1/$f$ noise in the intensity fluctuations of vertical-cavity surface-emitting lasers subject to parallel optical injection

A first analysis of fluctuations of the light intensity of vertical-cavity surface-emitting lasers operating in a bistable regime reveals the presence of $1/f$ noise. In this regime the intensity fluctuates between two recently characterized states with residence times $\left\{{\tau }_1\right\}$ and $\left\{{\tau }_2\right\}$. We identify three distinct processes. One of them presents a coherence enhancement phenomenon, and in the other two the distribution of residence times in one of the states follows either a power law $P\left({\tau }_1\right)\sim {\tau }_1^{-2}$ or $P\left({\tau }_2\right)\sim {\tau }_2^{-2}$, and this is the cause of the $1/f$ shape in the spectral density of the intensity. The process at the coherence enhancement zone shows $1/f$ fluctuations in the light intensity and also in the time residence process. It is shown that the origin of these fluctuations is due to a power-law distribution in the time separation between pulses observed in the time residence series.

Figure 1

PSDs of series of pulses with interpulse processes distributed following a power law $P\left(\tau \right)\sim {\tau }^{-2}$. (a) $T=2^{14}$ with exponential (green), Gaussian (red), and flat (black) pulses (hopping process). (b) Hopping process with distinct sizes, $T=2^{12}$ (black), $T=2^{14}$ (red), and $T=2^{16}$ (green). In the inset, collapse of spectra following the scaling law (1). (c) $T=2^{14}$ but now with truncated power laws with ${\tau }_c={10}^5$ (black), ${10}^4$ (red), ${10}^3$ (green), and ${10}^2$ (blue). The inset shows the corresponding saturation sizes $T_c$. (d) Collapse of PSDs with truncated power laws following the scaling law (1) but now with $T_c$ instead of $T$. As an indication for the eye, dashed lines in (a), (b), and (d) follow an exact $1/f$ law.

Figure 2

Upper panel: Random (down, red) and correlated (up, blue) series of residence times distributed following a power law $P\left(\tau \right)\sim {\tau }^{-2}$. Lower panel: PSDs of the corresponding hopping processes averaged over 1000 samples. The $1/f$ shape is robust against correlations.

Figure 3

Experimental time traces of the orthogonal polarization signal corresponding to the P1 & IL+PS dynamics (a)–(c) and the zoomed parts of the time series (d)–(f). The system is in the P1 (IL+PS) state when the signal is small (large). The bias current is $I_{\text{bias}}=5$ mA. Left panel: ${\nu }_i=0.8$ GHz, $P_i=48.01\mu \mathrm{W}$. Center panel: ${\nu }_i=0.9$ GHz, $P_i=53.83\mu \mathrm{W}$. Right panel: ${\nu }_i=1.1$ GHz, ${\mathrm{P}}_{\mathrm{i}}=64.71\mu \mathrm{W}$.

Figure 4

Log-log plot of the probability density of experimental residence times $P\left({\tau }_2\right)$ for each of the 24 samples (dot colored curves) and their average (diamond curve) with its fitted line in the interval $({10}^{-5},{10}^{-2})$ s (thick dashed line) showing a decay, $P\left({\tau }_2\right)\sim {\tau }_2^{-d}$, with $d=1.85$. Thin dashed lines are reference lines to guide the eye. Inset: fitted exponents of the probability density of each of the individual samples, giving $d=1.83\pm 0.17$.

Figure 5

(a) Residence time series $\left\{{\tau }_{2i}\right\}$ of the hopping process observed in case (c) of Fig. 3, and (b) synthetically generated $\left\{{\varepsilon }_{2i}\right\}$. (c) Probability density of the residence time series: ${\tau }_2$ (blue circles), ${\tau }_1$ (cyan squares), ${\varepsilon }_2$ (dashed line, red), and ${\varepsilon }_1$ (black continuous line). (d) Spectral densities of the four residence time processes, $S_{{\tau }_2}$ (upper, blue), $S_{{\tau }_1}$ (lower, violet), $S_{{\varepsilon }_2}$ (upper,red), and $S_{{\varepsilon }_1}$ (lower, brown). $S_{\tau }$ spectra are averaged over 24 samples while $S_{\varepsilon }$ spectra are averaged over 1000. (e) Detail of the peaked structure of the spectrum of ${\tau }_2$ in frequency units of $f^{\prime }=f/\langle \tau \rangle =f\times 3.7$ kHz. The dashed line corresponds to the expected most prominent peak around $f=100$ Hz.

Figure 6

PSD of the hopping process observed in the case (c) of Fig. 3 (blue spectrum) compared with that of a synthetically generated hopping process with the same distribution of residence times but in a random sequence (red spectrum). In both cases, the number of samples is 24.

Figure 7

(a) Time residence series $\left\{{\tau }_{2i}\right\}$ of the hopping process observed in case (b) of our experiment. (b) Time residence series $\left\{{\varepsilon }_{2i}\right\}$ obtained synthetically with the model shown in the text. (c) Method for obtaining the interpulse time series $\left\{h_i\right\}$ for a given threshold. (d) Probability density of residence times of each one of the states: ${\tau }_2$ (green squares), ${\tau }_1$ (violet circles), and their combined process (dashed black line). (e) Probability density of residence times of experimental (blue solid line) and synthetic (red solid line) signals. (f) Probability density of interpulse time series $\left\{h_i\right\}$ for several thresholds: 3.0 (blue circles), 2.8 (violet squares), 2.6 (green diamonds), and 2.4 (brown triangles).

Figure 8

PSDs of the time residence processes $\left\{{\tau }_2\right\}$ observed in the case (b) of Fig. 3 (blue spectrum) compared with that of a synthetically generated process with the same distribution of interpulse times but in a random sequence (red spectrum). To compare with PSDs of the hopping process, the frequency is taken as $f^{\prime }=f/\langle {\tau }_2\rangle =0.4\times f\mathrm{MHz}$. The number of samples is 24 in both cases.

Figure 9

PSDs of the hopping processes generated from the time residence series of the previous figure.