Multiscale permutation entropy analysis of laser beam wandering in isotropic turbulence

We have experimentally quantified the temporal structural diversity from the coordinate fluctuations of a laser beam propagating through isotropic optical turbulence. The main focus here is on the characterization of the long-range correlations in the wandering of a thin Gaussian laser beam over a screen after propagating through a turbulent medium. To fulfill this goal, a laboratory-controlled experiment was conducted in which coordinate fluctuations of the laser beam were recorded at a sufficiently high sampling rate for a wide range of turbulent conditions. Horizontal and vertical displacements of the laser beam centroid were subsequently analyzed by implementing the symbolic technique based on ordinal patterns to estimate the well-known permutation entropy. We show that the permutation entropy estimations at multiple time scales evidence an interplay between different dynamical behaviors. More specifically, a crossover between two different scaling regimes is observed. We confirm a transition from an integrated stochastic process contaminated with electronic noise to a fractional Brownian motion with a Hurst exponent $H=5/6$ as the sampling time increases. Besides, we are able to quantify, from the estimated entropy, the amount of electronic noise as a function of the turbulence strength. We have also demonstrated that these experimental observations are in very good agreement with numerical simulations of noisy fractional Brownian motions with a well-defined crossover between two different scaling regimes.

Figure 1

Schematic diagram of the laboratory experimental setup.

Figure 2

Mean and SD (displayed as error bars) of the normalized PE as a function of the embedding delay $\tau$, using an embedding dimension $D=5$, for the integrated fluctuations of the centroid coordinates of the laser beam for five representative turbulence strengths $C_n^2$ ($m^{-2/3}\times {10}^{-9}$): (a) 5.7, (b) 6.7, (c) 8.0, (d) 14.1, (e) 40.2. Qualitative similar results are found with $D=3$, 4, and 6.

Figure 3

Mean and SD (displayed as error bars) of the normalized PE as a function of the embedding delay $\tau$, using an embedding dimension $D=5$, for a simulated fBm of length $M=20000$ with a well-defined crossover at time scales (a) 10, (b) 20, and (c) 30. Different NSRs are also included. Qualitative similar results are found when $D=3$, 4, and 6 are used.

Figure 4

Mean and SD (displayed as error bars) of the noise quantifier ${\mathrm{\Gamma }}_5$ and NSR for the horizontal and vertical coordinates of the laser beam centroid as a function of the turbulence strength. In each case, both quantifiers are averaged over 21 realizations. The inset plot shows ${\mathrm{\Gamma }}_5$ vs NSR (only mean values are depicted for better visualization). Qualitative similar results are found with $D=3$, 4, and 6.

Figure 5

Mean and standard deviation of ${\widehat{\mathcal{H}}}_5$ from the interval $\tau \in [30,50]$, as a function of the turbulent strength $C_n^2$ for the (a) horizontal coordinate and (b) vertical coordinate. The gray region indicates ${\mathcal{H}}_5\pm 3\sigma$ from a simulated fBm with $H=5/6$ contaminated with noise according to the NSR computed from the experiments (see Fig. 4). The entropy ${\widehat{\mathcal{H}}}_5$ estimated from the shuffled data is also depicted. Qualitative similar results are found with $D=3$, 4, and 6.

Figure 6

Averaged normalized PE estimated with $D=5$ for the horizontal and vertical experimental records and for the numerical simulations. In the former case, the analysis is developed as a function of the embedding delay $\tau$ and the turbulence intensity, while in the latter case the NSR is implemented for quantifying the turbulence strength.