales.

Figure 1

Allan variance of a chaotic laser time series. (a), (c) Example of a laser chaos time series, normalized with respect to the intensity in the single mode; $I_0$ is the intensity of a laser intensity without optical feedback ($\kappa =0$). The Allan variance considers the variability of a time series for a given timescale (${\tau }_A$). The red dashed lines indicate the mean values of the original time series, with ${\tau }_A$ being 0.1 ns in (a) and 1.0 ns in (c). The difference between consecutive timeslots, i.e., $\mathrm{\Delta }{\overline{s}}_k^{\left({\tau }_A\right)}$ is indicated by blue arrows. (b) Allan variance is defined as the variance of $\mathrm{\Delta }{\overline{s}}_k^{\left({\tau }_A\right)}$ with ${\tau }_A$ ranging from ${10}^{-2}$ to ${10}^2$ ns. Both axes are logarithmic. The green and purple dots represent the Allan variance when ${\tau }_A$ is 0.1 and 1.0 ns, respectively.

Figure 2

Example of an LFF time series. (a) Original time series. (b) Time series obtained by applying an ideal low-pass filter with a cutoff frequency of 100 MHz to the time series of (a). Sudden dropouts with gradual recovery are observed.

Figure 3

Bifurcation diagram of the light intensity levels $I/I_0$ with respect to the optical feedback strength indicated by the dots. As the feedback strength increases, the dynamics undergo stable emission (blue dots), periodic oscillation, quasiperiodic oscillation (green dots), CC, and LFFs (red dots). The LFFs are not clearly distinguishable from the other dynamics in the bifurcation diagram. The black curve represents a figure of merit calculated using the Allan variance. A detailed analysis is presented in Section 3. See also Fig. 5.

Figure 4

Comparison of CC and LFFs based on their PSDs, autocorrelation (AC), and Allan variance. (a), (b) Time-domain snapshot, (c), (d) PSD up to 1 GHz, (e), (f) PSD up to 5 MHz, (g), (h) AC up to 1000 ns, and (i), (j) Allan variance for time series generated with $\kappa =1.55{\mathrm{ns}}^{-1}$ for CC and $\kappa =6.21{\mathrm{ns}}^{-1}$ for LFFs. For the PSD, the red arrow indicates the global peak apart from the direct current (DC) component, and the blue arrow indicates a local peak in the low-frequency regime apart from the DC component. The low-frequency regime is defined as 0–5 MHz. For the Allan variance, the red and blue arrows indicate the peaks in the fast and slow regimes, respectively. The slow regime is defined as ${\tau }_A>{\tau }_D$, as the timescale of dropout dynamics of LFFs are longer than ${\tau }_D$. Insets show magnified views of the Allan variance at approximately 20 ns, corresponding to ${\tau }_D$. Local minima are observed, indicating that the Allan variance can reveal ${\tau }_D$.

Figure 5

Characterizing fast and slow peaks in the Allan variance. (a) Representative features. The Allan variance curve is for $\kappa =6.21{\mathrm{ns}}^{-1}$. (b) Ratio of the slow peak to the fast peak. (c) Ratio of the slow peak to the minimum between the two peaks. (d) Width of the slow peak. Large outliers are saturated and displayed as large dots for visibility.

Figure 6

Allan variance of the LFF time series considering experimental limitations. The black curve is the Allan variance from the original simulation. The curve is identical to that of Fig. 4. Colored curves indicate the Allan variance for 8-bit quantized and downsampled LFF time series. Dashed and dotted curves indicate the Allan variance for the low-pass filtered LFF time series. Offsets are introduced to each curve except for the original line (black solid curve) for visibility.

Figure 7

PSD-based characterization. The ratio of the LFF peak power to the overall peak power was evaluated as a function of feedback strength. Large outliers are saturated and displayed as large dots for visibility. In a previous study [15], when the ratio exceeded $-15$ dB, as indicated by the red line, the time-series data were classified as LFFs. The ratio was less stable and fluctuates more significantly compared to the Allan variance-based measures (see Fig. 5).

Figure 8

Resolution of the Allan variance. (a) Allan variance of the sinusoidal function with additive Gaussian noise. (b) Allan variance of sine waves of different frequencies added together at a fixed rate. The red and blue arrows indicate the peaks corresponding to the 5- and 1-GHz components, respectively.