Extended Self-similarity in Multimode Optical Fiber Speckles

TL;DR

This work addresses whether Extended Self-Similarity (ESS) can arise in linear wave systems, specifically speckle patterns from coherent light in multimode fibers. It combines comprehensive experiments across four wavelengths and three core diameters with numerical simulations based on coupled-mode theory to analyze intensity structure functions $S_p(r)$ and their ESS representation. The main finding is that the ESS exponents satisfy $\beta_p \approx p/3$ for low orders, consistent with Kolmogorov scaling, and that intermittency is weak, quantified by a small KO62 parameter $\kappa$; simulations reproduce these results with $\kappa \approx -7.3\times 10^{-4}$. This demonstrates the universality of ESS beyond nonlinear turbulence and positions linear MMF speckle as a controllable platform for studying scaling in complex wavefields, with potential applications in wavelength sensing and beyond.

Abstract

Extended Self-Similarity (ESS) is a widely used tool for uncovering universal power-law scaling in systems dominated by nonlinear interactions. This work demonstrates that ESS scaling can also emerge in a system governed by purely linear physics: the propagation of coherent light in a multimode fiber. The system produces complex speckle patterns arising solely from deterministic linear mode interference. We analyze the intensity structure functions of these speckles and observe a robust extended scaling range. The measured scaling exponents align with the classical Kolmogorov scaling exponents. This finding establishes that the statistical signatures captured by ESS are not exclusive to nonlinear systems, revealing a broader applicability of this scaling framework to complex linear systems.

Figures (12)

• Figure 1: Experimental setup and representative speckle patterns. (a) Schematic of the optical setup used to generate and record speckle patterns. L: lens; M: mirror. (b-d) Representative speckle patterns measured at the output of 2-meter-long MMFs with core diameters of (b) $200\ {\rm{\mu m}}$, (c) $105\ {\rm{\mu m}}$, and (d) $50\ {\rm{\mu m}}$, respectively, using a $532\,\rm{nm}$ laser.
• Figure 2: Structure functions and ESS for different fiber core diameters $D$ at $\lambda = 532$ nm. In each subplot, order $p$ increases from bottom to top. The top row (a, b) corresponds to $D = 50\,{\rm{\mu m}}$ (triangles), the middle row (c, d) to $D = 105\,{\rm{\mu m}}$ (squares), and the bottom row (e, f) to $D = 200\,{\rm{\mu m}}$ (circles). Left column (a, c, e): $S_p(r)$ versus spatial scale $r$. Right column (b, d, f): $S_p(r)$ versus $S_3(r)$. Solid lines are linear fits in log-log scale. Error bars represent $2\times$ standard errors of the mean.
• Figure 3: ESS scalings. (a) Measured ESS scaling exponent $\beta_p$ as a function of order $p$ for the three fiber core diameters [$50\ \rm {\mu m}$ (triangle), $105\ \rm {\mu m}$ (square), and $200\ \rm {\mu m}$ (circle)] with $532\,\rm{nm}$ laser. The black line represents the classical K41 scaling relation $\beta_p = p/3$. (b) Histograms of the deviation $\beta_6$ from the K41 prediction ($\beta_6 = 2$), obtained from 100 independent speckle realizations for each core diameter.
• Figure 4: Intermittency analysis of fiber speckle patterns. (a) Deviation of the ESS scaling exponents from the K41 prediction, $\beta_p - p/3$, plotted against order $p$ for $\lambda = 532\,\text{nm}$. Data for core diameters of $200\,{\rm{\mu m}}$, $105\,{\rm{\mu m}}$, and $50\,{\rm{\mu m}}$ are shown as circles, squares, and triangles, respectively. The solid lines are parabolic fits of the form $\kappa \, p(p-3)$, from which the intermittency strength $\kappa$ is extracted. (b) Summary of the fitted $\kappa$ values for all experimental conditions (4 wavelengths $\times$ 3 core diameters). Colors denote wavelength: $532\,\text{nm}$ (green), $660\,\text{nm}$ (red), $780\,\text{nm}$ (brown), and $980\,\text{nm}$ (gray). Error bars represent the standard error from the fitting procedure.
• Figure 5: Numerical simulation of optical mode evolution in an MMF. (a) Simulated intensity patterns $I(z,x,y)=|E(z,x,y)|^{2}$ at propagation distances $z = 0, 10^{-6}, 10^{-5}, 10^{-4}, 10^{-3}, 10^{-2}, 10^{-1}, 1~\mathrm{m}$. The simulation parameters are $\lambda = 980\,\mathrm{nm}$ and $D=105\,\mathrm{\mu m}$. (b) Distribution of mode weights $\rho_{q}$ at $10^{-6}\,\mathrm{m}$ (red), and $10^{-4}\,\mathrm{m}$ (blue). (c) Evolution of the von Neumann entropy $S(z)$ as a function of propagation distance.